Fix #511: Live Corners on any path's corners (stars, polygons, pen paths)

Corner widgets only ever showed on live rectangles: CornerWidgets::of bailed
out unless the selection's live shape was LiveShape::Rectangle, and
object.setLiveShape only edited rectangle radii. A star is a plain path, so it
had no widgets with either selection tool.

- geom::corners: the corners of any path (anchors without handles between two
  straight sides) and the path with them cut round, inverted round or
  chamfered, each corner a circle tangent to both sides, its radius no larger
  than takes the cut halfway along its shorter side. Rectangles now cut their
  corners with it (bit-identical anchors and layout).
- doc::corners::LiveCorners/set_corners: one model for a live rectangle's,
  a live polygon's (new radii/kinds) and any other path's corners (new
  LiveShape::Path keeping the uncut outline, plain again once nothing is cut);
  radii scale with transforms (Scale Corners) and stay circular.
- object.setLiveShape takes anchor indices for `corners` on any path; polygons
  keep a shared radius as their sides change.
- Widgets: Direct Selection on any path, Selection on live rectangles and
  polygons; placed on each corner's bisector, the drag follows it, the red
  limit outline marks corners at their own limit; the angle preference and
  View > Hide Corner Widget apply. Corners dialog and the Properties and
  Transform panels' Corner Radius (now on polygons too) work on them.
- Tests for the star, the polygon and a pen path in geom, doc, tools, engine
  and ui-egui; docs and ROADMAP.

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

ROADMAP.md

@@ -4,7 +4,7 @@ VectorCraft is a clean-room, open-source, pure-Rust reimplementation of the Adob
 
 This file tracks **how far we are and what's left**. Time estimates are wall-clock hours of continuous Claude Opus 5.5 agent work (including builds and the CI gate), given both for **one agent** and for **4–6 parallel agents** on disjoint crates. They are counted from the remaining work (see [Parity estimate](#parity-estimate)), calibrated against measured throughput, and updated as work lands.
 
-_Last updated: 2026-10-08 (after M4.14–M4.98, M8.1–M8.20, M14.4–M14.7, the MCP prompts, resource templates, completions and logging, the 2026-10-07 issue fixes, the Layers panel rework and the 2026-10-08 community contributions: Pen and shape-tool modifiers, path editing, Layers, Artboards, Swatches/Brushes and dialogs, Japanese composition and PDF text, Hebrew/Arabic type, interface languages (Spanish among them), the font menu, saved selections, MCP and EPS, the missing-font export warning (#319), open paths in the Shape Builder (#374), PDF/`.ai` import keeping strokes live and hidden and nested layers (#372), per-corner Live Corners (#381), SVG type opening at the size it draws at (#396), the Wayland clipboard with copied-file paste (#398), ruler guides that can be selected, moved and deleted (#414), two batches of Preferences options that were stored but unused (#394), skeleton reads with `document.find` for large documents over MCP (#416) and anchors dragged with Direct Selection snapping to points, segments and smart guides, clipped SVG layers opening as clipping layers (#420), PDF/`.ai` layers read on pages of tens of millions of operators (#419), Document Color Mode putting RGB greys on the black plate on request and converting pattern tiles (#421) and movable dialogs with Offset Path's Joins dropdown (#423) and numeric fields scrubbed by dragging their label (#400), stroke weight presets in the stroke unit (#426), the Appearance panel's Opacity popup (#430), type on a path brackets, flip and Type on a Path Options that keep the type in place (#429), Layers rows dragged with a ghost and dropped on the trash or New Layer (#435), tool flyouts opening on a right or corner press (#436), tool flyouts torn off into floating toolbars (#437), the Shape Builder's erase mode deleting edges too (#439), installed fonts registered outside the font folders and fonts installed while the app runs listed in the font menus (#443), alike circular corners on rectangles that aren't square (#442), tool cursors as OS cursors so the pointer no longer trails over the canvas (#444), artboards copied and pasted with their art (#446), guides dragged out of the rulers snapping to the art's side midpoints and the artboards, artboard guides (#451), the toolbar's double arrow and Window › Toolbars › Double Column switching between one and two columns in any window height (#455), the Smart Guides display options and Snapping Tolerance (#394), Font Size and Leading combo boxes with preset lists in the Character and Properties panels and the Control bar (#458), screenshots pasted from the clipboard on macOS, Windows and the web (#457), the Properties panel's Fill and Stroke matching the Control bar's, Illustrator's Window menu panel keys (Shift+F7 for Align…) and Alt-wheel zoom whatever the wheel preference (Discord feedback), the point/area type widget (Discord feedback), groups in legacy Illustrator .ai files (Discord feedback), a rotating desktop log file (#465), image controls in the Control bar and Properties panel: Image Trace with its presets, Mask, Crop Image, Embed/Unembed, and an Image Trace object's preset, Expand and Release (Discord feedback), the Pen working as the Anchor Point tool with Alt held over a selected path's handles and anchors, and anchor buttons that convert, connect and cut paths at the selected anchors (Discord feedback), Japanese hanging punctuation (burasagari, #463), Escape leaving Presentation Mode with no screen mode saved across restarts, documents dropped on the desktop app opening as tabs of their own and a note that an `.ai` file's art off its artboards isn't in the part read (#472), Add/Delete Anchor Point tool icons drawn from the Pen tool's (#479), EPS files from Illustrator 2026, Inkscape/cairo, matplotlib, CorelDRAW, Affinity, LibreOffice and Ghostscript opening as vectors, and a preview that stands in saying why (Discord feedback, #466, #469, #474), chords with modifiers recorded in Edit › Keyboard Shortcuts (#487), Type Object Selection by Path Only (#394), Anti-aliased Artwork (#394), Show Pixel Grid (#394), pen and touch taps pressed and lifted within one frame clicking where they tapped and the Linux/Wayland pen-tablet workaround (#491), bounding-box resizes landing on the other objects' anchors and centres with labels, and the Rotate, Scale, Shear and Reflect tools snapping their drags (#482), Shift-drag marquees that toggle the objects or anchors they reach and a Lasso whose Alt subtraction keeps the rest of the selection (#483), the mouse wheel stepping the focused numeric field, the dialogs' and preferences' drag values made such fields (#485), Show The Home Screen When No Documents Are Open (#394), Numbers Without Units Are Points (#394), Auto-Collapse Iconic Panels (#394), the Contextual Task Bar moved by its handle and its More Options menu pinning and resetting its position, a designed macOS DMG window (#493), panels, panel groups and the Tools panel undocking intLine truncated
+_Last updated: 2026-10-08 (after M4.14–M4.98, M8.1–M8.20, M14.4–M14.7, the MCP prompts, resource templates, completions and logging, the 2026-10-07 issue fixes, the Layers panel rework and the 2026-10-08 community contributions: Pen and shape-tool modifiers, path editing, Layers, Artboards, Swatches/Brushes and dialogs, Japanese composition and PDF text, Hebrew/Arabic type, interface languages (Spanish among them), the font menu, saved selections, MCP and EPS, the missing-font export warning (#319), open paths in the Shape Builder (#374), PDF/`.ai` import keeping strokes live and hidden and nested layers (#372), per-corner Live Corners (#381), SVG type opening at the size it draws at (#396), the Wayland clipboard with copied-file paste (#398), ruler guides that can be selected, moved and deleted (#414), two batches of Preferences options that were stored but unused (#394), skeleton reads with `document.find` for large documents over MCP (#416) and anchors dragged with Direct Selection snapping to points, segments and smart guides, clipped SVG layers opening as clipping layers (#420), PDF/`.ai` layers read on pages of tens of millions of operators (#419), Document Color Mode putting RGB greys on the black plate on request and converting pattern tiles (#421) and movable dialogs with Offset Path's Joins dropdown (#423) and numeric fields scrubbed by dragging their label (#400), stroke weight presets in the stroke unit (#426), the Appearance panel's Opacity popup (#430), type on a path brackets, flip and Type on a Path Options that keep the type in place (#429), Layers rows dragged with a ghost and dropped on the trash or New Layer (#435), tool flyouts opening on a right or corner press (#436), tool flyouts torn off into floating toolbars (#437), the Shape Builder's erase mode deleting edges too (#439), installed fonts registered outside the font folders and fonts installed while the app runs listed in the font menus (#443), alike circular corners on rectangles that aren't square (#442), tool cursors as OS cursors so the pointer no longer trails over the canvas (#444), artboards copied and pasted with their art (#446), guides dragged out of the rulers snapping to the art's side midpoints and the artboards, artboard guides (#451), the toolbar's double arrow and Window › Toolbars › Double Column switching between one and two columns in any window height (#455), the Smart Guides display options and Snapping Tolerance (#394), Font Size and Leading combo boxes with preset lists in the Character and Properties panels and the Control bar (#458), screenshots pasted from the clipboard on macOS, Windows and the web (#457), the Properties panel's Fill and Stroke matching the Control bar's, Illustrator's Window menu panel keys (Shift+F7 for Align…) and Alt-wheel zoom whatever the wheel preference (Discord feedback), the point/area type widget (Discord feedback), groups in legacy Illustrator .ai files (Discord feedback), a rotating desktop log file (#465), image controls in the Control bar and Properties panel: Image Trace with its presets, Mask, Crop Image, Embed/Unembed, and an Image Trace object's preset, Expand and Release (Discord feedback), the Pen working as the Anchor Point tool with Alt held over a selected path's handles and anchors, and anchor buttons that convert, connect and cut paths at the selected anchors (Discord feedback), Japanese hanging punctuation (burasagari, #463), Escape leaving Presentation Mode with no screen mode saved across restarts, documents dropped on the desktop app opening as tabs of their own and a note that an `.ai` file's art off its artboards isn't in the part read (#472), Add/Delete Anchor Point tool icons drawn from the Pen tool's (#479), EPS files from Illustrator 2026, Inkscape/cairo, matplotlib, CorelDRAW, Affinity, LibreOffice and Ghostscript opening as vectors, and a preview that stands in saying why (Discord feedback, #466, #469, #474), chords with modifiers recorded in Edit › Keyboard Shortcuts (#487), Type Object Selection by Path Only (#394), Anti-aliased Artwork (#394), Show Pixel Grid (#394), pen and touch taps pressed and lifted within one frame clicking where they tapped and the Linux/Wayland pen-tablet workaround (#491), bounding-box resizes landing on the other objects' anchors and centres with labels, and the Rotate, Scale, Shear and Reflect tools snapping their drags (#482), Shift-drag marquees that toggle the objects or anchors they reach and a Lasso whose Alt subtraction keeps the rest of the selection (#483), the mouse wheel stepping the focused numeric field, the dialogs' and preferences' drag values made such fields (#485), Show The Home Screen When No Documents Are Open (#394), Numbers Without Units Are Points (#394), Auto-Collapse Iconic Panels (#394), the Contextual Task Bar moved by its handle and its More Options menu pinning and resetting its position, a designed macOS DMG window (#493), panels, panel groups and the Tools panel undocking intLine truncated
 
 ## Where we are
 
@@ -121,7 +121,7 @@ grade by behaviour against `plan/illustrator/`, not by whether a menu item exist
   - **Shapes:** all shape tools (including Flare) and the line, arc, spiral and grid tools. Space held while dragging out a shape, line, arc, spiral or grid moves it at its current size. While a grid is dragged out, Up/Down change its rows (concentric dividers) and Left/Right its columns (radial dividers).
   - **Cutting:** Eraser, Scissors, Knife, Mirror & Cut, Line Cut and Rectangle Cut (real geometry, compound paths keep their holes).
   - **Transform:** Rotate, Reflect, Scale, Shear (click or Alt-click snaps the reference point to anchors and centres; Alt-click opens the tool's dialog, and a drag with Alt down at the release, from the press or pressed mid-drag, transforms a copy), Reshape, Free Transform (free and perspective distort and shear, switched by the keys held during a handle drag).
-  - **Live Corners:** drag a live rectangle's corner widgets (Selection or Direct Selection) to round all corners, with a radius readout; with Direct Selection, select corner anchors to show and round just those corners (#381). Each corner keeps its own radius and kind (Round, Inverted Round, Chamfer): Alt-click a widget to cycle the kind, double-click it for the Corners dialog, and the Properties and Transform radius fields set the selected corners (`object.setLiveShape {corners, radius, kind}`). Corners stay circular when the rectangle is scaled unevenly or along its own sides when rotated (the radius keeps its size, or scales by the mean scale with Scale Corners on); only a shear distorts them. On a rectangle that isn't square every corner is the same circle: a radius past half the shorter side stops there for all of them, the widgets sit on the corners as drawn and turn the edited corners red at that limit, CSS export writes the drawn radii, and rectangles saved stretched by older versions get circular corners when their corners are next set (#442).
+  - **Live Corners:** drag a live rectangle's corner widgets (Selection or Direct Selection) to round all corners, with a radius readout; with Direct Selection, select corner anchors to show and round just those corners (#381). Each corner keeps its own radius and kind (Round, Inverted Round, Chamfer): Alt-click a widget to cycle the kind, double-click it for the Corners dialog, and the Properties and Transform radius fields set the selected corners (`object.setLiveShape {corners, radius, kind}`). Corners stay circular when the rectangle is scaled unevenly or along its own sides when rotated (the radius keeps its size, or scales by the mean scale with Scale Corners on); only a shear distorts them. On a rectangle that isn't square every corner is the same circle: a radius past half the shorter side stops there for all of them, the widgets sit on the corners as drawn and turn the edited corners red at that limit, CSS export writes the drawn radii, and rectangles saved stretched by older versions get circular corners when their corners are next set (#442). Live Corners work on any path's corners too, as on a rectangle's (#511): a star's, a live polygon's (which also shows them to the Selection tool, has a Corner Radius field in the Properties and Transform panels and keeps its radius as its sides change) and a pen path's, wherever two straight sides meet at an anchor. Direct Selection shows the widgets of every corner, or of the corners holding selected anchors; each corner is a circle tangent to both sides, drawn no larger than takes its cut halfway along its shorter side; a path keeps its uncut outline so its corners stay editable, and the same command, dialog, angle preference and View › Hide Corner Widget apply.
   - **Pen:** Alt pressed in the middle of dragging out an anchor splits its handles and keeps the curve already shaped into it (only the outgoing handle follows the pointer); Space held while dragging out an anchor moves it, handles and all. A press on the first anchor of the path being drawn closes it whatever the drag then does: the drag shapes the closing curve (symmetric handles, Alt breaks them) and the release closes the path wherever it ends (#501). Clicking the last anchor of the path being drawn retracts its outgoing handle (convert cursor), and dragging from it pulls out a new one on its own. Auto Add/Delete: over a selected path, a click on a segment adds an anchor and a click on an anchor deletes it (add and delete cursors); Shift held, or Preferences › General › Disable Auto Add/Delete, starts a new path instead. Alt held over a handle end or an anchor of a selected path (the one being drawn too) works as the Anchor Point tool, with its convert cursor: dragging a handle moves it alone, clicking a smooth anchor makes it a corner, dragging an anchor pulls out new symmetric handles, dragging a segment reshapes it; drawing then goes on. Cmd (Ctrl) held drags with the selection tool chosen last, Direct Selection until one is (handles, anchors and segments of the path being drawn or another), and the Pen then goes on drawing its path while that stays selected; the pointer shows that tool's while Cmd is held. The same holds for agents (`pointer_gesture` with `mods.cmd`), as the borrowing now lives in the engine (#494).
   - **Path editing (#494):** the handles shown are the handles that drag: Direct Selection drags those of a path selected as a whole (as the Pen leaves it), the Anchor Point tool and Pen+Alt no longer grab handles that Show handles when multiple anchors are selected hides, a handle shorter than the tolerance leaves its anchor to be picked, and the anchor tools pick within Selection & Anchor Display › Tolerance. A dragged segment keeps a smooth anchor at either end smooth. With Direct Selection, the Pen, the Curvature and the anchor tools, a path selected as a whole shows all its anchors selected, and the Control bar and the Properties panel offer Convert (corner, smooth) for them too; Remove and Cut wait for anchors direct-selected (#504). A handle drag gathers its snap targets once per drag (it took 21 ms per pointer move on a 20,000-path document, now 1 ms; `vectorcraft-cli perf` measures handle and segment drags on a 5,000-anchor path).
   - **Graphs:** Column, Stacked Column, Bar, Stacked Bar, Line, Area, Scatter, Pie and Radar graph tools with Graph Data and Graph Type.
@@ -167,7 +167,7 @@ grade by behaviour against `plan/illustrator/`, not by whether a menu item exist
 |---|---|---|---|
 | M0 | Skeleton + vertical slice | ✅ done | — |
 | M1 | Selection, transform, layers, MCP | ✅ mostly done (transform reference point snaps to anchors/centres; rotated objects keep a rotated bounding box and their angle; Free Transform, Scale and Reflect along a rotated box pending) | 4–6 |
-| M2 | Drawing tools + smart guides | ✅ mostly done (Flare, Reshape, Live Corners widget dragging and per-corner radii and kinds landed; Shaper, the remaining Pen modifier nuances) | 15–20 |
+| M2 | Drawing tools + smart guides | ✅ mostly done (Flare, Reshape, Live Corners widget dragging and per-corner radii and kinds landed, on any path's corners; Shaper, the remaining Pen modifier nuances) | 15–20 |
 | M3 | Paint & appearance (swatches, color, gradient, stroke, appearance, transparency, styles) | ✅ done (M3.7–M3.98): swatches, groups, libraries, tints, Lab spots and Registration; Color Picker, Color panel, Color Guide, Color Themes, Edit Colors and Recolor Artwork; gradients that follow every transform, the annotator, linked stops, freeform gradients, focal points, gradients on strokes and Expand; stroke geometry, arrowheads, dashes fitted to corners, stroke on type, width profiles and the Width tool, Scale Strokes and preview bounds; Appearance targeting, container appearance, target circles, Expand Appearance; knockout, isolation, blend accuracy, CMYK blending, overprint, Attributes, masks and clipping, Flatten Transparency; graphic styles, links and libraries. Left: freeform and mixed spot/process gradients export as stops or process colours to SVG/PDF; confirm the Scale Strokes & Effects default | 4–8 |
 | M4 | Files (native, SVG, PDF, raster, Export for Screens, clipboard interop) | ✅ done (M4.14–M4.98 on 2026-10-05): Save As/Save a Copy/Revert/templates, Data Recovery and background save, SVG import and export fidelity (units, text, symbols, filters, SVGZ, Preserve Editing, SVG Options), PDF import (colours, layers, masks, editable text, security) and export (presets, marks, bleed, ICC and output intent, raster effects, subset fonts, PDF/X-1a, X-3 and X-4, PDF layers, overprint, page thumbnails, fast web view, PDF 1.3), Place and Links, Package, EPS and DXF in and out, EMF/WMF, TIFF/BMP/Targa/PSD/GIF/PNG-8 export, Print and print presets, slices, Save for Web, Asset Export, clipboard flavours, File Info. Left: DWG (use DXF), PSD placement as layers, overprint read back from PDF, polish | 3–6 |
 | M5 | Performance | 🟡 background render + caches + MT done; `vectorcraft-cli bench` and `vectorcraft-cli perf` (budget suite); file format v2 opens 3× faster (50k paths: 722 → 244 ms); raster effects (glows, shadows, blur, feather) no longer force the whole frame single-threaded (filtered offscreen per effect, verified equal to the single-threaded reference); effect-heavy demos need a clean-machine benchmark; dirty-region rendering, GPU backend spike pending | 15–25 |
@@ -199,7 +199,7 @@ about 1–1.5 agent-hours each, so the other rows (estimated on the older scale)
 | Area | Weight | Done | Missing (main items) | One agent (h) |
 |---|---:|---:|---|---:|
 | Selection, transform & align tools | 6 | 85% | Free Transform/Scale/Reflect along a rotated box, Start Global Edit, transform nuances (the rotated persistent bounding box is done) | 4–6 |
-| Drawing tools | 7 | 85% | Shaper Groups (merge/punch overlapping shapes), pencil and the remaining pen modifier nuances, the freehand Tool Options' Keep selected, Alt to the Smooth tool, Blob Brush Merge Only with Selection and brush angle/roundness, Touch Type, Live Corners on plain paths and polygons (and Relative rounding) | 10–15 |
+| Drawing tools | 7 | 85% | Shaper Groups (merge/punch overlapping shapes), pencil and the remaining pen modifier nuances, the freehand Tool Options' Keep selected, Alt to the Smooth tool, Blob Brush Merge Only with Selection and brush angle/roundness, Touch Type, Live Corners on corners between curved sides (and Relative rounding) | 10–15 |
 | Path operations, Pathfinder, Shape Builder, Live Paint | 5 | 86% | Live Paint gap options, Shape Builder gap detection, options (Tool Options dialog, clicking a stroke in merge mode) and Shift-drag marquee | 3–6 |
 | Colour, swatches, gradients, patterns, mesh, recolor | 7 | 96% | freeform and mixed spot/process gradients in SVG/PDF (exported as stops or process colours), pattern fills in Expand | 2–4 |
 | Strokes, brushes, width profiles | 5 | 88% | brush options depth, brush libraries (generated in code), brushes driven by pen tilt, bearing and barrel rotation (6D art pens) | 12–20 |

crates/doc/src/corners.rs

@@ -0,0 +1,271 @@
+//! Live Corners on any path: the corners of its uncut outline (a live rectangle's or polygon's, a
+//! live path's, or a plain path's own), each one's radius and kind, and the anchors of the path as
+//! drawn that belong to each. [`set_corners`] edits them.
+
+use std::borrow::Cow;
+use std::collections::BTreeSet;
+
+use vectorcraft_geom::corners::{Corner, cut_corners, path_corners};
+use vectorcraft_geom::shapes::{self, CornerKind};
+use vectorcraft_geom::{Affine, PathData, Rect};
+
+use crate::node::{LiveShape, Node, NodeKind};
+use crate::selection::AnchorRef;
+
+/// The Live Corners of a path.
+#[derive(Clone, Debug)]
+pub struct LiveCorners<'a> {
+    /// The path with its corners uncut, in the corners' own space: a live rectangle's (its size
+    /// and radii in document units, see [`LiveShape::folded`]), else the document's.
+    pub base: Cow<'a, PathData>,
+    /// Maps `base` into the document.
+    pub xf: Affine,
+    /// The corners that can be cut, in anchor order ([`path_corners`] of `base`).
+    pub corners: Vec<Corner>,
+    radii: Vec<f64>,
+    kinds: Vec<CornerKind>,
+    /// For each anchor of the path as drawn (per subpath), the anchor of `base` it comes from.
+    sources: Vec<Vec<usize>>,
+}
+
+impl<'a> LiveCorners<'a> {
+    /// The Live Corners of `path` and its live shape; `None` for shapes without corners (ellipses,
+    /// lines).
+    pub fn new(path: &'a PathData, live: Option<&'a LiveShape>) -> Option<Self> {
+        let (base, xf, radii, kinds) = match live {
+            None => (Cow::Borrowed(path), Affine::IDENTITY, vec![], vec![]),
+            Some(l @ LiveShape::Rectangle { .. }) => {
+                let LiveShape::Rectangle { w, h, radii, kinds, xf } = l.folded() else { return None };
+                (Cow::Owned(shapes::rectangle(Rect::new(0.0, 0.0, w, h))), xf, radii.to_vec(), kinds.to_vec())
+            }
+            Some(l @ LiveShape::Polygon { radii, kinds, .. }) => (Cow::Owned(l.polygon_outline()), Affine::IDENTITY, radii.clone(), kinds.clone()),
+            Some(LiveShape::Path { base, radii, kinds }) => (Cow::Borrowed(base), Affine::IDENTITY, radii.clone(), kinds.clone()),
+            Some(LiveShape::Ellipse { .. } | LiveShape::Line { .. }) => return None,
+        };
+        let sources = if radii.iter().any(|r| *r > 0.0) {
+            cut_corners(&base, &radii, &kinds).1
+        } else {
+            let mut first = 0;
+            base.subpaths
+                .iter()
+                .map(|sp| {
+                    first += sp.anchors.len();
+                    (first - sp.anchors.len()..first).collect()
+                })
+                .collect()
+        };
+        Some(Self { corners: path_corners(&base), base, xf, radii, kinds, sources })
+    }
+
+    /// The Live Corners of a path object.
+    pub fn of(node: &'a Node) -> Option<Self> {
+        let NodeKind::Path { path, live, .. } = &node.kind else { return None };
+        Self::new(path, live.as_ref())
+    }
+
+    /// The corner at anchor `index` of `base`.
+    pub fn corner(&self, index: usize) -> Option<&Corner> {
+        self.corners.binary_search_by_key(&index, |c| c.index).ok().and_then(|i| self.corners.get(i))
+    }
+
+    /// The radius corner `index` is set to, in document units (drawn no larger than fits).
+    pub fn radius(&self, index: usize) -> f64 {
+        self.radii.get(index).copied().unwrap_or(0.0)
+    }
+
+    /// The kind corner `index` is cut with.
+    pub fn kind(&self, index: usize) -> CornerKind {
+        self.kinds.get(index).copied().unwrap_or_default()
+    }
+
+    /// Every corner.
+    pub fn all(&self) -> BTreeSet<usize> {
+        self.corners.iter().map(|c| c.index).collect()
+    }
+
+    /// The corners holding one of `anchors` (of the path as drawn).
+    pub fn corners_of(&self, anchors: &BTreeSet<AnchorRef>) -> BTreeSet<usize> {
+        anchors.iter().filter_map(|(si, ai)| self.sources.get(*si)?.get(*ai).copied()).filter(|k| self.corner(*k).is_some()).collect()
+    }
+
+    /// The corners Live Corners edit for a selection: those holding a selected anchor when the
+    /// path is partly selected (Direct Selection), else every corner.
+    pub fn picked(&self, partial: Option<&BTreeSet<AnchorRef>>) -> BTreeSet<usize> {
+        let picked = partial.map(|a| self.corners_of(a)).unwrap_or_default();
+        if picked.is_empty() { self.all() } else { picked }
+    }
+
+    /// The anchors of the path as drawn that belong to `corners` (to keep them selected as the
+    /// path changes).
+    pub fn anchors_of(&self, corners: &BTreeSet<usize>) -> BTreeSet<AnchorRef> {
+        let mut out = BTreeSet::new();
+        for (si, from) in self.sources.iter().enumerate() {
+            out.extend(from.iter().enumerate().filter(|(_, k)| corners.contains(k)).map(|(ai, _)| (si, ai)));
+        }
+        out
+    }
+
+    /// For each anchor of the path as drawn (per subpath), the anchor of `base` it comes from.
+    pub fn sources(&self) -> &[Vec<usize>] {
+        &self.sources
+    }
+
+    /// The radius (document units) and kind `corners` share (each `None` when they differ, or
+    /// for no corners).
+    pub fn style(&self, corners: &BTreeSet<usize>) -> (Option<f64>, Option<CornerKind>) {
+        fn shared<T: Copy>(mut it: impl Iterator<Item = T>, same: impl Fn(T, T) -> bool) -> Option<T> {
+            let first = it.next()?;
+            it.all(|x| same(x, first)).then_some(first)
+        }
+        let radius = shared(corners.iter().map(|k| self.radius(*k)), |a, b| (a - b).abs() < 1e-9);
+        (radius, shared(corners.iter().map(|k| self.kind(*k)), |a, b| a == b))
+    }
+}
+
+/// Set `radius` (document units; negative: 0) and/or `kind` on the `corners` (anchor indices of
+/// the uncut outline, see [`LiveCorners`]) of a path and its live shape, then cut the path again.
+/// A path that isn't a live shape keeps its uncut outline as a [`LiveShape::Path`], which is a
+/// plain path again once no corner is cut. Ellipses and lines have no corners: they stay as they
+/// are.
+pub fn set_corners(path: &mut PathData, live: &mut Option<LiveShape>, corners: &BTreeSet<usize>, radius: Option<f64>, kind: Option<CornerKind>) {
+    fn edit(radii: &mut [f64], kinds: &mut [CornerKind], corners: &BTreeSet<usize>, radius: Option<f64>, kind: Option<CornerKind>) {
+        for k in corners {
+            if let (Some(r), Some(slot)) = (radius, radii.get_mut(*k)) {
+                *slot = r.max(0.0);
+            }
+            if let (Some(kind), Some(slot)) = (kind, kinds.get_mut(*k)) {
+                *slot = kind;
+            }
+        }
+    }
+    /// One radius and kind per corner of an outline of `n` anchors while editing; none saved
+    /// when all are sharp or round.
+    fn edit_vec(
+        n: usize,
+        radii: &mut Vec<f64>,
+        kinds: &mut Vec<CornerKind>,
+        corners: &BTreeSet<usize>,
+        radius: Option<f64>,
+        kind: Option<CornerKind>,
+    ) {
+        radii.resize(n, 0.0);
+        kinds.resize(n, CornerKind::Round);
+        edit(radii, kinds, corners, radius, kind);
+        if radii.iter().all(|r| *r <= 0.0) {
+            radii.clear();
+        }
+        if kinds.iter().all(|k| *k == CornerKind::Round) {
+            kinds.clear();
+        }
+    }
+    let l = live.get_or_insert_with(|| LiveShape::Path { base: path.clone(), radii: vec![], kinds: vec![] });
+    // Radii are document lengths and corners circular, also on a rectangle from a file that kept
+    // an uneven scale in its transform (#442).
+    l.fold_scale();
+    match l {
+        LiveShape::Rectangle { radii, kinds, .. } => edit(radii, kinds, corners, radius, kind),
+        LiveShape::Polygon { sides, radii, kinds, .. } => edit_vec(*sides as usize, radii, kinds, corners, radius, kind),
+        LiveShape::Path { base, radii, kinds } => edit_vec(base.anchor_count(), radii, kinds, corners, radius, kind),
+        LiveShape::Ellipse { .. } | LiveShape::Line { .. } => return,
+    }
+    match live {
+        Some(LiveShape::Path { base, radii, kinds }) if radii.is_empty() && kinds.is_empty() => {
+            *path = std::mem::take(base);
+            *live = None;
+        }
+        Some(l) => *path = l.to_path(),
+        None => {}
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use vectorcraft_geom::{Point, SubPath};
+
+    fn set(v: &[usize]) -> BTreeSet<usize> {
+        v.iter().copied().collect()
+    }
+
+    fn anchors(v: &[usize]) -> BTreeSet<AnchorRef> {
+        v.iter().map(|ai| (0, *ai)).collect()
+    }
+
+    #[test]
+    fn rectangle_corners_map_to_their_anchors() {
+        let l = LiveShape::Rectangle { w: 100.0, h: 50.0, radii: [0.0, 10.0, 0.0, 0.0], kinds: Default::default(), xf: Affine::IDENTITY };
+        let path = l.to_path();
+        let c = LiveCorners::new(&path, Some(&l)).unwrap();
+        // Anchors: top-left, the top-right corner's two, bottom-right, bottom-left.
+        assert_eq!(c.sources(), [vec![0, 1, 1, 2, 3]]);
+        assert_eq!(path.anchor_count(), 5);
+        assert_eq!(c.corners_of(&anchors(&[2, 4])), set(&[1, 3]));
+        assert!(c.corners_of(&anchors(&[9])).is_empty(), "out of range");
+        assert_eq!(c.anchors_of(&set(&[1, 3])), anchors(&[1, 2, 4]));
+        // Partly selected: the corners of the selected anchors; wholly (or no anchor): all four.
+        assert_eq!(c.picked(Some(&anchors(&[0]))), set(&[0]));
+        assert_eq!(c.picked(None), set(&[0, 1, 2, 3]));
+        assert_eq!(c.picked(Some(&anchors(&[]))), set(&[0, 1, 2, 3]));
+        assert_eq!(c.style(&set(&[1])), (Some(10.0), Some(CornerKind::Round)));
+        assert_eq!(c.style(&set(&[0, 1])), (None, Some(CornerKind::Round)));
+        assert_eq!(c.style(&set(&[0, 2, 3])), (Some(0.0), Some(CornerKind::Round)));
+        let e = LiveShape::Ellipse { w: 1.0, h: 1.0, pie: (0.0, 360.0), xf: Affine::IDENTITY };
+        assert!(LiveCorners::new(&e.to_path(), Some(&e)).is_none());
+    }
+
+    /// #511: a plain path (a star) rounds its corners and keeps them editable; with none cut it
+    /// is plain again.
+    #[test]
+    fn a_plain_path_keeps_its_cut_corners_live() {
+        let star = shapes::star(Point::new(100.0, 100.0), 50.0, 25.0, 5, 0.0);
+        let (mut path, mut live) = (star.clone(), None);
+        set_corners(&mut path, &mut live, &set(&[0, 2]), Some(5.0), None);
+        let Some(LiveShape::Path { base, radii, kinds }) = &live else { panic!("live corners: {live:?}") };
+        assert_eq!((base, radii.len(), kinds.len()), (&star, 10, 0));
+        assert_eq!(path.anchor_count(), 12, "two corners cut");
+        let c = LiveCorners::new(&path, live.as_ref()).unwrap();
+        assert_eq!(c.style(&set(&[0, 2])), (Some(5.0), Some(CornerKind::Round)));
+        assert_eq!(c.corners_of(&anchors(&[0, 11])), set(&[0]), "the first corner's cut ends the path");
+        // Kinds alone keep it live; squaring every corner makes it the plain star again.
+        set_corners(&mut path, &mut live, &set(&[0, 2]), Some(0.0), Some(CornerKind::Chamfer));
+        assert!(matches!(&live, Some(LiveShape::Path { radii, .. }) if radii.is_empty()));
+        set_corners(&mut path, &mut live, &set(&[0, 2]), None, Some(CornerKind::Round));
+        assert_eq!((path, live), (star, None));
+    }
+
+    /// A polygon's corners are cut in the document and survive a change of sides when shared.
+    #[test]
+    fn polygon_corners_follow_their_sides() {
+        let mut l = LiveShape::Polygon { radius: 50.0, sides: 6, xf: Affine::translate((100.0, 100.0)), radii: vec![], kinds: vec![] };
+        let (mut path, mut live) = (l.to_path(), Some(l.clone()));
+        set_corners(&mut path, &mut live, &set(&[0, 1, 2, 3, 4, 5]), Some(8.0), None);
+        assert_eq!(path.anchor_count(), 12);
+        l = live.clone().unwrap();
+        l.set_sides(8);
+        assert!(matches!(&l, LiveShape::Polygon { radii, .. } if *radii == vec![8.0; 8]));
+        assert_eq!(l.to_path().anchor_count(), 16);
+        // Mixed radii: the new corners are sharp.
+        set_corners(&mut path, &mut live, &set(&[1]), Some(3.0), None);
+        let mut l = live.unwrap();
+        l.set_sides(7);
+        assert!(matches!(&l, LiveShape::Polygon { radii, .. } if *radii == [8.0, 3.0, 8.0, 8.0, 8.0, 8.0, 0.0]));
+        // An uneven scale keeps the cuts circular: the radius scales by the mean scale.
+        let mut p = l.clone();
+        assert!(p.transform(Affine::scale_non_uniform(4.0, 1.0)));
+        assert!(matches!(&p, LiveShape::Polygon { radii, .. } if (radii[0] - 16.0).abs() < 1e-9));
+        assert!(!p.transform(Affine::translate((5.0, 0.0))), "a move needs no new cuts");
+    }
+
+    #[test]
+    fn open_paths_and_curves_keep_their_ends_and_smooth_anchors() {
+        let pen = PathData::single(SubPath::polyline(&[Point::new(0.0, 0.0), Point::new(100.0, 0.0), Point::new(100.0, 80.0)], false));
+        let c = LiveCorners::new(&pen, None).unwrap();
+        assert_eq!(c.all(), set(&[1]));
+        assert_eq!(c.picked(Some(&anchors(&[0]))), set(&[1]), "an end is no corner: every corner");
+        let (mut path, mut live) = (pen.clone(), None);
+        set_corners(&mut path, &mut live, &c.all(), Some(10.0), None);
+        assert_eq!(path.anchor_count(), 4);
+        assert_eq!(path.subpaths[0].anchors[0].p, Point::new(0.0, 0.0));
+        assert_eq!(path.subpaths[0].anchors[3].p, Point::new(100.0, 80.0));
+    }
+}

crates/doc/src/lib.rs

@@ -10,6 +10,7 @@ pub mod assets;
 pub mod blend;
 pub mod clipnest;
 pub mod cmyk;
+pub mod corners;
 pub mod graph;
 pub mod hit;
 pub mod inks;
@@ -58,6 +59,7 @@ pub use appearance::{
     StrokeLayer, WidthProfile,
 };
 pub use assets::ExportAsset;
+pub use corners::LiveCorners;
 pub use graph::{GraphKind, GraphSpec};
 pub use hit::{Hit, HitKind};
 pub use links::{LinkInfo, PlacementOptions};

crates/doc/src/node.rs

@@ -1,17 +1,16 @@
 //! Document nodes.
 
-use std::collections::BTreeSet;
 use std::sync::Arc;
 
 use serde::{Deserialize, Serialize};
 use vectorcraft_color::BlendMode;
+use vectorcraft_geom::corners::cut_corners;
 use vectorcraft_geom::shapes::{self, CornerKind};
 use vectorcraft_geom::{Affine, BezPath, FillRule, PathData, Point, Rect};
 
 use crate::appearance::Appearance;
 use crate::live::{BlendSpec, EnvelopeKind, GradientMesh, Outliner};
 use crate::pattern::RepeatSpec;
-use crate::selection::AnchorRef;
 use crate::text::TextObject;
 
 /// Stable per-document object id. Never reused.
@@ -104,11 +103,30 @@ pub enum LiveShape {
         radius: f64,
         sides: u32,
         xf: Affine,
+        /// Corner radii in document units, by vertex from the first one clockwise (none: sharp).
+        #[serde(default, skip_serializing_if = "Vec::is_empty")]
+        radii: Vec<f64>,
+        /// Corner kinds, in the same order (none: round).
+        #[serde(default, skip_serializing_if = "all_round")]
+        kinds: Vec<CornerKind>,
     },
     Line {
         a: Point,
         b: Point,
     },
+    /// Any other path whose corners Live Corners cut (a star, a pen path): it keeps its uncut
+    /// outline so the corners stay editable. Without a cut corner it is a plain path again.
+    Path {
+        /// The path with its corners uncut, in the document.
+        base: PathData,
+        /// Corner radii in document units, by anchor index of `base` (counting every subpath's
+        /// anchors in order; missing: sharp).
+        #[serde(default, skip_serializing_if = "Vec::is_empty")]
+        radii: Vec<f64>,
+        /// Corner kinds, in the same order (missing: round).
+        #[serde(default, skip_serializing_if = "all_round")]
+        kinds: Vec<CornerKind>,
+    },
 }
 
 impl LiveShape {
@@ -119,6 +137,7 @@ impl LiveShape {
             LiveShape::Ellipse { .. } => "Ellipse",
             LiveShape::Polygon { .. } => "Polygon",
             LiveShape::Line { .. } => "Line",
+            LiveShape::Path { .. } => "Path",
         }
     }
     /// Regenerate the path.
@@ -128,10 +147,33 @@ impl LiveShape {
                 shapes::rectangle_with_corners(Rect::new(0.0, 0.0, *w, *h), *radii, *kinds).transformed(*xf)
             }
             LiveShape::Ellipse { w, h, xf, .. } => shapes::ellipse(Rect::new(0.0, 0.0, *w, *h)).transformed(*xf),
-            LiveShape::Polygon { radius, sides, xf } => shapes::polygon(Point::ZERO, *radius, *sides, 0.0).transformed(*xf),
+            LiveShape::Polygon { radii, kinds, .. } => cut_corners(&self.polygon_outline(), radii, kinds).0,
             LiveShape::Line { a, b } => shapes::line(*a, *b),
+            LiveShape::Path { base, radii, kinds } => cut_corners(base, radii, kinds).0,
         }
     }
+    /// A polygon's outline with its corners uncut, in the document (empty for other shapes).
+    pub fn polygon_outline(&self) -> PathData {
+        match self {
+            LiveShape::Polygon { radius, sides, xf, .. } => shapes::polygon(Point::ZERO, *radius, *sides, 0.0).transformed(*xf),
+            _ => PathData::default(),
+        }
+    }
+    /// Give a polygon `n` sides (3–1000). Its corners keep the radius and kind they shared; else
+    /// the corners past the old last one are sharp.
+    pub fn set_sides(&mut self, n: u64) {
+        let LiveShape::Polygon { sides, radii, kinds, .. } = self else { return };
+        *sides = n.clamp(3, 1000) as u32;
+        fn spread<T: Copy + PartialEq + Default>(v: &mut Vec<T>, n: usize) {
+            match v.first().copied() {
+                Some(x) if v.iter().all(|y| *y == x) => *v = vec![x; n],
+                Some(_) => v.resize(n, T::default()),
+                None => {}
+            }
+        }
+        spread(radii, *sides as usize);
+        spread(kinds, *sides as usize);
+    }
     /// Apply `a` to the shape. True when the shape is no longer `a` applied to the old path, so
     /// the caller must regenerate the path with [`LiveShape::to_path`].
     ///
@@ -140,19 +182,27 @@ impl LiveShape {
     /// only the rotation, reflection and position. The radii scale by the mean scale (the square
     /// root of the determinant), which [`LiveShape::keep_corners`] undoes for Scale Corners off.
     /// A shear can't keep circular corners: `xf` takes it, as do moves (so files saved with a
-    /// scale in `xf` keep their exact geometry until they're next transformed).
+    /// scale in `xf` keep their exact geometry until they're next transformed). A polygon's or a
+    /// path's corner radii are document lengths: they scale by the mean scale and the corners are
+    /// cut again, circular whatever `a` does.
     pub fn transform(&mut self, a: Affine) -> bool {
         match self {
             LiveShape::Rectangle { xf, .. } => {
                 *xf = a * *xf;
-                let [a0, a1, a2, a3, _, _] = a.as_coeffs();
-                let moves_only = (a0 - 1.0).abs() < 1e-12 && a1.abs() < 1e-12 && a2.abs() < 1e-12 && (a3 - 1.0).abs() < 1e-12;
-                !moves_only && self.fold_scale()
+                !moves_only(a) && self.fold_scale()
             }
-            LiveShape::Ellipse { xf, .. } | LiveShape::Polygon { xf, .. } => {
+            LiveShape::Ellipse { xf, .. } => {
                 *xf = a * *xf;
                 false
             }
+            LiveShape::Polygon { xf, radii, .. } => {
+                *xf = a * *xf;
+                scale_radii(radii, a)
+            }
+            LiveShape::Path { base, radii, .. } => {
+                base.transform(a);
+                scale_radii(radii, a)
+            }
             LiveShape::Line { a: p, b } => {
                 *p = a * *p;
                 *b = a * *b;
@@ -192,52 +242,16 @@ impl LiveShape {
     /// Divide live corner radii by `k`, the mean scale of a transform just applied, so the corners
     /// keep their size (Scale Corners off). False when nothing changed.
     pub fn keep_corners(&mut self, k: f64) -> bool {
-        match self {
-            LiveShape::Rectangle { radii, .. } if radii.iter().any(|r| *r > 0.0) && k > 1e-12 => {
-                radii.iter_mut().for_each(|r| *r /= k);
-                true
-            }
-            _ => false,
+        let radii: &mut [f64] = match self {
+            LiveShape::Rectangle { radii, .. } => radii,
+            LiveShape::Polygon { radii, .. } | LiveShape::Path { radii, .. } => radii,
+            LiveShape::Ellipse { .. } | LiveShape::Line { .. } => return false,
+        };
+        if !radii.iter().any(|r| *r > 0.0) || k <= 1e-12 {
+            return false;
         }
-    }
-    /// The corner (radii order) of each anchor of a rectangle's path; empty for other shapes.
-    pub fn anchor_corners(&self) -> Vec<usize> {
-        match self {
-            LiveShape::Rectangle { w, h, radii, .. } => shapes::rectangle_anchor_corners(Rect::new(0.0, 0.0, *w, *h), *radii),
-            _ => vec![],
-        }
-    }
-    /// The corners of a rectangle that hold one of `anchors` (of its only subpath).
-    pub fn corners_of(&self, anchors: &BTreeSet<AnchorRef>) -> [bool; 4] {
-        let map = self.anchor_corners();
-        let mut out = [false; 4];
-        for k in anchors.iter().filter(|(si, _)| *si == 0).filter_map(|(_, ai)| map.get(*ai)) {
-            if let Some(c) = out.get_mut(*k) {
-                *c = true;
-            }
-        }
-        out
-    }
-    /// The corners Live Corners edit for a selection: those holding a selected anchor when the
-    /// shape is partly selected (Direct Selection), else all four.
-    pub fn picked_corners(&self, partial: Option<&BTreeSet<AnchorRef>>) -> [bool; 4] {
-        let picked = partial.map_or([true; 4], |a| self.corners_of(a));
-        if picked.contains(&true) { picked } else { [true; 4] }
-    }
-    /// The anchors of a rectangle's `corners` (to keep them selected as its path changes).
-    pub fn corner_anchors(&self, corners: [bool; 4]) -> BTreeSet<AnchorRef> {
-        self.anchor_corners().into_iter().enumerate().filter(|(_, k)| corners.get(*k) == Some(&true)).map(|(ai, _)| (0, ai)).collect()
-    }
-    /// The radius (in document units, see [`LiveShape::folded`]) and kind the `corners` of a
-    /// rectangle share (each `None` when they differ).
-    pub fn corner_style(&self, corners: [bool; 4]) -> (Option<f64>, Option<CornerKind>) {
-        let LiveShape::Rectangle { radii, kinds, .. } = self.folded() else { return (None, None) };
-        fn shared<T: Copy>(v: [T; 4], corners: [bool; 4], same: impl Fn(T, T) -> bool) -> Option<T> {
-            let mut it = v.into_iter().zip(corners).filter(|(_, on)| *on).map(|(x, _)| x);
-            let first = it.next()?;
-            it.all(|x| same(x, first)).then_some(first)
-        }
-        (shared(radii, corners, |a, b| (a - b).abs() < 1e-9), shared(kinds, corners, |a, b| a == b))
+        radii.iter_mut().for_each(|r| *r /= k);
+        true
     }
     /// Rotation angle of the live shape in degrees (shown in the Properties panel).
     pub fn angle_deg(&self) -> f64 {
@@ -247,14 +261,33 @@ impl LiveShape {
                 c[1].atan2(c[0]).to_degrees()
             }
             LiveShape::Line { a, b } => (b.y - a.y).atan2(b.x - a.x).to_degrees(),
+            LiveShape::Path { .. } => 0.0,
         }
     }
 }
 
-fn all_round(kinds: &[CornerKind; 4]) -> bool {
+fn all_round(kinds: &[CornerKind]) -> bool {
     kinds.iter().all(|k| *k == CornerKind::Round)
 }
 
+/// Whether `a` only moves (no rotation, scale, shear or reflection).
+fn moves_only(a: Affine) -> bool {
+    let [a0, a1, a2, a3, _, _] = a.as_coeffs();
+    (a0 - 1.0).abs() < 1e-12 && a1.abs() < 1e-12 && a2.abs() < 1e-12 && (a3 - 1.0).abs() < 1e-12
+}
+
+/// Scale corner radii in document units by the mean scale of `a`, applied to their path. True
+/// when the path must be cut again: some corner is cut and `a` does more than move it (an
+/// uneven scale would otherwise make the cuts elliptical).
+fn scale_radii(radii: &mut [f64], a: Affine) -> bool {
+    if moves_only(a) || !radii.iter().any(|r| *r > 0.0) {
+        return false;
+    }
+    let k = a.determinant().abs().sqrt();
+    radii.iter_mut().for_each(|r| *r *= k);
+    true
+}
+
 /// What a transform scales besides geometry, by its mean scale (the square root of its
 /// determinant). `Scaling::default()` (and `false`) scales nothing else; `true` scales stroke
 /// weights and dashes only. Transforms the user asks for take theirs from Scale Strokes & Effects
@@ -1214,7 +1247,7 @@ impl Node {
 
     /// Shapes drawn with the shape tools (live rectangles, ellipses and polygons) show their centre.
     pub fn shows_center_by_default(&self) -> bool {
-        matches!(&self.kind, NodeKind::Path { live: Some(l), .. } if !matches!(l, LiveShape::Line { .. }))
+        matches!(&self.kind, NodeKind::Path { live: Some(l), .. } if !matches!(l, LiveShape::Line { .. } | LiveShape::Path { .. }))
     }
 
     /// Change this object's attributes with `f`; all-default attributes are dropped.
@@ -1276,23 +1309,9 @@ mod tests {
     }
 
     #[test]
-    fn rectangle_corners_map_to_their_anchors() {
+    fn corner_kinds_save_only_when_some_corner_isnt_round() {
         let mut l = LiveShape::Rectangle { w: 100.0, h: 50.0, radii: [0.0, 10.0, 0.0, 0.0], kinds: Default::default(), xf: Affine::IDENTITY };
-        // Anchors: top-left, the top-right corner's two, bottom-right, bottom-left.
-        assert_eq!(l.anchor_corners(), [0, 1, 1, 2, 3]);
-        assert_eq!(l.to_path().anchor_count(), 5);
-        let set = |v: &[usize]| v.iter().map(|ai| (0, *ai)).collect::<BTreeSet<AnchorRef>>();
-        assert_eq!(l.corners_of(&set(&[2, 4])), [false, true, false, true]);
-        assert_eq!(l.corners_of(&set(&[9])), [false; 4], "out of range");
-        assert_eq!(l.corner_anchors([false, true, false, true]), set(&[1, 2, 4]));
-        // Partly selected: the corners of the selected anchors; wholly (or no anchor): all four.
-        assert_eq!(l.picked_corners(Some(&set(&[0]))), [true, false, false, false]);
-        assert_eq!(l.picked_corners(None), [true; 4]);
-        assert_eq!(l.picked_corners(Some(&set(&[]))), [true; 4]);
-        assert_eq!(l.corner_style([false, true, false, false]), (Some(10.0), Some(CornerKind::Round)));
-        assert_eq!(l.corner_style([true, true, false, false]), (None, Some(CornerKind::Round)));
-        assert_eq!(l.corner_style([true, false, true, true]), (Some(0.0), Some(CornerKind::Round)));
-        // Corner kinds save only when some corner isn't round, and older files read as round.
+        // Older files read as round.
         let json = serde_json::to_value(&l).unwrap();
         assert!(json.get("kinds").is_none(), "{json}");
         assert_eq!(serde_json::from_value::<LiveShape>(json).unwrap(), l);
@@ -1302,7 +1321,16 @@ mod tests {
         let json = serde_json::to_value(&l).unwrap();
         assert_eq!(json["kinds"], serde_json::json!(["round", "chamfer", "round", "round"]));
         assert_eq!(serde_json::from_value::<LiveShape>(json).unwrap(), l);
-        assert!(LiveShape::Ellipse { w: 1.0, h: 1.0, pie: (0.0, 360.0), xf: Affine::IDENTITY }.anchor_corners().is_empty());
+        // Polygons saved before Live Corners cut theirs read as sharp; a live path round-trips.
+        let p: LiveShape =
+            serde_json::from_value(serde_json::json!({"shape": "polygon", "radius": 10.0, "sides": 5, "xf": [1, 0, 0, 1, 0, 0]})).unwrap();
+        assert_eq!(p, LiveShape::Polygon { radius: 10.0, sides: 5, xf: Affine::IDENTITY, radii: vec![], kinds: vec![] });
+        let star = shapes::star(Point::new(0.0, 0.0), 20.0, 10.0, 5, 0.0);
+        let path = LiveShape::Path { base: star, radii: vec![2.0; 10], kinds: vec![] };
+        let json = serde_json::to_value(&path).unwrap();
+        assert_eq!((json["shape"].as_str(), json.get("kinds")), (Some("path"), None));
+        assert_eq!(serde_json::from_value::<LiveShape>(json).unwrap(), path);
+        assert_eq!(path.label(), "Path");
     }
 
     /// A node holding a live rectangle (`w` × `h`, all radii `r`, placed by `xf`) and its path.
@@ -1427,7 +1455,8 @@ mod tests {
         assert!(near_xf(*xf, Affine::translate((5.0, 5.0))));
         let (a, b) = (folded.to_path().bounds().unwrap(), live.to_path().bounds().unwrap());
         assert!(near(a.x0, b.x0) && near(a.y0, b.y0) && near(a.x1, b.x1) && near(a.y1, b.y1), "same place and size: {a:?} {b:?}");
-        assert_eq!(live.corner_style([true; 4]).0, Some(40.0), "the radius in document units");
+        let corners = crate::LiveCorners::of(&n).unwrap();
+        assert_eq!(corners.style(&corners.all()).0, Some(40.0), "the radius in document units");
         // Nothing to fold: a moved, rotated or reflected rectangle and a sheared one stay as they are.
         let place = Affine::translate((5.0, 5.0)) * Affine::rotate(0.5) * Affine::scale_non_uniform(-1.0, 1.0);
         let shear = Affine::new([1.0, 0.0, 0.5, 1.0, 0.0, 0.0]);

crates/engine/src/cmd/create.rs

@@ -188,7 +188,8 @@ fn polygon(s: &mut Session, p: &Value) -> Result<Value> {
     let r = f64_req(p, "radius", "shape.polygon")?.abs();
     let sides = p.get("sides").and_then(Value::as_u64).unwrap_or(6).clamp(3, 1000) as u32;
     let rot = f64_or(p, "rotation", 0.0);
-    let live = LiveShape::Polygon { radius: r, sides, xf: Affine::translate(c.to_vec2()) * Affine::rotate(rot.to_radians()) };
+    let xf = Affine::translate(c.to_vec2()) * Affine::rotate(rot.to_radians());
+    let live = LiveShape::Polygon { radius: r, sides, xf, radii: vec![], kinds: vec![] };
     add_art(s, "Polygon", path_kind(live.to_path(), Some(live)), None)
 }
 

crates/engine/src/cmd/object.rs

@@ -1,11 +1,13 @@
 //! Object menu: transforms, arrange, group, lock/hide, compound paths, clipping masks, isolation,
 //! align & distribute, object properties.
 
+use std::collections::BTreeSet;
 use std::sync::Arc;
 
 use serde_json::{Value, json};
 use vectorcraft_color::{BlendMode, Paint};
-use vectorcraft_doc::{Appearance, Document, Knockout, Node, NodeId, NodeKind, OrientedBox, Scaling};
+use vectorcraft_doc::corners::set_corners;
+use vectorcraft_doc::{Appearance, Document, Knockout, LiveCorners, LiveShape, Node, NodeId, NodeKind, OrientedBox, Scaling};
 use vectorcraft_geom::shapes::CornerKind;
 use vectorcraft_geom::{Affine, FillRule, Point, Rect, Vec2};
 
@@ -194,7 +196,7 @@ pub fn specs() -> Vec<CommandSpec> {
             "Live Shape Properties",
             [],
             None,
-            "{id?, ids?, radius?: pt, kind?: \"round\"|\"invertedRound\"|\"chamfer\", corners?: [0..3…], sides?: n} (Live Corners: radius and kind set a rectangle's corners: `corners` (0 top-left, 1 top-right, 2 bottom-right, 3 bottom-left), else the corners with a Direct-Selected anchor, else all four; sides: a polygon's)",
+            "{id?, ids?, radius?: pt, kind?: \"round\"|\"invertedRound\"|\"chamfer\", corners?: [i…], sides?: n} (Live Corners on any path: radius and kind set its corners (anchors without handles between two straight sides): `corners` (anchor indices of the path with its corners uncut, counting every subpath's anchors in order: a rectangle's 0 top-left, 1 top-right, 2 bottom-right, 3 bottom-left; a polygon's from the first vertex clockwise), else the corners with a Direct-Selected anchor, else every corner; each radius is drawn no larger than half the corner's shorter side allows; a path that isn't a live shape keeps its uncut outline so its corners stay editable, and is plain again once none is cut; sides: a polygon's, which keep the radius they shared)",
             has_selection,
             set_live_shape
         ),
@@ -1006,21 +1008,12 @@ fn expand_shape(s: &mut Session, _: &Value) -> Result<Value> {
     ok()
 }
 
-/// The `corners` of `object.setLiveShape`: a mask of corner indices 0–3.
-fn corners_param(p: &Value) -> Result<Option<[bool; 4]>> {
+/// The `corners` of `object.setLiveShape`: anchor indices of a path's uncut outline.
+fn corners_param(p: &Value) -> Result<Option<BTreeSet<usize>>> {
     let Some(v) = p.get("corners").filter(|v| !v.is_null()) else { return Ok(None) };
-    let err = || {
-        bad(
-            "object.setLiveShape",
-            format!("`corners` must list corner indices 0–3 (0 top-left, 1 top-right, 2 bottom-right, 3 bottom-left), not {v}"),
-        )
-    };
-    let mut mask = [false; 4];
-    for k in v.as_array().ok_or_else(err)? {
-        let slot = k.as_u64().and_then(|k| mask.get_mut(usize::try_from(k).ok()?)).ok_or_else(err)?;
-        *slot = true;
-    }
-    Ok(Some(mask))
+    let err = || bad("object.setLiveShape", format!("`corners` must list anchor indices (0, 1, 2…), not {v}"));
+    let list = v.as_array().ok_or_else(err)?;
+    list.iter().map(|k| k.as_u64().and_then(|k| usize::try_from(k).ok()).ok_or_else(err)).collect::<Result<_>>().map(Some)
 }
 
 fn set_live_shape(s: &mut Session, p: &Value) -> Result<Value> {
@@ -1035,39 +1028,37 @@ fn set_live_shape(s: &mut Session, p: &Value) -> Result<Value> {
         })
         .transpose()?;
     let radius = p.get("radius").and_then(Value::as_f64);
+    let sides = p.get("sides").and_then(Value::as_u64);
     s.edit("Live Shape", |d, sel| {
         for id in &ids {
-            let Some(NodeKind::Path { path, live: Some(live), .. }) = d.node_mut(*id).map(|n| &mut n.kind) else { continue };
-            // Radii are document lengths and corners circular, also on a rectangle from a file
-            // that kept an uneven scale in its transform (#442).
-            live.fold_scale();
-            let partial = sel.anchors.get_mut(id);
-            let picked = corners.unwrap_or_else(|| live.picked_corners(partial.as_deref()));
-            let (layout, selected) = (live.anchor_corners(), partial.as_deref().map(|a| live.corners_of(a)));
-            match live {
-                vectorcraft_doc::LiveShape::Rectangle { radii, kinds, .. } => {
-                    for ((_, r), k) in picked.iter().zip(radii.iter_mut()).zip(kinds.iter_mut()).filter(|((on, _), _)| **on) {
-                        if let Some(radius) = radius {
-                            *r = radius.max(0.0);
-                        }
-                        if let Some(kind) = kind {
-                            *k = kind;
-                        }
-                    }
-                }
-                vectorcraft_doc::LiveShape::Polygon { sides, .. } => {
-                    if let Some(n) = p.get("sides").and_then(Value::as_u64) {
-                        *sides = n.clamp(3, 1000) as u32;
-                    }
-                }
-                _ => {}
+            let Some(NodeKind::Path { path, live, .. }) = d.node_mut(*id).map(|n| &mut n.kind) else { continue };
+            if let (Some(l @ LiveShape::Polygon { .. }), Some(n)) = (live.as_mut(), sides) {
+                l.set_sides(n);
+                *path = l.to_path();
             }
-            *path = live.to_path();
+            if radius.is_none() && kind.is_none() {
+                continue;
+            }
+            let partial = sel.anchors.get_mut(id);
+            // The corners to edit, the Direct-Selected ones and the anchor layout before the edit.
+            let Some(c) = LiveCorners::new(path, live.as_ref()) else { continue };
+            let picked = match &corners {
+                Some(k) if k.last().is_some_and(|k| *k >= c.base.anchor_count()) => {
+                    let n = c.base.anchor_count();
+                    let past: Vec<_> = k.range(n..).collect();
+                    return Err(bad("object.setLiveShape", format!("object {id} has anchors 0–{} for `corners`, not {past:?}", n.saturating_sub(1))));
+                }
+                Some(k) => k.clone(),
+                None => c.picked(partial.as_deref()),
+            };
+            let (selected, layout) = (partial.as_deref().map(|a| c.corners_of(a)), c.sources().to_vec());
+            set_corners(path, live, &picked, radius, kind);
             // Direct-Selected corners stay selected as they gain or lose anchors.
             if let (Some(anchors), Some(selected)) = (partial, selected)
-                && live.anchor_corners() != layout
+                && let Some(c) = LiveCorners::new(path, live.as_ref())
+                && c.sources() != layout
             {
-                *anchors = live.corner_anchors(selected);
+                *anchors = c.anchors_of(&selected);
             }
         }
         Ok(())

crates/engine/src/tests_livecorners.rs

@@ -1,10 +1,11 @@
-//! Live Corners on live rectangles: `object.setLiveShape` rounds the given corners, the
-//! Direct-Selected ones or all four, sets corner kinds, and keeps selected corners selected.
+//! Live Corners: `object.setLiveShape` rounds the given corners of a live rectangle, the
+//! Direct-Selected ones or all four, sets corner kinds, and keeps selected corners selected; and
+//! does the same on any path's corners: a star, a live polygon, a pen path (#511).
 
 use std::collections::BTreeSet;
 
 use serde_json::{Value, json};
-use vectorcraft_doc::{AnchorRef, LiveShape, NodeKind};
+use vectorcraft_doc::{AnchorRef, LiveCorners, LiveShape, NodeKind};
 use vectorcraft_geom::Affine;
 use vectorcraft_geom::shapes::CornerKind;
 
@@ -150,3 +151,131 @@ fn corners_of_a_non_square_rectangle_are_alike_circles() {
     run(&mut s, json!({"radius": 8}));
     assert_spans(&s, id, [8.0; 4]);
 }
+
+/// A selected five-pointed star at (200, 200) of radii 60 and 30: a plain path.
+fn star(s: &mut Session) -> NodeId {
+    NodeId(s.execute("shape.star", &json!({"cx": 200, "cy": 200, "radius1": 60, "radius2": 30})).unwrap()["id"].as_u64().unwrap())
+}
+
+fn node_path(s: &Session, id: NodeId) -> (vectorcraft_geom::PathData, Option<LiveShape>) {
+    let NodeKind::Path { path, live, .. } = &s.doc().unwrap().doc.node(id).unwrap().kind else { panic!("not a path") };
+    (path.clone(), live.clone())
+}
+
+/// The radius and kind every corner of `id` shares, and how many corners it has.
+fn style(s: &Session, id: NodeId) -> (Option<f64>, Option<CornerKind>, usize) {
+    let c = LiveCorners::of(s.doc().unwrap().doc.node(id).unwrap()).unwrap();
+    let (r, k) = c.style(&c.all());
+    (r, k, c.corners.len())
+}
+
+/// The radius set on each of the first `N` corners of `id`.
+fn radii<const N: usize>(s: &Session, id: NodeId) -> [f64; N] {
+    let c = LiveCorners::of(s.doc().unwrap().doc.node(id).unwrap()).unwrap();
+    std::array::from_fn(|k| c.radius(k))
+}
+
+/// #511: a star (a plain path) rounds all ten corners as one step, keeps them editable (a new
+/// radius cuts the outline again rather than the cut path), and is the plain star again at 0.
+#[test]
+fn a_star_rounds_its_corners_and_keeps_them_live() {
+    let (mut s, _) = session();
+    let id = star(&mut s);
+    let (sharp, _) = node_path(&s, id);
+    assert_eq!(style(&s, id), (Some(0.0), Some(CornerKind::Round), 10));
+    let undo = s.doc().unwrap().history.undo.len();
+    run(&mut s, json!({"radius": 4}));
+    assert_eq!(s.doc().unwrap().history.undo.len(), undo + 1);
+    let (path, live) = node_path(&s, id);
+    assert_eq!(path.anchor_count(), 20);
+    assert!(matches!(&live, Some(LiveShape::Path { base, .. }) if *base == sharp), "{live:?}");
+    assert_eq!(style(&s, id), (Some(4.0), Some(CornerKind::Round), 10));
+    assert_eq!(s.doc().unwrap().doc.node(id).unwrap().kind_label(), "Path");
+    run(&mut s, json!({"radius": 6, "kind": "chamfer"}));
+    assert_eq!((node_path(&s, id).0.anchor_count(), style(&s, id)), (20, (Some(6.0), Some(CornerKind::Chamfer), 10)));
+    run(&mut s, json!({"radius": 0, "kind": "round"}));
+    assert_eq!(node_path(&s, id), (sharp, None));
+    s.execute("edit.undo", &json!({})).unwrap();
+    assert_eq!(style(&s, id).0, Some(6.0));
+}
+
+/// Each corner stops at its own limit: its cut reaches halfway along its shorter side, so a
+/// star's neighbouring corners meet without overlapping.
+#[test]
+fn star_corners_stop_at_their_own_limit() {
+    let (mut s, _) = session();
+    let id = star(&mut s);
+    run(&mut s, json!({"radius": 1000}));
+    let (path, live) = node_path(&s, id);
+    let Some(LiveShape::Path { base, .. }) = live else { panic!("live corners") };
+    // Every cut ends at the middle of a side: each side's two cuts meet there.
+    let sp = &base.subpaths[0];
+    for (i, a) in sp.anchors.iter().enumerate() {
+        let mid = a.p.midpoint(sp.anchors[(i + 1) % 10].p);
+        assert!(path.anchors().any(|(_, _, b)| b.p.distance(mid) < 1e-9), "a cut ends at {mid:?}");
+    }
+    assert_eq!(style(&s, id).0, Some(1000.0), "the radius set is kept; it draws as large as fits");
+}
+
+/// #511: Direct Selection picks one of the star's anchors: that corner alone rounds, and both of
+/// its new anchors stay selected.
+#[test]
+fn a_direct_selected_star_anchor_rounds_its_corner_alone() {
+    let (mut s, _) = session();
+    let id = star(&mut s);
+    s.execute("select.anchors", &json!({"id": id.0, "anchors": [[0, 0]], "mode": "set"})).unwrap();
+    run(&mut s, json!({"radius": 5}));
+    assert_eq!(node_path(&s, id).0.anchor_count(), 11);
+    // The top tip's cut ends the path and starts it.
+    assert_eq!(selected_anchors(&s, id), Some(anchors(&[0, 10])));
+    assert_eq!(radii::<2>(&s, id), [5.0, 0.0]);
+    // Explicit corners, as MCP and the widgets give them.
+    run(&mut s, json!({"id": id.0, "corners": [1, 3], "radius": 2}));
+    assert_eq!(radii::<4>(&s, id), [5.0, 2.0, 0.0, 2.0]);
+    // Past the outline's last anchor is an error, and nothing changes.
+    assert!(s.execute("object.setLiveShape", &json!({"id": id.0, "corners": [10], "radius": 1})).is_err());
+    assert_eq!(node_path(&s, id).0.anchor_count(), 13);
+}
+
+/// A live polygon stays live with cut corners, which keep their radius as its sides change and
+/// stay circular when it is scaled unevenly (the radius scales by the mean scale).
+#[test]
+fn a_polygon_keeps_its_corners_through_sides_and_scales() {
+    let (mut s, _) = session();
+    let id = NodeId(s.execute("shape.polygon", &json!({"cx": 200, "cy": 200, "radius": 80, "sides": 6})).unwrap()["id"].as_u64().unwrap());
+    run(&mut s, json!({"radius": 10}));
+    let (path, live) = node_path(&s, id);
+    assert_eq!(path.anchor_count(), 12);
+    assert!(matches!(live, Some(LiveShape::Polygon { sides: 6, .. })));
+    run(&mut s, json!({"sides": 8}));
+    assert_eq!((node_path(&s, id).0.anchor_count(), style(&s, id)), (16, (Some(10.0), Some(CornerKind::Round), 8)));
+    s.execute("object.scale", &json!({"sx": 400, "sy": 100, "corners": true})).unwrap();
+    assert!(matches!(node_path(&s, id).1, Some(LiveShape::Polygon { .. })));
+    assert!((style(&s, id).0.unwrap() - 20.0).abs() < 1e-9);
+    // Every cut is a circle of that radius: its ends are as far from its centre.
+    let path = node_path(&s, id).0;
+    let c = LiveCorners::of(s.doc().unwrap().doc.node(id).unwrap()).unwrap();
+    for k in &c.corners {
+        let centre = k.on_bisector(20.0 / k.sin());
+        let ends = path.anchors().filter(|(_, _, a)| (a.p.distance(centre) - 20.0).abs() < 1e-6).count();
+        assert_eq!(ends, 2, "corner {} is circular", k.index);
+    }
+}
+
+/// #511: a pen path's corners round; its ends don't. An ellipse has no corners to round.
+#[test]
+fn a_pen_path_rounds_its_corners_but_not_its_ends() {
+    let (mut s, _) = session();
+    let pen = json!({"anchors": [{"x": 10, "y": 200}, {"x": 110, "y": 100}, {"x": 210, "y": 200}, {"x": 310, "y": 100}]});
+    let id = NodeId(s.execute("path.create", &pen).unwrap()["id"].as_u64().unwrap());
+    s.execute("select.set", &json!({"ids": [id.0]})).unwrap();
+    run(&mut s, json!({"radius": 8}));
+    let path = node_path(&s, id).0;
+    assert_eq!(path.anchor_count(), 6);
+    let sp = &path.subpaths[0];
+    assert_eq!((sp.anchors[0].p.x, sp.anchors[5].p.x), (10.0, 310.0), "the ends stay");
+    let e = NodeId(s.execute("shape.ellipse", &json!({"x": 0, "y": 0, "width": 50, "height": 30})).unwrap()["id"].as_u64().unwrap());
+    let before = node_path(&s, e);
+    run(&mut s, json!({"id": e.0, "radius": 8}));
+    assert_eq!(node_path(&s, e), before);
+}

crates/geom/src/corners.rs

@@ -0,0 +1,283 @@
+//! Live Corners on any path: the anchors where two straight sides meet at an angle, and the path
+//! with those corners cut round, inverted round or chamfered. Rectangles cut theirs the same way.
+
+use kurbo::{Point, Vec2};
+
+use crate::path::{Anchor, AnchorKind, PathData, SubPath};
+use crate::shapes::{CornerKind, KAPPA};
+
+/// Lengths and radii at or below this are none at all.
+const EPS: f64 = 1e-9;
+/// Sides closer than this to straight on (or folded back), as the sine of their angle, make no corner.
+const MIN_SINE: f64 = 1e-6;
+
+/// A corner Live Corners can cut: an anchor without handles between two straight sides that
+/// meet at an angle.
+#[derive(Clone, Copy, Debug, PartialEq)]
+pub struct Corner {
+    /// The anchor's index in its path, counting the anchors of every subpath in order.
+    pub index: usize,
+    /// Where the sides meet.
+    pub at: Point,
+    /// Unit directions along the sides: towards the previous anchor, then towards the next one.
+    pub u: Vec2,
+    pub v: Vec2,
+    /// The sides' lengths, in the same order.
+    pub lu: f64,
+    pub lv: f64,
+}
+
+impl Corner {
+    /// The corner of anchor `i` of `sp`, whose index in the path is `index`; `None` when the
+    /// anchor is an end of an open subpath, has handles or is smooth, when a side curves or has
+    /// no length, or when the sides run straight on.
+    fn of(sp: &SubPath, i: usize, index: usize) -> Option<Self> {
+        let n = sp.anchors.len();
+        let (prev, next) = if sp.closed { ((i + n).checked_sub(1)? % n, (i + 1) % n) } else { (i.checked_sub(1)?, i + 1) };
+        let (a, prev, next) = (sp.anchors.get(i)?, sp.anchors.get(prev)?, sp.anchors.get(next)?);
+        if a.kind == AnchorKind::Smooth || a.has_in() || a.has_out() || prev.has_out() || next.has_in() {
+            return None;
+        }
+        let (du, dv) = (prev.p - a.p, next.p - a.p);
+        let (lu, lv) = (du.hypot(), dv.hypot());
+        if !(lu > EPS && lv > EPS && lu.is_finite() && lv.is_finite()) {
+            return None;
+        }
+        let c = Self { index, at: a.p, u: du / lu, v: dv / lv, lu, lv };
+        (c.sin() > MIN_SINE).then_some(c)
+    }
+
+    /// The cosine of the angle between the sides.
+    pub fn cos(&self) -> f64 {
+        self.u.dot(self.v)
+    }
+
+    /// The sine of the angle between the sides.
+    pub fn sin(&self) -> f64 {
+        self.u.cross(self.v).abs()
+    }
+
+    /// The angle between the sides, in degrees (90 at a rectangle's corners).
+    pub fn angle_deg(&self) -> f64 {
+        self.sin().atan2(self.cos()).to_degrees()
+    }
+
+    /// How far from the corner a round cut of radius `r` meets each side: r / tan(angle / 2).
+    pub fn setback(&self, r: f64) -> f64 {
+        r * (1.0 + self.cos()) / self.sin()
+    }
+
+    /// The largest radius the corner draws: its cut then reaches halfway along the shorter side,
+    /// so neighbouring corners at their largest meet without overlapping (half the shorter side
+    /// at a rectangle's corners).
+    pub fn max_radius(&self) -> f64 {
+        self.lu.min(self.lv) / 2.0 * self.sin() / (1.0 + self.cos())
+    }
+
+    /// The radius the corner is drawn with: `radius`, no larger than [`Corner::max_radius`], 0 when
+    /// negative or not a number.
+    pub fn fitted(&self, radius: f64) -> f64 {
+        radius.max(0.0).min(self.max_radius())
+    }
+
+    /// The point `k` along the bisector, measured in (u + v): the centre of a round cut of radius
+    /// `r` is at `k = r / sin`.
+    pub fn on_bisector(&self, k: f64) -> Point {
+        self.at + (self.u + self.v) * k
+    }
+
+    /// How much the radius grows when the centre of the round cut moves by `d`: `d`'s part along
+    /// the bisector, in radius units.
+    pub fn radius_change(&self, d: Vec2) -> f64 {
+        d.dot(self.u + self.v) * self.sin() / (2.0 + 2.0 * self.cos())
+    }
+
+    /// The two anchors replacing the corner when cut with `radius` (> 0, fitted): where the cut
+    /// meets the arriving side, then the leaving one.
+    fn cut(&self, radius: f64, kind: CornerKind) -> [Anchor; 2] {
+        let t = self.setback(radius);
+        let (start, end) = (self.at + self.u * t, self.at + self.v * t);
+        let (h_out, h_in) = match kind {
+            // An arc of `radius` tangent to both sides, turning by 180° less the angle.
+            CornerKind::Round => {
+                let h = radius * arc_handle(-self.cos());
+                (start - self.u * h, end - self.v * h)
+            }
+            // An arc centred on the corner through both ends, spanning the angle.
+            CornerKind::InvertedRound => {
+                let h = t * arc_handle(self.cos());
+                (start + square_to(self.u, self.v) * h, end + square_to(self.v, self.u) * h)
+            }
+            CornerKind::Chamfer => (start, end),
+        };
+        [Anchor { p: start, h_in: start, h_out, kind: AnchorKind::Corner }, Anchor { p: end, h_in, h_out: end, kind: AnchorKind::Corner }]
+    }
+}
+
+/// The unit direction square to unit `a`, on `b`'s side.
+fn square_to(a: Vec2, b: Vec2) -> Vec2 {
+    let w = b - a * a.dot(b);
+    w / w.hypot()
+}
+
+/// The handle length, per unit of radius, of a cubic drawing a circular arc that spans an angle
+/// whose cosine is `cos`: 4/3 · tan(angle / 4) ([`KAPPA`] for a quarter circle).
+fn arc_handle(cos: f64) -> f64 {
+    if cos.abs() < 1e-12 {
+        return KAPPA;
+    }
+    let (sin_half, cos_half) = (((1.0 - cos) / 2.0).max(0.0).sqrt(), ((1.0 + cos) / 2.0).max(0.0).sqrt());
+    4.0 / 3.0 * sin_half / (1.0 + cos_half)
+}
+
+/// The corners of `path` Live Corners can cut, in anchor order.
+pub fn path_corners(path: &PathData) -> Vec<Corner> {
+    let mut out = vec![];
+    let mut first = 0;
+    for sp in &path.subpaths {
+        out.extend((0..sp.anchors.len()).filter_map(|i| Corner::of(sp, i, first + i)));
+        first += sp.anchors.len();
+    }
+    out
+}
+
+/// `path` with each of its corners ([`path_corners`]) cut by the radius in `radii` and the kind in
+/// `kinds` at its anchor index (missing: 0 and round), each radius no larger than fits
+/// ([`Corner::fitted`]). A cut corner becomes two anchors; a closed subpath whose first corner
+/// is cut starts where that cut ends, so the cut closes it. Also returns, per subpath, the
+/// anchor index in `path` each anchor comes from.
+pub fn cut_corners(path: &PathData, radii: &[f64], kinds: &[CornerKind]) -> (PathData, Vec<Vec<usize>>) {
+    let corners = path_corners(path);
+    let mut corners = corners.iter().peekable();
+    let (mut subpaths, mut sources) = (Vec::with_capacity(path.subpaths.len()), Vec::with_capacity(path.subpaths.len()));
+    let mut first = 0;
+    for sp in &path.subpaths {
+        let (mut anchors, mut from) = (Vec::with_capacity(sp.anchors.len()), Vec::with_capacity(sp.anchors.len()));
+        let mut closing = None;
+        for (i, a) in sp.anchors.iter().enumerate() {
+            let index = first + i;
+            let corner = corners.next_if(|c| c.index == index);
+            let radius = corner.map_or(0.0, |c| c.fitted(radii.get(index).copied().unwrap_or(0.0)));
+            match corner {
+                Some(c) if radius > EPS => {
+                    let [start, end] = c.cut(radius, kinds.get(index).copied().unwrap_or_default());
+                    if i == 0 && sp.closed {
+                        closing = Some(start);
+                    } else {
+                        anchors.push(start);
+                        from.push(index);
+                    }
+                    anchors.push(end);
+                    from.push(index);
+                }
+                _ => {
+                    anchors.push(*a);
+                    from.push(index);
+                }
+            }
+        }
+        if let Some(a) = closing {
+            anchors.push(a);
+            from.push(first);
+        }
+        first += sp.anchors.len();
+        subpaths.push(SubPath::new(anchors, sp.closed));
+        sources.push(from);
+    }
+    (PathData::new(subpaths), sources)
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::shapes;
+    use kurbo::Rect;
+
+    fn near(a: Point, b: Point) -> bool {
+        a.distance(b) < 1e-9
+    }
+
+    /// A five-pointed star's tips are acute corners, its inner corners obtuse: all ten cut, each
+    /// a circle of the radius tangent to both sides.
+    #[test]
+    fn a_star_has_ten_corners_cut_as_circles() {
+        let star = shapes::star(Point::new(100.0, 100.0), 50.0, 25.0, 5, 0.0);
+        let corners = path_corners(&star);
+        assert_eq!(corners.len(), 10);
+        let angles: Vec<_> = corners.iter().map(|c| c.angle_deg()).collect();
+        assert!(angles.iter().step_by(2).all(|a| *a < 90.0), "tips are acute: {angles:?}");
+        assert!(angles.iter().skip(1).step_by(2).all(|a| *a > 90.0), "inner corners are obtuse: {angles:?}");
+        let (cut, from) = cut_corners(&star, &[4.0; 10], &[]);
+        assert_eq!(cut.anchor_count(), 20);
+        assert_eq!(from, [vec![0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 0]]);
+        for c in &corners {
+            // Both ends of the cut lie `setback` along the sides, as far from the arc's centre as
+            // the radius, and that centre is `radius` from either side.
+            let t = c.setback(4.0);
+            let o = c.on_bisector(4.0 / c.sin());
+            for p in [c.at + c.u * t, c.at + c.v * t] {
+                assert!((p.distance(o) - 4.0).abs() < 1e-9);
+                assert!(cut.anchors().any(|(_, _, a)| near(a.p, p)), "{p:?} is an anchor");
+            }
+        }
+    }
+
+    #[test]
+    fn radii_stop_at_half_the_shorter_side_per_corner() {
+        let tri = PathData::single(SubPath::polyline(&[Point::new(0.0, 0.0), Point::new(100.0, 0.0), Point::new(0.0, 40.0)], true));
+        let corners = path_corners(&tri);
+        assert_eq!(corners.len(), 3);
+        for c in &corners {
+            let max = c.max_radius();
+            assert!((c.setback(max) - c.lu.min(c.lv) / 2.0).abs() < 1e-9);
+            assert_eq!(c.fitted(1e6), max);
+            assert_eq!(c.fitted(-3.0), 0.0);
+            assert_eq!(c.fitted(f64::NAN), 0.0);
+        }
+        // The right angle at (0, 0) takes up to 20, half its shorter side.
+        assert!((corners[0].max_radius() - 20.0).abs() < 1e-9);
+    }
+
+    #[test]
+    fn ends_curves_smooth_and_straight_anchors_are_no_corners() {
+        // An open zig-zag: its ends aren't corners, nor is the anchor its sides run straight on through.
+        let open = PathData::single(SubPath::polyline(
+            &[Point::new(0.0, 0.0), Point::new(50.0, 0.0), Point::new(100.0, 0.0), Point::new(100.0, 50.0), Point::new(150.0, 80.0)],
+            false,
+        ));
+        assert_eq!(path_corners(&open).iter().map(|c| c.index).collect::<Vec<_>>(), [2, 3]);
+        // A curve on either side, or handles on the anchor, leave it alone.
+        let mut sp = SubPath::polyline(&[Point::new(0.0, 0.0), Point::new(100.0, 0.0), Point::new(100.0, 100.0), Point::new(0.0, 100.0)], true);
+        sp.anchors[1].h_out = Point::new(110.0, 10.0);
+        let p = PathData::single(sp);
+        assert_eq!(path_corners(&p).iter().map(|c| c.index).collect::<Vec<_>>(), [0, 3]);
+        assert!(path_corners(&shapes::ellipse(Rect::new(0.0, 0.0, 10.0, 10.0))).is_empty());
+        // A second subpath counts on from the first one's anchors.
+        let two = PathData::new(vec![
+            SubPath::polyline(&[Point::new(0.0, 0.0), Point::new(10.0, 0.0), Point::new(0.0, 10.0)], true),
+            SubPath::polyline(&[Point::new(50.0, 0.0), Point::new(60.0, 0.0), Point::new(50.0, 10.0)], true),
+        ]);
+        let (cut, from) = cut_corners(&two, &[0.0, 0.0, 0.0, 1.0], &[]);
+        assert_eq!(from, [vec![0, 1, 2], vec![3, 4, 5, 3]]);
+        assert_eq!(cut.anchor_count(), 7);
+    }
+
+    /// Inverted round and chamfer cuts end where the round one does, at any angle.
+    #[test]
+    fn kinds_share_their_ends() {
+        let star = shapes::star(Point::new(0.0, 0.0), 50.0, 25.0, 5, 0.0);
+        let ends = |k: CornerKind| -> Vec<Point> { cut_corners(&star, &[3.0; 10], &[k; 10]).0.anchors().map(|(_, _, a)| a.p).collect() };
+        let round = ends(CornerKind::Round);
+        assert_eq!(ends(CornerKind::InvertedRound), round);
+        assert_eq!(ends(CornerKind::Chamfer), round);
+        // The inverted arc stays on the circle about the corner.
+        let c = path_corners(&star)[0];
+        let [s, e] = c.cut(3.0, CornerKind::InvertedRound);
+        let bez = kurbo::CubicBez::new(s.p, s.h_out, e.h_in, e.p);
+        let r = c.setback(3.0);
+        for t in [0.25, 0.5, 0.75] {
+            use kurbo::ParamCurve;
+            assert!((bez.eval(t).distance(c.at) - r).abs() < r * 1e-3);
+        }
+    }
+}

crates/geom/src/lib.rs

@@ -9,6 +9,7 @@
 
 pub mod arc;
 pub mod bez;
+pub mod corners;
 pub mod hit;
 pub mod path;
 pub mod projective;

crates/geom/src/shapes.rs

@@ -62,52 +62,10 @@ pub fn fitted_corner_radius(w: f64, h: f64, radius: f64) -> f64 {
 /// Rectangle whose corners (top-left, top-right, bottom-right, bottom-left, y down) each have
 /// their own radius and kind, every radius clamped to half the shorter side. A corner without a
 /// radius is one anchor, the others two (where the cut meets each side). Clockwise from the
-/// top-left corner, its anchor on the top side first.
+/// top-left corner, its anchor on the top side first. Cut as any path's corners are
+/// ([`crate::corners::cut_corners`]).
 pub fn rectangle_with_corners(r: Rect, radii: [f64; 4], kinds: [CornerKind; 4]) -> PathData {
-    PathData::single(SubPath::new(corner_anchors(r, radii, kinds).into_iter().map(|(_, a)| a).collect(), true))
-}
-
-/// The corner (index into `radii`) of each anchor of [`rectangle_with_corners`].
-pub fn rectangle_anchor_corners(r: Rect, radii: [f64; 4]) -> Vec<usize> {
-    corner_anchors(r, radii, [CornerKind::Round; 4]).into_iter().map(|(k, _)| k).collect()
-}
-
-/// The anchors of [`rectangle_with_corners`], each with its corner.
-fn corner_anchors(r: Rect, radii: [f64; 4], kinds: [CornerKind; 4]) -> Vec<(usize, Anchor)> {
-    let r = r.abs();
-    // Each corner, with the directions of the side arriving at it and of the side leaving it.
-    let corners = [
-        (Point::new(r.x0, r.y0), Vec2::new(0.0, -1.0), Vec2::new(1.0, 0.0)),
-        (Point::new(r.x1, r.y0), Vec2::new(1.0, 0.0), Vec2::new(0.0, 1.0)),
-        (Point::new(r.x1, r.y1), Vec2::new(0.0, 1.0), Vec2::new(-1.0, 0.0)),
-        (Point::new(r.x0, r.y1), Vec2::new(-1.0, 0.0), Vec2::new(0.0, -1.0)),
-    ];
-    let mut out = Vec::with_capacity(8);
-    let mut top_left_start = None;
-    for (k, ((c, din, dout), (rad, kind))) in corners.into_iter().zip(radii.into_iter().zip(kinds)).enumerate() {
-        let rad = fitted_corner_radius(r.width(), r.height(), rad);
-        if rad <= 1e-9 {
-            out.push((k, Anchor::corner(c)));
-            continue;
-        }
-        // The cut runs from `start` on the arriving side to `end` on the leaving one.
-        let (start, end, h) = (c - din * rad, c + dout * rad, rad * KAPPA);
-        let (h_out, h_in) = match kind {
-            CornerKind::Round => (start + din * h, end - dout * h),
-            CornerKind::InvertedRound => (start + dout * h, end - din * h),
-            CornerKind::Chamfer => (start, end),
-        };
-        let start = Anchor { p: start, h_in: start, h_out, kind: AnchorKind::Corner };
-        let end = Anchor { p: end, h_in, h_out: end, kind: AnchorKind::Corner };
-        if k == 0 {
-            top_left_start = Some(start);
-        } else {
-            out.push((k, start));
-        }
-        out.push((k, end));
-    }
-    out.extend(top_left_start.map(|a| (0, a)));
-    out
+    crate::corners::cut_corners(&rectangle(r), &radii, &kinds).0
 }
 
 /// Ellipse inscribed in `r` with 4 smooth anchors (left, top, right, bottom).
@@ -244,6 +202,11 @@ pub fn polar_grid(r: Rect, concentric: u32, radial: u32) -> Vec<PathData> {
 mod tests {
     use super::*;
 
+    /// The corner (index into `radii`) of each anchor of [`rectangle_with_corners`].
+    fn corner_sources(r: Rect, radii: [f64; 4]) -> Vec<usize> {
+        crate::corners::cut_corners(&rectangle(r), &radii, &[]).1.concat()
+    }
+
     #[test]
     fn rect_is_4_corners_clockwise() {
         let p = rectangle(Rect::new(0.0, 0.0, 100.0, 50.0));
@@ -293,7 +256,7 @@ mod tests {
             a((x0, y0 + rad), (x0, y0 + rad), (x0, y0 + rad - k)),
         ];
         assert_eq!(rounded_rectangle(r, rad).subpaths[0].anchors, want);
-        assert_eq!(rectangle_anchor_corners(r, [rad; 4]), [0, 1, 1, 2, 2, 3, 3, 0]);
+        assert_eq!(corner_sources(r, [rad; 4]), [0, 1, 1, 2, 2, 3, 3, 0]);
         assert_eq!(rounded_rectangle(r, 0.0), rectangle(r));
     }
 
@@ -305,11 +268,11 @@ mod tests {
         let p = rectangle_with_corners(r, radii, [CornerKind::Round; 4]);
         let pts: Vec<Point> = p.subpaths[0].anchors.iter().map(|a| a.p).collect();
         assert_eq!(pts, [Point::new(0.0, 0.0), Point::new(90.0, 0.0), Point::new(100.0, 10.0), Point::new(100.0, 60.0), Point::new(0.0, 60.0)]);
-        assert_eq!(rectangle_anchor_corners(r, radii), [0, 1, 1, 2, 3]);
+        assert_eq!(corner_sources(r, radii), [0, 1, 1, 2, 3]);
         let quarter = PI * 25.0;
         assert!((p.subpaths[0].area() - (6000.0 - 100.0 + quarter)).abs() < 0.1, "a quarter circle off one corner");
         // The top-left corner rounded: its top-side anchor first, its left-side one last.
-        assert_eq!(rectangle_anchor_corners(r, [10.0, 0.0, 0.0, 200.0]), [0, 1, 2, 3, 3, 0]);
+        assert_eq!(corner_sources(r, [10.0, 0.0, 0.0, 200.0]), [0, 1, 2, 3, 3, 0]);
         // Inverted round cuts a quarter circle out; a chamfer cuts a triangle.
         let inverted = rectangle_with_corners(r, radii, [CornerKind::InvertedRound; 4]);
         assert!((inverted.subpaths[0].area() - (6000.0 - quarter)).abs() < 0.1);

crates/svg/src/css.rs

@@ -621,7 +621,9 @@ fn corners(path: &PathData, live: Option<&LiveShape>) -> Option<Corners> {
             let sweep = (pie.1 - pie.0).abs();
             (sweep < 1e-6 || (sweep - 360.0).abs() < 1e-6).then_some(Corners::Ellipse)
         }
-        Some(LiveShape::Rectangle { .. } | LiveShape::Ellipse { .. } | LiveShape::Polygon { .. } | LiveShape::Line { .. }) => None,
+        Some(
+            LiveShape::Rectangle { .. } | LiveShape::Ellipse { .. } | LiveShape::Polygon { .. } | LiveShape::Line { .. } | LiveShape::Path { .. },
+        ) => None,
         None => is_upright_rectangle(path).then_some(Corners::Square),
     }
 }

crates/tools/src/corners.rs

@@ -1,187 +1,231 @@
-//! Live Corners: the widgets inside the corners of a selected live rectangle. Dragging one rounds
-//! (or sharpens) the corners whose widgets show together: all four when the whole shape is
-//! selected, those with a selected anchor when Direct Selection picked some. Alt-clicking a widget
-//! cycles their corner kind (round, inverted round, chamfer); double-clicking one opens the
-//! Corners dialog. The Selection and Direct Selection tools share this, and the canvas draws the
-//! widgets from the same geometry.
+//! Live Corners: the widgets inside the corners of a selected path. The Selection tool shows them
+//! on a live rectangle or polygon, Direct Selection on any path (a star, a pen path) at each corner
+//! anchor: one without handles between two straight sides. Dragging one rounds (or sharpens) the
+//! corners whose widgets show together: every corner when the whole path is selected, those with a
+//! selected anchor when Direct Selection picked some. Alt-clicking a widget cycles their corner
+//! kind (round, inverted round, chamfer); double-clicking one opens the Corners dialog. The
+//! Selection and Direct Selection tools share this, and the canvas draws the widgets from the same
+//! geometry.
 
 use serde_json::{Map, Value, json};
-use vectorcraft_doc::{Document, LiveShape, NodeId, NodeKind, Selection};
-use vectorcraft_geom::shapes::{self, CornerKind};
-use vectorcraft_geom::{Affine, BezPath, Point, Rect, Vec2};
+use vectorcraft_doc::{Document, LiveCorners, LiveShape, NodeId, NodeKind, Selection};
+use vectorcraft_geom::corners::{Corner, cut_corners};
+use vectorcraft_geom::shapes::CornerKind;
+use vectorcraft_geom::{Affine, BezPath, PathData, Point, Vec2};
 
 use crate::{Action, Overlay, PointerEvent, ToolContext};
 
 /// The dialog a double-click on a corner widget opens: `{id, corners}`.
 pub const DIALOG: &str = "corners";
 
-/// Widgets sit at least this far inside their corner (screen px), further in once the radius is.
+/// Widgets sit at least this far inside their corner along each side of a right angle (screen px;
+/// √2 times it along the bisector, at any angle), further in once the radius is.
 const MIN_INSET_PX: f64 = 10.0;
-/// Shapes whose shorter side is smaller than this on screen (px) hide their widgets.
+/// Corners with a side shorter than this on screen (px) hide their widgets.
 const MIN_SIDE_PX: f64 = 3.0 * MIN_INSET_PX;
-/// Corners in radii order (top-left, top-right, bottom-right, bottom-left): position as factors
-/// of (w, h) and the inward diagonal.
-const CORNERS: [((f64, f64), (f64, f64)); 4] =
-    [((0.0, 0.0), (1.0, 1.0)), ((1.0, 0.0), (-1.0, 1.0)), ((1.0, 1.0), (-1.0, -1.0)), ((0.0, 1.0), (1.0, -1.0))];
 
-/// The corner widgets of a selection that is exactly one editable live rectangle.
+/// One corner's widget.
 #[derive(Clone, Copy, Debug, PartialEq)]
+struct Widget {
+    /// The corner, in the corners' own space.
+    corner: Corner,
+    /// The radius it is set to (document units) and its kind.
+    radius: f64,
+    kind: CornerKind,
+    /// The angle between its sides in the document, in degrees.
+    angle: f64,
+    /// The widget's centre in the document.
+    at: Point,
+}
+
+/// The corner widgets of a selection that is exactly one editable path with corners.
+#[derive(Clone, Debug, PartialEq)]
 pub struct CornerWidgets {
     pub id: NodeId,
-    w: f64,
-    h: f64,
-    radii: [f64; 4],
-    kinds: [CornerKind; 4],
+    /// Maps the corners' space into the document (a live rectangle's own; else none).
     xf: Affine,
-    /// Widget centres in document coordinates, in radii order.
-    pub points: [Point; 4],
-    /// The corners whose widgets show (and which a drag edits).
-    pub shown: [bool; 4],
+    /// The widgets that show (and which a drag edits), in anchor order.
+    widgets: Vec<Widget>,
+    /// Every corner of the path shows its widget.
+    all: bool,
 }
 
 impl CornerWidgets {
-    /// The widgets at `zoom` (screen px per document point), if the selection has them.
-    pub fn of(doc: &Document, selection: &Selection, zoom: f64) -> Option<Self> {
+    /// The widgets at `zoom` (screen px per document point), if the selection has them: on any
+    /// path with Direct Selection (`any_path`), on live rectangles and polygons with Selection.
+    pub fn of(doc: &Document, selection: &Selection, zoom: f64, any_path: bool) -> Option<Self> {
         let [id] = selection.objects[..] else { return None };
         if !doc.is_editable(id) {
             return None;
         }
-        let Some(NodeKind::Path { live: Some(live), .. }) = doc.node(id).map(|n| &n.kind) else { return None };
-        // In document units, as `object.setLiveShape` edits them.
-        let LiveShape::Rectangle { w, h, radii, kinds, xf } = live.folded() else { return None };
-        // Screen pixels per shape unit along each side (the shape may be scaled or skewed).
-        let c = xf.as_coeffs();
-        let (px, py) = (Vec2::new(c[0], c[1]).hypot() * zoom, Vec2::new(c[2], c[3]).hypot() * zoom);
-        if xf.determinant().abs() < 1e-12 || (w * px).min(h * py) < MIN_SIDE_PX {
+        let NodeKind::Path { path, live, .. } = &doc.node(id)?.kind else { return None };
+        if !any_path && !matches!(live, Some(LiveShape::Rectangle { .. } | LiveShape::Polygon { .. })) {
             return None;
         }
-        // Each widget sits on its corner's diagonal, at the centre of the arc as drawn.
-        let points = std::array::from_fn(|k| {
-            let ((fx, fy), (sx, sy)) = CORNERS[k];
-            let r = shapes::fitted_corner_radius(w, h, radii[k]);
-            let ix = r.max(MIN_INSET_PX / px).min(w / 2.0);
-            let iy = r.max(MIN_INSET_PX / py).min(h / 2.0);
-            xf * Point::new(fx * w + sx * ix, fy * h + sy * iy)
-        });
-        Some(Self { id, w, h, radii, kinds, xf, points, shown: live.picked_corners(selection.partial(id)) })
+        let lc = LiveCorners::new(path, live.as_ref())?;
+        let det = lc.xf.determinant().abs();
+        if det < 1e-12 {
+            return None;
+        }
+        // Screen pixels per unit of the corners' space (a rectangle's may be skewed).
+        let scale = zoom * det.sqrt();
+        let picked = lc.picked(selection.partial(id));
+        let widgets: Vec<_> = lc
+            .corners
+            .iter()
+            .filter(|c| picked.contains(&c.index) && c.lu.min(c.lv) * scale >= MIN_SIDE_PX)
+            .map(|c| {
+                // On the bisector at the centre of the arc as drawn, at least the minimum inset
+                // in and at most half the shorter side.
+                let radius = lc.radius(c.index);
+                let k = (c.fitted(radius) / c.sin()).max(MIN_INSET_PX / (scale * (1.0 + c.cos()).sqrt())).min(c.lu.min(c.lv) / 2.0);
+                Widget { corner: *c, radius, kind: lc.kind(c.index), angle: angle_in(lc.xf, c), at: lc.xf * c.on_bisector(k) }
+            })
+            .collect();
+        let all = widgets.len() == lc.corners.len();
+        (!widgets.is_empty()).then_some(Self { id, xf: lc.xf, widgets, all })
     }
 
     /// Selection & Anchor Display → Hide Corner Widget for angles greater than: the widgets of
-    /// corners wider than `max` degrees (a skewed rectangle's obtuse ones) hide; none if all do.
+    /// corners wider than `max` degrees (in the document) hide; none if all do.
     pub fn within_angle(mut self, max: f64) -> Option<Self> {
-        // The angle between the shape's sides where they meet at the top-left corner; the
-        // top-right and bottom-left corners are its supplement.
-        let c = self.xf.as_coeffs();
-        let (u, v) = (Vec2::new(c[0], c[1]), Vec2::new(c[2], c[3]));
-        let a = u.cross(v).abs().atan2(u.dot(v)).to_degrees();
-        for (k, shown) in self.shown.iter_mut().enumerate() {
-            let angle = if k % 2 == 0 { a } else { 180.0 - a };
-            *shown &= angle <= max + 1e-9;
-        }
-        self.shown.contains(&true).then_some(self)
+        let n = self.widgets.len();
+        self.widgets.retain(|w| w.angle <= max + 1e-9);
+        self.all &= self.widgets.len() == n;
+        (!self.widgets.is_empty()).then_some(self)
     }
 
-    /// The widgets the active tool can drag (View → Show Corner Widget on).
-    pub fn for_tool(cx: &ToolContext) -> Option<Self> {
-        if !cx.corner_widgets {
+    /// The widgets on show: [`Self::of`] with View → Show Corner Widget `on`, less those wider
+    /// than `max_angle` ([`Self::within_angle`]).
+    pub fn showing(doc: &Document, selection: &Selection, zoom: f64, any_path: bool, on: bool, max_angle: f64) -> Option<Self> {
+        if !on {
             return None;
         }
-        Self::of(cx.doc, cx.selection, cx.zoom)?.within_angle(cx.corner_widget_max_angle)
+        Self::of(doc, selection, zoom, any_path)?.within_angle(max_angle)
+    }
+
+    /// The widgets the active tool can drag (Direct Selection: `any_path`).
+    pub fn for_tool(cx: &ToolContext, any_path: bool) -> Option<Self> {
+        Self::showing(cx.doc, cx.selection, cx.zoom, any_path, cx.corner_widgets, cx.corner_widget_max_angle)
     }
 
     /// The centres of the widgets that show.
     pub fn visible(&self) -> impl Iterator<Item = Point> + '_ {
-        self.points.iter().zip(self.shown).filter(|(_, on)| *on).map(|(p, _)| *p)
+        self.widgets.iter().map(|w| w.at)
     }
 
-    /// The indices of the shown corners.
-    fn shown_indices(&self) -> Value {
-        (0..4).filter(|k| self.shown[*k]).collect()
+    /// The corners whose widgets show: anchor indices of the path's uncut outline.
+    pub fn corners(&self) -> Vec<usize> {
+        self.widgets.iter().map(|w| w.corner.index).collect()
     }
 
-    /// Index of the shown widget nearest to `p` within `tol` (document units).
-    pub fn hit(&self, p: Point, tol: f64) -> Option<usize> {
-        let d = |k: &usize| self.points[*k].distance(p);
-        (0..4).filter(|k| self.shown[*k] && d(k) <= tol).min_by(|a, b| d(a).total_cmp(&d(b)))
+    /// The widget nearest to `p` within `tol` (document units).
+    fn hit(&self, p: Point, tol: f64) -> Option<usize> {
+        let d = |k: &usize| self.widgets.get(*k).map_or(f64::INFINITY, |w| w.at.distance(p));
+        (0..self.widgets.len()).filter(|k| d(k) <= tol).min_by(|a, b| d(a).total_cmp(&d(b)))
     }
 
-    /// `object.setLiveShape` params setting `key` on the shown corners (all four unless some are
-    /// hidden, which `corners` then leaves out).
+    /// `object.setLiveShape` params setting `key` on the shown corners (every corner unless some
+    /// are hidden, which `corners` then leaves out).
     fn command(&self, key: &str, value: Value) -> Value {
         let mut p = Map::new();
         p.insert("id".into(), json!(self.id.0));
         p.insert(key.into(), value);
-        if self.shown != [true; 4] {
-            p.insert("corners".into(), self.shown_indices());
+        if !self.all {
+            p.insert("corners".into(), json!(self.corners()));
         }
         Value::Object(p)
     }
 }
 
-/// Is `p` over a corner widget the active tool would drag?
-pub fn over_widget(cx: &ToolContext, p: Point) -> bool {
-    CornerWidgets::for_tool(cx).and_then(|w| w.hit(p, cx.tol(5.0))).is_some()
+/// The angle between corner `c`'s sides once `xf` maps them into the document, in degrees.
+fn angle_in(xf: Affine, c: &Corner) -> f64 {
+    let [a, b, cc, d, _, _] = xf.as_coeffs();
+    let map = |v: Vec2| Vec2::new(a * v.x + cc * v.y, b * v.x + d * v.y);
+    let (u, v) = (map(c.u), map(c.v));
+    u.cross(v).abs().atan2(u.dot(v)).to_degrees()
+}
+
+/// Is `p` over a corner widget the active tool would drag (Direct Selection: `any_path`)?
+pub fn over_widget(cx: &ToolContext, p: Point, any_path: bool) -> bool {
+    CornerWidgets::for_tool(cx, any_path).and_then(|w| w.hit(p, cx.tol(5.0))).is_some()
 }
 
 /// A double-click on a corner widget opens the Corners dialog for the shown corners.
-pub fn double_click(cx: &ToolContext, p: Point) -> Option<Action> {
-    let w = CornerWidgets::for_tool(cx)?;
+pub fn double_click(cx: &ToolContext, p: Point, any_path: bool) -> Option<Action> {
+    let w = CornerWidgets::for_tool(cx, any_path)?;
     w.hit(p, cx.tol(5.0))?;
-    Some(Action::Dialog(DIALOG.into(), json!({ "id": w.id.0, "corners": w.shown_indices() })))
+    Some(Action::Dialog(DIALOG.into(), json!({ "id": w.id.0, "corners": w.corners() })))
 }
 
-/// Dragging a corner widget: the radius follows the pointer along the corner's diagonal. An
+/// Dragging a corner widget: the radius follows the pointer along the corner's bisector. An
 /// Alt-click (no drag) cycles the corner kind instead.
-#[derive(Clone, Copy, Debug)]
+#[derive(Clone, Debug)]
 pub struct CornerDrag {
     widgets: CornerWidgets,
-    corner: usize,
+    /// The dragged widget (index into the shown ones).
+    widget: usize,
+    /// The path's uncut outline with its corners' radii and kinds (to outline the corners at
+    /// their largest).
+    base: PathData,
+    radii: Vec<f64>,
+    kinds: Vec<CornerKind>,
     start: Point,
     began: bool,
     alt: bool,
     radius: f64,
-    /// The pointer went past fully round: the corners are at their largest radius.
+    /// The pointer went past fully round: the dragged corner is at its largest radius.
     at_limit: bool,
     at: Point,
 }
 
 impl CornerDrag {
-    /// Start a drag when the press `ev` is on a widget of the selection.
-    pub fn hit(cx: &ToolContext, ev: &PointerEvent) -> Option<Self> {
-        let (widgets, p) = (CornerWidgets::for_tool(cx)?, ev.pos);
-        let corner = widgets.hit(p, cx.tol(5.0))?;
-        Some(Self { widgets, corner, start: p, began: false, alt: ev.mods.alt, radius: widgets.radii[corner], at_limit: false, at: p })
+    /// Start a drag when the press `ev` is on a widget of the selection (Direct Selection:
+    /// `any_path`).
+    pub fn hit(cx: &ToolContext, ev: &PointerEvent, any_path: bool) -> Option<Self> {
+        let (widgets, p) = (CornerWidgets::for_tool(cx, any_path)?, ev.pos);
+        let widget = widgets.hit(p, cx.tol(5.0))?;
+        let radius = widgets.widgets.get(widget)?.radius;
+        let lc = cx.doc.node(widgets.id).and_then(LiveCorners::of)?;
+        let n = lc.base.anchor_count();
+        let (radii, kinds) = ((0..n).map(|k| lc.radius(k)).collect(), (0..n).map(|k| lc.kind(k)).collect());
+        let base = lc.base.into_owned();
+        Some(Self { widgets, widget, base, radii, kinds, start: p, began: false, alt: ev.mods.alt, radius, at_limit: false, at: p })
     }
 
     /// The start radius (as drawn: no larger than fits) plus the pointer's travel along the
-    /// corner's inward diagonal (in the shape's own units), from square to fully round; and
-    /// whether the travel reached fully round (the limit every corner shares).
+    /// corner's bisector (in the corners' own units), from square to fully round; and whether the
+    /// travel reached fully round.
     fn radius_at(&self, p: Point) -> (f64, bool) {
-        let w = &self.widgets;
-        let inv = w.xf.inverse();
-        let d = inv * p - inv * self.start;
-        let (_, (sx, sy)) = CORNERS[self.corner];
-        let max = shapes::max_corner_radius(w.w, w.h);
-        let r = shapes::fitted_corner_radius(w.w, w.h, w.radii[self.corner]) + (d.x * sx + d.y * sy) / 2.0;
+        let Some(w) = self.widgets.widgets.get(self.widget) else { return (self.radius, self.at_limit) };
+        let inv = self.widgets.xf.inverse();
+        let max = w.corner.max_radius();
+        let r = w.corner.fitted(w.radius) + w.corner.radius_change(inv * p - inv * self.start);
         (r.min(max).max(0.0), max > 0.0 && r >= max)
     }
 
-    /// The edited corners' cuts at the dragged radius, in document coordinates.
-    fn edited_corners(&self) -> BezPath {
-        let w = &self.widgets;
-        let rect = Rect::new(0.0, 0.0, w.w, w.h);
-        let radii = std::array::from_fn(|k| if w.shown[k] { self.radius } else { w.radii[k] });
-        let corner_of = shapes::rectangle_anchor_corners(rect, radii);
-        let path = shapes::rectangle_with_corners(rect, radii, w.kinds).transformed(w.xf);
+    /// The cuts of the edited corners that are at their largest radius, in document coordinates.
+    fn corners_at_limit(&self) -> BezPath {
+        let shown = &self.widgets.widgets;
+        let mut radii = self.radii.clone();
+        for w in shown {
+            if let Some(r) = radii.get_mut(w.corner.index) {
+                *r = self.radius;
+            }
+        }
+        let full = |k: &usize| shown.iter().any(|w| w.corner.index == *k && self.radius >= w.corner.max_radius() - 1e-9);
+        let (path, sources) = cut_corners(&self.base, &radii, &self.kinds);
         let mut out = BezPath::new();
-        let Some(sp) = path.subpaths.first() else { return out };
-        // A corner's cut is the segment between its two anchors (the last one, for the top-left).
-        for i in 0..sp.segment_count() {
-            let (a, b) = (corner_of.get(i), corner_of.get(i + 1).or(corner_of.first()));
-            if a == b && a.is_some_and(|k| w.shown.get(*k) == Some(&true)) {
-                let c = sp.segment(i);
-                out.move_to(c.p0);
-                out.curve_to(c.p1, c.p2, c.p3);
+        for (sp, from) in path.transformed(self.widgets.xf).subpaths.iter().zip(&sources) {
+            // A corner's cut is the segment between its two anchors (the last one, for a closed
+            // subpath's first corner).
+            for i in 0..sp.segment_count() {
+                let (a, b) = (from.get(i), from.get(i + 1).or(from.first()));
+                if a == b && a.is_some_and(full) {
+                    let c = sp.segment(i);
+                    out.move_to(c.p0);
+                    out.curve_to(c.p1, c.p2, c.p3);
+                }
             }
         }
         out
@@ -205,22 +249,21 @@ impl CornerDrag {
     pub fn finish(self) -> Vec<Action> {
         if self.began {
             vec![Action::Commit]
-        } else if self.alt {
-            let kind = self.widgets.kinds[self.corner].next();
-            vec![Action::Exec("object.setLiveShape".into(), self.widgets.command("kind", json!(kind)))]
+        } else if let Some(w) = self.widgets.widgets.get(self.widget).filter(|_| self.alt) {
+            vec![Action::Exec("object.setLiveShape".into(), self.widgets.command("kind", json!(w.kind.next())))]
         } else {
             vec![]
         }
     }
 
-    /// The radius readout next to the pointer; at the largest radius, the edited corners in red.
+    /// The radius readout next to the pointer; at the largest radius, the corners there in red.
     pub fn overlays(&self, cx: &ToolContext) -> Vec<Overlay> {
         if !self.began {
             return vec![];
         }
         let mut out = vec![Overlay::Measure { p: self.at, text: format!("Radius: {}", cx.len(self.radius)) }];
         if self.at_limit {
-            out.push(Overlay::Path { path: self.edited_corners(), color: crate::builder::HIGHLIGHT_RED, width: 2.0, dashed: false });
+            out.push(Overlay::Path { path: self.corners_at_limit(), color: crate::builder::HIGHLIGHT_RED, width: 2.0, dashed: false });
         }
         out
     }
@@ -240,24 +283,38 @@ mod tests {
 
     /// A live `w` × `h` rectangle with corner radius `r`.
     fn live_shape(w: f64, h: f64, r: f64, xf: Affine) -> (Document, NodeId) {
+        let live = LiveShape::Rectangle { w, h, radii: [r; 4], kinds: Default::default(), xf };
+        doc_with(live.to_path(), Some(live))
+    }
+
+    /// A document holding one path with its live shape.
+    fn doc_with(path: PathData, live: Option<LiveShape>) -> (Document, NodeId) {
         let mut d = Document::new(500.0, 500.0);
         let l = d.layers[0].id;
         let id = d.alloc_id();
-        let live = LiveShape::Rectangle { w, h, radii: [r; 4], kinds: Default::default(), xf };
-        let mut n = Node::path(id, live.to_path(), Appearance::default_art());
+        let mut n = Node::path(id, path, Appearance::default_art());
         if let NodeKind::Path { live: slot, .. } = &mut n.kind {
-            *slot = Some(live);
+            *slot = live;
         }
         d.insert(Some(l), 0, n).unwrap();
         (d, id)
     }
 
+    /// A five-pointed star at (200, 200) of radii 60 and 30 (#511): a plain path.
+    fn star() -> (Document, NodeId) {
+        doc_with(vectorcraft_geom::shapes::star(Point::new(200.0, 200.0), 60.0, 30.0, 5, 0.0), None)
+    }
+
     fn selected(id: NodeId) -> Selection {
         let mut s = Selection::default();
         s.add(id);
         s
     }
 
+    fn pts(w: &CornerWidgets) -> Vec<Point> {
+        w.visible().collect()
+    }
+
     fn radius(a: &Action) -> f64 {
         let Action::Preview(c, v) = a else { panic!("not a preview: {a:?}") };
         assert_eq!(c, "object.setLiveShape");
@@ -267,17 +324,19 @@ mod tests {
     #[test]
     fn widgets_sit_inside_each_corner() {
         let (d, id) = live_rect(0.0, Affine::translate((100.0, 100.0)));
-        let w = CornerWidgets::of(&d, &selected(id), 1.0).unwrap();
-        assert_eq!(w.points, [Point::new(110.0, 110.0), Point::new(190.0, 110.0), Point::new(190.0, 190.0), Point::new(110.0, 190.0)]);
+        let w = CornerWidgets::of(&d, &selected(id), 1.0, false).unwrap();
+        assert_eq!(pts(&w), [Point::new(110.0, 110.0), Point::new(190.0, 110.0), Point::new(190.0, 190.0), Point::new(110.0, 190.0)]);
         // A larger radius moves them to the arc centres; zooming in keeps a 10 px minimum.
         let (d, id) = live_rect(20.0, Affine::translate((100.0, 100.0)));
-        assert_eq!(CornerWidgets::of(&d, &selected(id), 1.0).unwrap().points[0], Point::new(120.0, 120.0));
-        assert_eq!(CornerWidgets::of(&d, &selected(id), 4.0).unwrap().points[2], Point::new(180.0, 180.0));
-        // Hidden when the shape is tiny on screen, for plain paths and without a single selection.
-        assert!(CornerWidgets::of(&d, &selected(id), 0.2).is_none());
+        assert_eq!(pts(&CornerWidgets::of(&d, &selected(id), 1.0, false).unwrap())[0], Point::new(120.0, 120.0));
+        assert_eq!(pts(&CornerWidgets::of(&d, &selected(id), 4.0, false).unwrap())[2], Point::new(180.0, 180.0));
+        // Hidden when the shape is tiny on screen and without a single selection; a plain path
+        // shows them with Direct Selection only.
+        assert!(CornerWidgets::of(&d, &selected(id), 0.2, true).is_none());
         let (d, id) = doc_with_rect();
-        assert!(CornerWidgets::of(&d, &selected(id), 1.0).is_none());
-        assert!(CornerWidgets::of(&d, &Selection::default(), 1.0).is_none());
+        assert!(CornerWidgets::of(&d, &selected(id), 1.0, false).is_none());
+        assert_eq!(pts(&CornerWidgets::of(&d, &selected(id), 1.0, true).unwrap())[0], Point::new(110.0, 110.0));
+        assert!(CornerWidgets::of(&d, &Selection::default(), 1.0, true).is_none());
     }
 
     #[test]
@@ -307,9 +366,9 @@ mod tests {
     fn widgets_of_a_non_square_rectangle_follow_the_drawn_corners() {
         let (d, id) = live_shape(200.0, 80.0, 60.0, Affine::translate((100.0, 100.0)));
         let s = selected(id);
-        let w = CornerWidgets::of(&d, &s, 1.0).unwrap();
+        let w = CornerWidgets::of(&d, &s, 1.0, false).unwrap();
         let centres = [Point::new(140.0, 140.0), Point::new(260.0, 140.0), Point::new(260.0, 140.0), Point::new(140.0, 140.0)];
-        assert_eq!(w.points, centres);
+        assert_eq!(pts(&w), centres);
         let p = paint();
         let cx = cx(&d, &s, &p);
         let red = |t: &dyn crate::Tool| {
@@ -339,8 +398,8 @@ mod tests {
         // 50 × 80 with 10 pt corners stretched 4 × 1: drawn 200 × 80, a mean radius of 20.
         let (d, id) = live_shape(50.0, 80.0, 10.0, Affine::translate((100.0, 100.0)) * Affine::scale_non_uniform(4.0, 1.0));
         let s = selected(id);
-        let w = CornerWidgets::of(&d, &s, 1.0).unwrap();
-        assert_eq!(w.points[0], Point::new(120.0, 120.0));
+        let w = CornerWidgets::of(&d, &s, 1.0, false).unwrap();
+        assert_eq!(pts(&w)[0], Point::new(120.0, 120.0));
         let p = paint();
         let cx = cx(&d, &s, &p);
         let mut t = crate::create("directSelection");
@@ -357,8 +416,8 @@ mod tests {
         let s = selected(id);
         let p = paint();
         let cx = cx(&d, &s, &p);
-        let w = CornerWidgets::of(&d, &s, 1.0).unwrap();
-        assert!(w.points[0].distance(Point::new(190.0, 110.0)) < 1e-9);
+        let w = CornerWidgets::of(&d, &s, 1.0, false).unwrap();
+        assert!(pts(&w)[0].distance(Point::new(190.0, 110.0)) < 1e-9);
         let mut t = crate::create("directSelection");
         t.pointer(&cx, &PointerEvent::new(PointerKind::Down, 190.0, 110.0));
         let a = t.pointer(&cx, &PointerEvent::new(PointerKind::Drag, 180.0, 120.0));
@@ -377,10 +436,10 @@ mod tests {
         s.anchors.insert(id, [(0, 2)].into());
         let p = paint();
         let cx = cx(&d, &s, &p);
-        let w = CornerWidgets::of(&d, &s, 1.0).unwrap();
-        assert_eq!(w.shown, [false, false, true, false]);
-        assert_eq!(w.visible().collect::<Vec<_>>(), [Point::new(190.0, 190.0)]);
-        assert!(!over_widget(&cx, Point::new(110.0, 110.0)), "the other corners hide theirs");
+        let w = CornerWidgets::of(&d, &s, 1.0, true).unwrap();
+        assert_eq!(w.corners(), [2]);
+        assert_eq!(pts(&w), [Point::new(190.0, 190.0)]);
+        assert!(!over_widget(&cx, Point::new(110.0, 110.0), true), "the other corners hide theirs");
         let mut t = crate::create("directSelection");
         assert!(t.pointer(&cx, &PointerEvent::new(PointerKind::Down, 190.0, 191.0)).is_empty());
         let a = t.pointer(&cx, &PointerEvent::new(PointerKind::Drag, 175.0, 176.0));
@@ -423,20 +482,20 @@ mod tests {
     #[test]
     fn corners_wider_than_the_preference_hide_their_widgets() {
         let (d, id) = live_rect(0.0, Affine::translate((100.0, 100.0)));
-        let w = CornerWidgets::of(&d, &selected(id), 1.0).unwrap();
-        assert_eq!(w.within_angle(177.0).map(|w| w.shown), Some([true; 4]));
-        assert_eq!(w.within_angle(90.0).map(|w| w.shown), Some([true; 4]));
+        let w = CornerWidgets::of(&d, &selected(id), 1.0, false).unwrap();
+        assert_eq!(w.clone().within_angle(177.0).map(|w| w.corners()), Some(vec![0, 1, 2, 3]));
+        assert_eq!(w.clone().within_angle(90.0).map(|w| w.corners()), Some(vec![0, 1, 2, 3]));
         assert!(w.within_angle(89.0).is_none());
         // Sheared by 30°: 60° at the top-left and bottom-right corners, 120° at the others.
         let shear = Affine::translate((100.0, 100.0)) * Affine::new([1.0, 0.0, 30f64.to_radians().tan(), 1.0, 0.0, 0.0]);
         let (d, id) = live_rect(0.0, shear);
-        let w = CornerWidgets::of(&d, &selected(id), 1.0).unwrap();
-        assert_eq!(w.within_angle(100.0).map(|w| w.shown), Some([true, false, true, false]));
+        let w = CornerWidgets::of(&d, &selected(id), 1.0, false).unwrap();
+        assert_eq!(w.clone().within_angle(100.0).map(|w| w.corners()), Some(vec![0, 2]));
         let s = selected(id);
         let p = paint();
         let c = ToolContext { corner_widget_max_angle: 100.0, ..cx(&d, &s, &p) };
-        assert!(over_widget(&c, w.points[0]));
-        assert!(!over_widget(&c, w.points[1]), "the 120° corner's widget is hidden");
+        assert!(over_widget(&c, pts(&w)[0], false));
+        assert!(!over_widget(&c, pts(&w)[1], false), "the 120° corner's widget is hidden");
     }
 
     #[test]
@@ -445,14 +504,113 @@ mod tests {
         let s = selected(id);
         let p = paint();
         let mut c = cx(&d, &s, &p);
-        assert!(over_widget(&c, Point::new(110.0, 110.0)));
-        assert!(!over_widget(&c, Point::new(150.0, 150.0)));
+        assert!(over_widget(&c, Point::new(110.0, 110.0), false));
+        assert!(!over_widget(&c, Point::new(150.0, 150.0), false));
         c.corner_widgets = false;
-        assert!(!over_widget(&c, Point::new(110.0, 110.0)));
+        assert!(!over_widget(&c, Point::new(110.0, 110.0), true));
         // With the widgets hidden, a drag from the same spot moves the object.
         let mut t = crate::create("selection");
         t.pointer(&c, &PointerEvent::new(PointerKind::Down, 110.0, 110.0));
         let a = t.pointer(&c, &PointerEvent::new(PointerKind::Drag, 130.0, 110.0));
         assert_eq!(a[0], Action::Begin("Move".into()));
     }
+
+    /// #511: Direct Selection shows a widget in each of a star's ten corners (the Selection tool
+    /// none: a star isn't a live shape); dragging a tip's inward rounds them all as one step.
+    #[test]
+    fn a_star_shows_widgets_in_every_corner_with_direct_selection() {
+        let (d, id) = star();
+        let s = selected(id);
+        let p = paint();
+        let cx = cx(&d, &s, &p);
+        assert!(CornerWidgets::for_tool(&cx, false).is_none());
+        let w = CornerWidgets::for_tool(&cx, true).expect("widgets on a plain path");
+        assert_eq!(w.corners(), (0..10).collect::<Vec<_>>());
+        // The top tip's widget sits 10 √2 px down its bisector.
+        let tip = pts(&w)[0];
+        assert!(tip.distance(Point::new(200.0, 140.0 + 10.0 * 2f64.sqrt())) < 1e-9, "{tip:?}");
+        let mut t = crate::create("directSelection");
+        assert!(t.pointer(&cx, &PointerEvent::new(PointerKind::Down, tip.x, tip.y)).is_empty());
+        let a = t.pointer(&cx, &PointerEvent::new(PointerKind::Drag, tip.x, tip.y + 10.0));
+        assert_eq!(a[0], Action::Begin("Corner Radius".into()));
+        // The widget follows the arc's centre: r / sin(angle / 2) along the bisector.
+        let c = vectorcraft_geom::corners::path_corners(&star_path(&d, id))[0];
+        let want = 10.0 * (c.angle_deg() / 2.0).to_radians().sin();
+        assert!((radius(&a[1]) - want).abs() < 1e-9, "{a:?} want {want}");
+        assert!(matches!(&a[1], Action::Preview(_, v) if v["id"] == id.0 && v.get("corners").is_none()), "every corner: {a:?}");
+        // The Selection tool drags the star instead.
+        let mut t = crate::create("selection");
+        t.pointer(&cx, &PointerEvent::new(PointerKind::Down, tip.x, tip.y));
+        assert_eq!(t.pointer(&cx, &PointerEvent::new(PointerKind::Drag, tip.x, tip.y + 10.0))[0], Action::Begin("Move".into()));
+    }
+
+    fn star_path(d: &Document, id: NodeId) -> PathData {
+        let Some(NodeKind::Path { path, .. }) = d.node(id).map(|n| &n.kind) else { panic!("not a path") };
+        path.clone()
+    }
+
+    /// #511: a Direct-Selected anchor of the star shows its corner's widget alone, and the
+    /// preference hides the star's wide inner corners (about 125°) below their angle.
+    #[test]
+    fn star_corners_follow_the_anchors_and_the_angle_preference() {
+        let (d, id) = star();
+        let mut s = selected(id);
+        s.anchors.insert(id, [(0, 3)].into());
+        let p = paint();
+        let c = cx(&d, &s, &p);
+        let w = CornerWidgets::for_tool(&c, true).unwrap();
+        assert_eq!(w.corners(), [3]);
+        let mut t = crate::create("directSelection");
+        let at = pts(&w)[0];
+        t.pointer(&c, &PointerEvent::new(PointerKind::Down, at.x, at.y));
+        let a = t.pointer(&c, &PointerEvent::new(PointerKind::Drag, at.x + 20.0, at.y + 20.0));
+        assert!(matches!(&a[1], Action::Preview(_, v) if v["corners"] == json!([3])), "{a:?}");
+        // Corners wider than 90° hide: the five tips keep theirs.
+        let whole = selected(id);
+        let c = ToolContext { corner_widget_max_angle: 90.0, ..cx(&d, &whole, &p) };
+        let w = CornerWidgets::for_tool(&c, true).unwrap();
+        assert_eq!(w.corners(), [0, 2, 4, 6, 8]);
+        assert_eq!(w.command("radius", json!(4.0))["corners"], json!([0, 2, 4, 6, 8]));
+    }
+
+    /// A live polygon shows its widgets to the Selection tool too; past every corner's limit the
+    /// cuts outline in red.
+    #[test]
+    fn a_live_polygon_rounds_with_either_tool() {
+        let live = LiveShape::Polygon { radius: 80.0, sides: 6, xf: Affine::translate((200.0, 200.0)), radii: vec![], kinds: vec![] };
+        let (d, id) = doc_with(live.to_path(), Some(live));
+        let s = selected(id);
+        let p = paint();
+        let cx = cx(&d, &s, &p);
+        for any_path in [false, true] {
+            assert_eq!(CornerWidgets::for_tool(&cx, any_path).unwrap().corners().len(), 6);
+        }
+        let tip = pts(&CornerWidgets::for_tool(&cx, false).unwrap())[0];
+        let mut t = crate::create("selection");
+        t.pointer(&cx, &PointerEvent::new(PointerKind::Down, tip.x, tip.y));
+        let a = t.pointer(&cx, &PointerEvent::new(PointerKind::Drag, 200.0, 300.0));
+        // A hexagon's 120° corners stop where their cuts meet: 40 · tan 60°.
+        assert!((radius(&a[1]) - 40.0 * 3f64.sqrt()).abs() < 1e-9, "{a:?}");
+        let red = t
+            .overlays(&cx)
+            .into_iter()
+            .any(|o| matches!(o, Overlay::Path { color, path, .. } if color == crate::builder::HIGHLIGHT_RED && path.elements().len() == 12));
+        assert!(red, "six arcs at the limit");
+    }
+
+    /// A pen path's middle corner shows a widget; its ends and curved corners don't.
+    #[test]
+    fn a_pen_path_shows_widgets_on_its_corners_only() {
+        let mut sp = vectorcraft_geom::SubPath::polyline(
+            &[Point::new(100.0, 100.0), Point::new(300.0, 100.0), Point::new(300.0, 300.0), Point::new(100.0, 300.0)],
+            false,
+        );
+        sp.anchors[2].h_out = Point::new(250.0, 350.0);
+        let (d, id) = doc_with(PathData::single(sp), None);
+        let s = selected(id);
+        let p = paint();
+        let w = CornerWidgets::for_tool(&cx(&d, &s, &p), true).unwrap();
+        assert_eq!(w.corners(), [1]);
+        assert_eq!(pts(&w), [Point::new(290.0, 110.0)]);
+    }
 }

crates/tools/src/direct.rs

@@ -6,7 +6,7 @@
 //! steps), drag a direction handle to
 //! reshape (Shift keeps it at 45° steps round its anchor, Alt moves it alone; smart guides snap
 //! it), marquee to select anchors (Shift-drag toggles them: the selected ones inside are
-//! deselected, the others selected), drag a live rectangle's corner widget to round its corners
+//! deselected, the others selected), drag a corner widget of any path (a star, a pen path) to round its corners
 //! (the selected ones when anchors are selected; Alt-click cycles their kind, double-click opens
 //! the Corners dialog).
 //! Group Selection: click selects the leaf; each further click on it adds the next enclosing group;
@@ -317,13 +317,13 @@ impl Tool for DirectSelectionTool {
                     vec![Action::Exec("select.set".into(), json!({"ids": [target.0]}))]
                 }
             }
-            (PointerKind::DoubleClick, _) if !self.group => corners::double_click(cx, p).into_iter().collect(),
+            (PointerKind::DoubleClick, _) if !self.group => corners::double_click(cx, p, true).into_iter().collect(),
             (PointerKind::Move, State::Idle) => {
                 self.hover = if self.group || !cx.highlight_anchors { None } else { hovered_anchor(cx, p) };
                 vec![]
             }
             (PointerKind::Down, _) => {
-                if let Some(c) = CornerDrag::hit(cx, ev) {
+                if let Some(c) = CornerDrag::hit(cx, ev, true) {
                     self.state = State::Corner(c);
                     return vec![];
                 }
@@ -597,7 +597,7 @@ impl Tool for DirectSelectionTool {
         }
     }
     fn cursor(&self, cx: &ToolContext, p: Point, _m: Mods) -> Cursor {
-        if !self.group && (matches!(self.state, State::Corner(_)) || over_widget(cx, p)) {
+        if !self.group && (matches!(self.state, State::Corner(_)) || over_widget(cx, p, true)) {
             return Cursor::CornerRadius;
         }
         if !self.group && (matches!(self.state, State::Bracket(_)) || over_bracket(cx, p)) {

crates/tools/src/lib.rs

@@ -234,7 +234,7 @@ pub struct ToolContext<'a> {
     /// points (a transform's reference point) land on anchors and ruler guides within
     /// [`Self::snap_tolerance`].
     pub snap_to_point: bool,
-    /// View → Show Corner Widget: live rectangles show draggable Live Corners widgets.
+    /// View → Show Corner Widget: paths show draggable Live Corners widgets in their corners.
     pub corner_widgets: bool,
     /// Which paint proxy is in front (true = Fill): the one the Gradient tool edits.
     pub fill_active: bool,

crates/tools/src/select.rs

@@ -2,7 +2,7 @@
 //! the selected ones are deselected, the others selected), move (Alt copies, Shift constrains; Smart
 //! Guides, or with them off View › Snap to Point, snap it),
 //! bounding-box scale (Shift proportional, Alt from centre) and rotate (outside corners, Shift 45°),
-//! drag a live rectangle's corner widget to round its corners (Alt-click cycles their kind,
+//! drag a live rectangle's or polygon's corner widget to round its corners (Alt-click cycles their kind,
 //! double-click opens the Corners dialog), double-click to enter isolation mode (Double Click To
 //! Isolate), Cmd/Ctrl-click to select the object behind (Command Click to Select Objects Behind),
 //! click or drag a ruler guide ([`crate::rulerguide`]), drag the brackets of type on a path
@@ -169,7 +169,7 @@ impl Tool for SelectionTool {
         match (ev.kind, self.state.clone()) {
             (PointerKind::DoubleClick, _) => {
                 self.state = State::Idle;
-                if let Some(a) = corners::double_click(cx, p).or_else(|| typewidget::double_click(cx, p)) {
+                if let Some(a) = corners::double_click(cx, p, false).or_else(|| typewidget::double_click(cx, p)) {
                     return vec![a];
                 }
                 if crate::rulerguide::guide_at(cx, p).is_some() {
@@ -191,7 +191,7 @@ impl Tool for SelectionTool {
             (PointerKind::Down, _) => {
                 // 1. Live Corners widgets and type on a path's brackets, then the bounding-box
                 // handles of the current selection.
-                if let Some(c) = CornerDrag::hit(cx, ev) {
+                if let Some(c) = CornerDrag::hit(cx, ev, false) {
                     self.state = State::Corner(c);
                     return vec![];
                 }
@@ -381,7 +381,7 @@ impl Tool for SelectionTool {
         if self.guide.busy() {
             return self.guide.cursor(cx, p).unwrap_or_default();
         }
-        if over_widget(cx, p) {
+        if over_widget(cx, p, false) {
             return Cursor::CornerRadius;
         }
         if over_bracket(cx, p) {

crates/ui-egui/src/canvas.rs

@@ -1419,11 +1419,16 @@ fn selection_overlay(app: &mut VectorcraftApp, p: &egui::Painter, xf: &Xf) {
             anchor_square(p, xf.to_screen(a.p), color, false, anchor(direct));
         }
     }
-    // Live Corners widgets (Selection / Direct Selection on a single live rectangle).
+    // Live Corners widgets (Selection on a live rectangle or polygon, Direct Selection on any path).
     if matches!(tool, "selection" | "directSelection")
-        && app.ui.view.corner_widgets
-        && let Some(w) = vectorcraft_tools::corners::CornerWidgets::of(&st.doc, &st.selection, xf.zoom)
-            .and_then(|w| w.within_angle(app.session.prefs.hide_corner_widget_above))
+        && let Some(w) = vectorcraft_tools::corners::CornerWidgets::showing(
+            &st.doc,
+            &st.selection,
+            xf.zoom,
+            tool == "directSelection",
+            app.ui.view.corner_widgets,
+            app.session.prefs.hide_corner_widget_above,
+        )
     {
         let color = c32(st.doc.layer_color(w.id));
         for sp in w.visible().map(|q| xf.to_screen(q)) {
@@ -2249,6 +2254,25 @@ mod tests {
         assert_eq!(widgets(&mut app), 0);
     }
 
+    /// #511: a star shows a widget in each of its ten corners with Direct Selection (none with
+    /// the Selection tool: it isn't a live shape); View → Hide Corner Widget hides them.
+    #[test]
+    fn a_star_shows_corner_widgets_with_direct_selection() {
+        let mut app = VectorcraftApp::new(Session::new(), Default::default());
+        app.session.execute("file.new", &json!({"width": 400, "height": 300})).unwrap();
+        app.session.execute("shape.star", &json!({"cx": 200, "cy": 150, "radius1": 80, "radius2": 40})).unwrap();
+        let ctx = egui::Context::default();
+        let widgets = |app: &mut VectorcraftApp| {
+            shapes(app, &ctx).iter().filter(|s| matches!(s, Shape::Circle(c) if c.radius == 3.0 && c.fill == Color32::WHITE)).count()
+        };
+        app.select_tool("selection");
+        assert_eq!(widgets(&mut app), 0);
+        app.select_tool("directSelection");
+        assert_eq!(widgets(&mut app), 10);
+        app.run("view.cornerWidget", json!({})).unwrap();
+        assert_eq!(widgets(&mut app), 0);
+    }
+
     /// The type widget beside selected type's bounding box: hollow (white) on point type, filled
     /// in the layer colour on area type.
     #[test]

crates/ui-egui/src/dialogs/corners.rs

@@ -1,13 +1,16 @@
 //! Corners: double-clicking a Live Corners widget (or `ui.corners`) sets the corner kind and
-//! radius of a live rectangle's corners: those whose widgets show (the Direct-Selected ones, else
-//! all four). OK runs `object.setLiveShape` for them, one undo step.
+//! radius of a path's corners: those whose widgets show (the Direct-Selected ones, else every
+//! corner). OK runs `object.setLiveShape` for them, one undo step.
 //!
-//! Fields: `id`, `corners` (indices 0–3: top-left, top-right, bottom-right, bottom-left), `kind`
-//! ("round", "invertedRound" or "chamfer") and `radius` (pt). While the corners differ in kind
-//! or radius, that field is absent and OK leaves it as it is.
+//! Fields: `id`, `corners` (anchor indices of the path with its corners uncut: a rectangle's 0–3
+//! from the top-left clockwise), `kind` ("round", "invertedRound" or "chamfer") and `radius`
+//! (pt). While the corners differ in kind or radius, that field is absent and OK leaves it as it
+//! is.
+
+use std::collections::BTreeSet;
 
 use serde_json::{Value, json};
-use vectorcraft_engine::doc::{NodeId, NodeKind};
+use vectorcraft_engine::doc::{LiveCorners, NodeId};
 use vectorcraft_geom::shapes::CornerKind;
 
 use super::swatch_options::{grid, label};
@@ -20,7 +23,7 @@ pub const KIND: &str = vectorcraft_tools::corners::DIALOG;
 
 pub(super) const SPEC: DialogSpec = DialogSpec { heading: |_| tl!("Corners").into(), body, confirm, min_width: 280.0, ..DialogSpec::FORM };
 
-/// Open Corners for `{id?, corners?}`: the live rectangle `id` (default: the selected one) and its
+/// Open Corners for `{id?, corners?}`: the path `id` (default: the selected one) and its
 /// `corners` (default: the selected corners), filled in with their kind and radius.
 pub fn open(app: &mut VectorcraftApp, p: &Value) -> Result<Value, String> {
     let st = app.session.active().ok_or("no document")?;
@@ -28,22 +31,21 @@ pub fn open(app: &mut VectorcraftApp, p: &Value) -> Result<Value, String> {
         Some(id) => NodeId(id),
         None => match st.selection.objects[..] {
             [id] => id,
-            _ => return Err("select one live rectangle, or give its `id`".into()),
+            _ => return Err("select one path, or give its `id`".into()),
         },
     };
-    let Some(NodeKind::Path { live: Some(live @ vectorcraft_engine::doc::LiveShape::Rectangle { .. }), .. }) = st.doc.node(id).map(|n| &n.kind)
-    else {
-        return Err("Corners edits a live rectangle".into());
+    let Some(live) = st.doc.node(id).and_then(LiveCorners::of) else {
+        return Err("Corners edits the corners of a path".into());
     };
-    let corners = match p.get("corners").and_then(Value::as_array) {
-        Some(a) => std::array::from_fn(|k| a.iter().any(|v| v.as_u64() == Some(k as u64))),
-        None => live.picked_corners(st.selection.partial(id)),
+    let corners: BTreeSet<usize> = match p.get("corners").and_then(Value::as_array) {
+        Some(a) => a.iter().filter_map(Value::as_u64).filter_map(|k| usize::try_from(k).ok()).filter(|k| live.corner(*k).is_some()).collect(),
+        None => live.picked(st.selection.partial(id)),
     };
-    if !corners.contains(&true) {
-        return Err("give corners 0–3".into());
+    if corners.is_empty() {
+        return Err("give corners of the path: anchors between two straight sides".into());
     }
-    let (radius, kind) = live.corner_style(corners);
-    let mut fields = json!({"id": id.0, "corners": (0..4).filter(|k| corners[*k]).collect::<Vec<_>>()});
+    let (radius, kind) = live.style(&corners);
+    let mut fields = json!({"id": id.0, "corners": corners});
     if let Some(k) = kind {
         fields["kind"] = json!(k);
     }
@@ -94,7 +96,7 @@ fn confirm(app: &mut VectorcraftApp, d: &Dialog) -> Result<Value, String> {
 mod tests {
     use super::*;
     use vectorcraft_engine::Session;
-    use vectorcraft_engine::doc::LiveShape;
+    use vectorcraft_engine::doc::{LiveShape, NodeKind};
     use vectorcraft_tools::corners::CornerWidgets;
     use vectorcraft_tools::{PointerEvent, PointerKind};
 
@@ -121,7 +123,7 @@ mod tests {
     /// The radius the Properties panel's Corner Radius field shows.
     fn panel_radius(app: &VectorcraftApp, id: u64) -> Option<f64> {
         let n = app.session.active().unwrap().doc.node(NodeId(id)).unwrap().clone();
-        crate::panels::corner_radius(app, &n, &live(app, id))
+        crate::panels::corner_radius(app, &n)
     }
 
     fn shown(app: &mut VectorcraftApp) -> Vec<String> {
@@ -151,7 +153,7 @@ mod tests {
         app.select_tool("directSelection");
         let view = app.view_info();
         let st = app.session.active().unwrap();
-        let at: Vec<_> = CornerWidgets::of(&st.doc, &st.selection, view.zoom).unwrap().visible().collect();
+        let at: Vec<_> = CornerWidgets::of(&st.doc, &st.selection, view.zoom, true).unwrap().visible().collect();
         assert_eq!(at.len(), 1);
         crate::canvas::dispatch(&mut app, &PointerEvent::new(PointerKind::DoubleClick, at[0].x, at[0].y), view);
         let d = app.ui.dialog.clone().expect("Corners opened");
@@ -192,8 +194,25 @@ mod tests {
         app.ui.dialog.as_mut().unwrap().fields.insert("radius".into(), json!("3 pt"));
         super::super::confirm(&mut app).unwrap();
         assert_eq!(corners(&app, id), ([3.0; 4], before.1));
-        // Only live rectangles, and only real corners.
+        // Only paths, and only real corners.
         assert!(app.run("ui.corners", json!({"id": 9999})).is_err());
         assert!(app.run("ui.corners", json!({"corners": [7]})).is_err());
+        let e = app.run("shape.ellipse", json!({"x": 0, "y": 0, "width": 50, "height": 30})).unwrap()["id"].as_u64().unwrap();
+        assert!(app.run("ui.corners", json!({"id": e})).is_err());
+    }
+
+    /// #511: a star's corners open in Corners too; OK rounds them.
+    #[test]
+    fn corners_edits_a_star() {
+        let (mut app, _) = app();
+        let id = app.run("shape.star", json!({"cx": 200, "cy": 200, "radius1": 60, "radius2": 30})).unwrap()["id"].as_u64().unwrap();
+        app.run("ui.corners", json!({})).unwrap();
+        let d = app.ui.dialog.as_mut().unwrap();
+        assert_eq!((d.fields.get("corners"), d.f64("radius", -1.0)), (Some(&json!((0..10).collect::<Vec<_>>())), 0.0));
+        d.fields.insert("radius".into(), json!(5));
+        super::super::confirm(&mut app).unwrap();
+        let n = app.session.active().unwrap().doc.node(NodeId(id)).unwrap().clone();
+        assert!(matches!(&n.kind, NodeKind::Path { live: Some(LiveShape::Path { .. }), .. }));
+        assert_eq!(crate::panels::corner_radius(&app, &n), Some(5.0));
     }
 }

crates/ui-egui/src/i18n/es.tsv

@@ -3072,7 +3072,7 @@ axis	Both	Ambos
 @msg	Convert to Artboards: selection has no area	Convertir en mesas de trabajo: la selección no tiene área
 @msg	Convert to Shape: no rectangles or ellipses selected	Convertir en forma: no hay rectángulos ni elipses seleccionados
 @msg	Copied the CSS of {rules} object(s)	Se copió el CSS de {rules} objeto(s)
-@msg	Corners edits a live rectangle	Esquinas edita un rectángulo interactivo
+@msg	Corners edits the corners of a path	Esquinas edita las esquinas de un trazado
 @msg	Couldn't copy to the system clipboard: {e}	No se pudo copiar al portapapeles del sistema: {e}
 @msg	Couldn't open {name}: {e}	No se pudo abrir {name}: {e}
 @msg	Couldn't place {name}: the document it was dropped on was closed	No se pudo colocar {name}: se cerró el documento en el que se soltó
@@ -3238,7 +3238,7 @@ axis	Both	Ambos
 @msg	file version {0} is newer than this VectorCraft supports ({VERSION})	la versión de archivo {0} es más reciente de lo que admite este VectorCraft ({VERSION})
 @msg	freeform point {i} needs `at` [x, y] and a valid `color`	el punto de forma libre {i} necesita `at` [x, y] y un `color` válido
 @msg	give both `start` and `end` [x, y] (or neither)	indica `start` y `end` [x, y] (o ninguno de los dos)
-@msg	give corners 0–3	indica esquinas de 0 a 3
+@msg	give corners of the path: anchors between two straight sides	indica esquinas del trazado: puntos de ancla entre dos lados rectos
 @msg	give the path `id` and the width point `index`	indica el `id` del trazado y el `index` del punto de anchura
 @msg	guides are locked	las guías están bloqueadas
 @msg	image '{key}' has no pixels and was left out	la imagen '{key}' no tiene píxeles y se omitió
@@ -3422,7 +3422,7 @@ axis	Both	Ambos
 @msg	select artwork (or a colour group) to recolor	selecciona la ilustración (o un grupo de colores) que quieres volver a colorear
 @msg	select at least two objects	selecciona al menos dos objetos
 @msg	select objects, not layers	selecciona objetos, no capas
-@msg	select one live rectangle, or give its `id`	selecciona un rectángulo interactivo o indica su `id`
+@msg	select one path, or give its `id`	selecciona un trazado o indica su `id`
 @msg	select paths to join	selecciona los trazados que quieres unir
 @msg	select the art to repeat	selecciona la ilustración que quieres repetir
 @msg	select the artwork to warp	selecciona la ilustración que quieres deformar

crates/ui-egui/src/menus.rs

@@ -485,7 +485,7 @@ pub const UI_COMMANDS: &[(&str, &str, &str, &str)] = &[
         "ui.corners",
         "Corners…",
         "",
-        "{id?, corners?: [0..3…]} open Corners for live rectangle `id` (default: the selected one) and its corners (0 top-left, 1 top-right, 2 bottom-right, 3 bottom-left; default: the Direct-Selected corners, else all four) (dialog `corners`: kind (round|invertedRound|chamfer), radius (pt); double-clicking a corner widget opens it too): OK runs object.setLiveShape",
+        "{id?, corners?: [i…]} open Corners for path `id` (default: the selected one) and its corners (anchor indices of the path with its corners uncut, as object.setLiveShape takes them: a rectangle's 0 top-left, 1 top-right, 2 bottom-right, 3 bottom-left; default: the Direct-Selected corners, else every corner) (dialog `corners`: kind (round|invertedRound|chamfer), radius (pt); double-clicking a corner widget opens it too): OK runs object.setLiveShape",
     ),
     (
         "ui.colorGuideLimit",

crates/ui-egui/src/panels/mod.rs

@@ -45,7 +45,7 @@ pub mod transparency;
 use egui::{Rect, Sense, Ui, vec2};
 use serde_json::{Value, json};
 use vectorcraft_color::{BlendMode, Color, Paint};
-use vectorcraft_doc::{LiveShape, Node, StrokeLayer};
+use vectorcraft_doc::{LiveCorners, Node, StrokeLayer};
 use vectorcraft_engine::inspect::StrokeMixed;
 
 use crate::theme::Tokens;
@@ -57,11 +57,24 @@ pub fn first_selected(app: &VectorcraftApp) -> Option<Node> {
     first_node(app).cloned()
 }
 
-/// The radius the Live Corners of live rectangle `n` show in the panels: that of the corners the
-/// panels set (the Direct-Selected ones, else all four), blank when they differ.
-pub(crate) fn corner_radius(app: &VectorcraftApp, n: &Node, live: &LiveShape) -> Option<f64> {
+/// The radius the Live Corners of path `n` show in the panels: that of the corners the panels set
+/// (the Direct-Selected ones, else every corner), blank when they differ.
+pub(crate) fn corner_radius(app: &VectorcraftApp, n: &Node) -> Option<f64> {
     let partial = app.session.active().and_then(|d| d.selection.partial(n.id));
-    live.corner_style(live.picked_corners(partial)).0
+    let corners = LiveCorners::of(n)?;
+    corners.style(&corners.picked(partial)).0
+}
+
+/// The Corner Radius field of a live rectangle's or polygon's properties: [`corner_radius`],
+/// which a new value sets on those corners.
+pub(crate) fn corner_radius_row(app: &mut VectorcraftApp, ui: &mut Ui, n: &Node, id: &str) {
+    let units = app.session.general_unit();
+    ui.horizontal(|ui| {
+        dim_label(ui, tl!("Corner Radius:"));
+        if let Some(r) = crate::widgets::num_field(ui, id, corner_radius(app, n), units, 80.0) {
+            app.run("object.setLiveShape", json!({"radius": r})).ok();
+        }
+    });
 }
 
 /// Number of selected objects.

crates/ui-egui/src/panels/properties.rs

@@ -2,9 +2,9 @@
 
 use egui::Ui;
 use serde_json::json;
-use vectorcraft_doc::{NodeKind, Unit};
+use vectorcraft_doc::{LiveShape, NodeKind, Unit};
 
-use super::{corner_radius, first_selected, pstate, set_pstate};
+use super::{corner_radius_row, first_selected, pstate, set_pstate};
 use crate::theme::Tokens;
 use crate::widgets::{self, dim_label, divider, section_header};
 use crate::{VectorcraftApp, icons};
@@ -93,7 +93,7 @@ pub fn show(app: &mut VectorcraftApp, ui: &mut Ui) {
     divider(ui);
     section_header(ui, tl!("Quick Actions"));
     let is_group = matches!(first.as_ref().map(|n| &n.kind), Some(NodeKind::Group { .. }));
-    let is_live = matches!(first.as_ref().map(|n| &n.kind), Some(NodeKind::Path { live: Some(_), .. }));
+    let is_live = matches!(first.as_ref().map(|n| &n.kind), Some(NodeKind::Path { live: Some(l), .. }) if !matches!(l, LiveShape::Path { .. }));
     let mut actions: Vec<(&str, &str)> = vec![];
     if n_sel > 1 {
         actions.push((tl!("Group"), "object.group"));
@@ -335,14 +335,7 @@ pub fn transform_section(app: &mut VectorcraftApp, ui: &mut Ui) {
         && let NodeKind::Path { live: Some(live), .. } = &n.kind
     {
         match live {
-            vectorcraft_doc::LiveShape::Rectangle { .. } => {
-                ui.horizontal(|ui| {
-                    dim_label(ui, tl!("Corner Radius:"));
-                    if let Some(r) = widgets::num_field(ui, "radius", corner_radius(app, &n, live), units, 80.0) {
-                        app.run("object.setLiveShape", json!({"radius": r})).ok();
-                    }
-                });
-            }
+            vectorcraft_doc::LiveShape::Rectangle { .. } => corner_radius_row(app, ui, &n, "radius"),
             vectorcraft_doc::LiveShape::Polygon { sides, .. } => {
                 ui.horizontal(|ui| {
                     dim_label(ui, tl!("Sides:"));
@@ -350,6 +343,7 @@ pub fn transform_section(app: &mut VectorcraftApp, ui: &mut Ui) {
                         app.run("object.setLiveShape", json!({"sides": s as u32})).ok();
                     }
                 });
+                corner_radius_row(app, ui, &n, "radius");
             }
             _ => {}
         }

crates/ui-egui/src/panels/transform.rs

@@ -7,7 +7,7 @@ use serde_json::json;
 use vectorcraft_doc::NodeKind;
 use vectorcraft_geom::Rect;
 
-use super::{corner_radius, first_selected, pstate, set_pstate};
+use super::{corner_radius_row, first_selected, pstate, set_pstate};
 use crate::theme::Tokens;
 use crate::widgets::{self, menu_item};
 use crate::{VectorcraftApp, icons};
@@ -166,20 +166,16 @@ pub fn show(app: &mut VectorcraftApp, ui: &mut Ui) {
             app.run("object.reflect", json!({"axis": "horizontal", "origin": origin})).ok();
         }
     });
-    // Live shape properties.
+    // Live shape properties (a path with live corners is no shape).
     if let Some(n) = first_selected(app)
         && let NodeKind::Path { live: Some(live), .. } = &n.kind
+        && !matches!(live, vectorcraft_doc::LiveShape::Path { .. })
     {
         widgets::divider(ui);
         match live {
             vectorcraft_doc::LiveShape::Rectangle { .. } => {
                 widgets::subheader(ui, tl!("Rectangle Properties:"));
-                ui.horizontal(|ui| {
-                    widgets::dim_label(ui, tl!("Corner Radius:"));
-                    if let Some(r) = widgets::num_field(ui, "xfp-radius", corner_radius(app, &n, live), units, 80.0) {
-                        app.run("object.setLiveShape", json!({"radius": r})).ok();
-                    }
-                });
+                corner_radius_row(app, ui, &n, "xfp-radius");
             }
             vectorcraft_doc::LiveShape::Polygon { sides, .. } => {
                 widgets::subheader(ui, tl!("Polygon Properties:"));
@@ -189,6 +185,7 @@ pub fn show(app: &mut VectorcraftApp, ui: &mut Ui) {
                         app.run("object.setLiveShape", json!({"sides": s.clamp(3.0, 20.0) as u32})).ok();
                     }
                 });
+                corner_radius_row(app, ui, &n, "xfp-radius");
             }
             _ => {
                 widgets::subheader(ui, tl!("Shape Properties:"));

docs/control-protocol.md

@@ -291,9 +291,9 @@ Width Point Edit: double-clicking a width point with the Width tool, or `ui.widt
 value: true}` then confirm (the Delete button) removes the point with `stroke.widthPoint.remove`.
 
 Corners: double-clicking a Live Corners widget with the Selection or Direct Selection tool, or `ui.corners {id?,
-corners?}`, opens the `corners` dialog for a live rectangle's corners (the Direct-Selected ones, else all four; fields
-`id`, `corners`: indices 0–3 clockwise from the top-left, `kind`: round, invertedRound or chamfer, `radius` in points;
-`kind` or `radius` is absent while the corners differ). `ui.dialog.confirm` runs `object.setLiveShape` with them.
+corners?}`, opens the `corners` dialog for a path's corners (the Direct-Selected ones, else every corner; fields `id`,
+`corners`: anchor indices of the path with its corners uncut, a rectangle's 0–3 clockwise from the top-left, `kind`:
+round, invertedRound or chamfer, `radius` in points; `kind` or `radius` is absent while the corners differ). `ui.dialog.confirm` runs `object.setLiveShape` with them.
 
 Perspective plane options: double-clicking a plane widget of the perspective grid, or `ui.perspectivePlane {plane}`,
 opens the `perspectivePlane` dialog (fields `plane`: left, right or ground; `location`: points along the plane's

docs/mcp.md

@@ -1135,21 +1135,32 @@ size. The journal entry of a scaling command records the `strokes` and `corners`
 
 ## Live Corners
 
-`object.setLiveShape {id?, ids?, radius?, kind?, corners?}` sets the corners of live rectangles (one undo step):
-`radius` (pt) and `kind` (`round`, `invertedRound` or `chamfer`) go to the `corners` given (0 top-left, 1 top-right,
-2 bottom-right, 3 bottom-left), else to the corners holding a Direct-Selected anchor (`select.anchors`), else to all
-four. Each corner keeps its own radius and kind (the shape's `live` in queries has `radii` and, when a corner isn't
-round, `kinds`); a corner with no radius is one anchor, a cut one two, and Direct-Selected corners stay selected as
-that changes. Every corner is a circular arc, whatever the rectangle's proportions: a radius past half the shorter
-side draws at half of it (the same limit for every corner), and a rectangle from a file that kept an uneven scale in its
-transform (elliptical corners) gets circular ones in document units when its corners are next set. With the Selection or
-Direct Selection tool, dragging a corner widget rounds the corners whose widgets show (all four, or the Direct-Selected
-ones), outlining them in red once they reach that limit; Alt-clicking one cycles their kind and double-clicking one
-opens Corners (`ui.corners {id?, corners?}`, dialog `corners`: `kind`, `radius`; OK runs `object.setLiveShape`).
+`object.setLiveShape {id?, ids?, radius?, kind?, corners?}` sets the corners of any path (one undo step): a live
+rectangle's or polygon's, a star's, a pen path's. A corner is an anchor without handles between two straight sides
+(not an open path's ends, not a smooth anchor, not one the sides run straight on through). `radius` (pt) and `kind`
+(`round`, `invertedRound` or `chamfer`) go to the `corners` given, else to the corners holding a Direct-Selected
+anchor (`select.anchors`), else to every corner. `corners` are anchor indices of the path with its corners uncut,
+counting every subpath's anchors in order: a rectangle's 0 top-left, 1 top-right, 2 bottom-right, 3 bottom-left; a
+polygon's from its first vertex clockwise; a star's from its first tip; an index past the last anchor is an error.
+Each corner keeps its own radius and kind (the shape's `live` in queries has `radii` and, when a corner isn't round,
+`kinds`); a corner with no radius is one anchor, a cut one two, and Direct-Selected corners stay selected as that
+changes. A path that isn't a live shape keeps its uncut outline (`live` `{"shape": "path", "base", "radii"}`) so its
+corners stay editable, and is a plain path again once no corner is cut; a polygon stays a live polygon, its corners
+keeping the radius they shared when `sides` changes. Every corner is a circular arc tangent to both sides: a radius
+draws no larger than takes the cut halfway along the corner's shorter side (half the shorter side of a rectangle, the
+same limit for its four corners), the corners stay circular through uneven scales (the radius scales by the mean scale,
+or keeps its size with Scale Corners off), and a rectangle from a file that kept an uneven scale in its transform
+(elliptical corners) gets circular ones in document units when its corners are next set. Dragging a corner widget
+rounds the corners whose widgets show (every corner, or the Direct-Selected ones), outlining in red those that reach
+their limit; the Selection tool shows the widgets of live rectangles and polygons, the Direct Selection tool those of
+any path. Alt-clicking one cycles their kind and double-clicking one opens Corners (`ui.corners {id?, corners?}`,
+dialog `corners`: `kind`, `radius`; OK runs `object.setLiveShape`).
 
 ```json
 {"name":"run_command","arguments":{"command":"object.setLiveShape","params":{"id":12,"corners":[1],"radius":16}}}
 {"name":"run_command","arguments":{"command":"object.setLiveShape","params":{"id":12,"corners":[0,3],"radius":8,"kind":"chamfer"}}}
+{"name":"run_command","arguments":{"command":"shape.star","params":{"cx":200,"cy":200,"radius1":60,"radius2":30}}}
+{"name":"run_command","arguments":{"command":"object.setLiveShape","params":{"radius":5}}}
 ```
 
 ## Use Preview Bounds