Merge pull request #1246 from storytold/prc-compressed

craft-prc: rebuild compressed tessellations whose normals are recalculated (#778)

crates/prc/src/bits.rs

@@ -69,7 +69,6 @@ impl<'a> BitReader<'a> {
     }
 
     /// Array elements this reader may still decode.
-    #[cfg(test)]
     pub(crate) fn elements_left(&self) -> usize {
         self.budget.elements
     }

crates/prc/src/tess.rs

@@ -2,12 +2,14 @@
 //! (172) with its `TESS_Face`s (174), `TESS_3D_Wire` (175), `TESS_Markup` (176) and
 //! `TESS_3D_Compressed` (173) [WD 7.3.7, 7.8].
 //!
-//! `TESS_3D_Compressed` is read field by field to its end [WD 7.8.9.7] so the entities after
-//! it decode; its triangles are not rebuilt.
+//! `TESS_3D_Compressed` is read field by field to its end [WD 7.8.9.7]. One whose normals are
+//! recalculated is rebuilt into triangles ([`compressed`]); one that stores its normals, or
+//! whose arrays do not fit the traversal, is not.
 //!
 //! The schema runs after the fields of each concrete type read here (201, 172, 173, 174, 175,
 //! 176, 305), not for the abstract levels (`PRCBase`, `TESS`).
 
+mod compressed;
 mod triangulate;
 
 use crate::arrays;
@@ -49,7 +51,8 @@ pub enum Tessellation {
     Wire(Vec<Vec<[f64; 3]>>),
     /// A `TESS_Markup` (PMI drawing), read past but not decoded.
     Markup,
-    /// A `TESS_3D_Compressed` mesh, read past but not rebuilt.
+    /// A `TESS_3D_Compressed` mesh, read past but not rebuilt: it stores its normals, or its
+    /// arrays do not fit the traversal. One that is rebuilt is a [`Tessellation::Mesh`].
     #[non_exhaustive]
     Compressed {
         /// The entity's triangle count.
@@ -65,7 +68,8 @@ pub struct TriangleMesh {
     /// The coordinate array as points.
     pub positions: Vec<[f64; 3]>,
     /// Counter-clockwise triangles seen from the outside, as indices into
-    /// [`Self::positions`] [WD 7.8.5.2].
+    /// [`Self::positions`] [WD 7.8.5.2]. A rebuilt `TESS_3D_Compressed` keeps its traversal's
+    /// winding, which is not oriented.
     pub triangles: Vec<[u32; 3]>,
     /// `triangles` split by `TESS_Face`: face `i` is `triangles[faces[i].clone()]`.
     pub faces: Vec<std::ops::Range<usize>>,
@@ -136,7 +140,7 @@ impl Ctx<'_, '_> {
                     self.markup()?;
                     Tessellation::Markup
                 }
-                TESS_3D_COMPRESSED => Tessellation::Compressed { triangles: self.tess_3d_compressed()? },
+                TESS_3D_COMPRESSED => self.tess_3d_compressed()?,
                 t => return Err(malformed(format!("entity type {t} in the tessellation array"))),
             });
         }
@@ -295,23 +299,25 @@ impl Ctx<'_, '_> {
         self.schema.skip_added_fields(TESS_MARKUP, &mut self.r)
     }
 
-    /// `TESS_3D_Compressed` (173) after its type code [WD 7.8.9.7], read past; returns its
-    /// triangle count. prc-rs: the field order and version gates follow its schema.
+    /// `TESS_3D_Compressed` (173) after its type code [WD 7.8.9.7]: rebuilt into a mesh when
+    /// its normals are recalculated and its arrays fit, else read past to its end. prc-rs: the
+    /// field order and version gates follow its schema.
     ///
     /// Sizes the WD leaves unstated: the face count is the largest face index + 1,
     /// `normal_is_reversed` is one bit per triangle, and the point-reference array's
     /// compressed flag is implicit (more than three references).
-    fn tess_3d_compressed(&mut self) -> Result<usize, PrcError> {
+    fn tess_3d_compressed(&mut self) -> Result<Tessellation, PrcError> {
         let r = &mut self.r;
         r.bit()?; // is_calculated
         r.bit()?; // has_faces
-        r.double()?; // tolerance
+        let tolerance = r.double()?;
+        let mut origin = [0.0f64; 3];
         if self.version >= ORIGIN_FROM {
-            for _ in 0..3 {
-                r.float_as_bytes()?;
+            for o in &mut origin {
+                *o = f64::from(r.float_as_bytes()?);
             }
         }
-        arrays::compressed_integer_array(r)?; // point_array
+        let points = arrays::compressed_integer_array(r)?; // point_array
         let edge_status = arrays::character_array(r, 2, None, false)?;
         let face_of = arrays::compressed_indice_array(r, None)?;
         let t = face_of.len();
@@ -320,42 +326,50 @@ impl Ctx<'_, '_> {
         }
         let faces = face_of.iter().max().map_or(0, |&m| (m as usize).saturating_add(1));
         let n = r.unsigned_integer()? as usize;
-        let references = arrays::bool_array(r, n)?.iter().filter(|&&b| b).count();
-        let refs = arrays::compressed_indice_array(r, Some(references > 3))?;
-        if refs.len() != references {
-            return Err(malformed(format!("{} point references for {references} flagged points", refs.len())));
+        let is_reference = arrays::bool_array(r, n)?;
+        let flagged = is_reference.iter().filter(|&&b| b).count();
+        let references = arrays::compressed_indice_array(r, Some(flagged > 3))?;
+        if references.len() != flagged {
+            return Err(malformed(format!("{} point references for {flagged} flagged points", references.len())));
         }
-        self.compressed_normals(t, faces)?;
-        let r = &mut self.r;
-        if r.bit()? {
-            arrays::bool_array(r, faces)?; // is_point_color_on_face
-            arrays::character_array(r, 8, None, false)?; // point_color_array
+        let recalc = self.compressed_normals(t, faces)?;
+        let attributes = self.compressed_attributes(faces)?;
+        if !recalc {
+            return Ok(Tessellation::Compressed { triangles: t });
         }
-        if r.bit()? {
-            arrays::bool_array(r, faces)?; // is_multiple_line_attribute_on_face
-        }
-        arrays::short_array(r, 16)?; // line_attributes
-        if !r.bit()? {
-            self.compressed_texture_parameter()?;
-            if !self.r.bit()? {
-                arrays::bool_array(&mut self.r, faces)?; // face_has_texture
-            }
-        }
-        if self.r.bit()? {
-            arrays::character_array(&mut self.r, 8, None, false)?; // behaviours
-        }
-        self.schema.skip_added_fields(TESS_3D_COMPRESSED, &mut self.r)?;
-        Ok(t)
+        let arrays = compressed::Arrays {
+            tolerance,
+            origin,
+            points: &points,
+            edge_status: &edge_status,
+            triangles: t,
+            is_reference: &is_reference,
+            references: &references,
+            limit: self.r.elements_left(),
+        };
+        // A walk that would outgrow the allowance left stops there, and the arrays stay
+        // compressed, as do arrays the walk does not fit.
+        let Ok(m) = compressed::rebuild(&arrays) else {
+            return Ok(Tessellation::Compressed { triangles: t });
+        };
+        // The rebuilt triangles, their styles and every face range are charged once the arrays
+        // fit and before any of them is made, so arrays that do not fit spend nothing.
+        let styled = attributes.styled(faces);
+        let kept = m.triangles.len().saturating_mul(triangle_elements(false, 0, styled));
+        self.r.take_elements(kept.saturating_add(faces.saturating_mul(RANGE_ELEMENTS)))?;
+        let styles = if styled { attributes.styles(&face_of, faces) } else { Vec::new() };
+        Ok(Tessellation::Mesh(by_face(m, &face_of, faces, &styles)))
     }
 
-    /// A compressed tessellation's normals, from `must_recalculate_normals` on, read past.
-    fn compressed_normals(&mut self, triangles: usize, faces: usize) -> Result<(), PrcError> {
+    /// A compressed tessellation's normals, from `must_recalculate_normals` on, read past;
+    /// returns whether they are recalculated.
+    fn compressed_normals(&mut self, triangles: usize, faces: usize) -> Result<bool, PrcError> {
         let r = &mut self.r;
         if r.bit()? {
             arrays::bool_array(r, triangles)?; // normal_is_reversed
             r.double()?; // crease_angle
             r.character()?; // normal_recalculation_flags
-            return Ok(());
+            return Ok(true);
         }
         let angle_bits = u32::from(r.character()?);
         if angle_bits > MAX_ANGLE_BITS {
@@ -365,7 +379,29 @@ impl Ctx<'_, '_> {
         arrays::bool_array(r, n)?; // is_binary_normal
         arrays::short_array(r, angle_bits)?; // normal angles
         arrays::bool_array(r, faces)?; // is_face_planar
-        Ok(())
+        Ok(false)
+    }
+
+    /// A compressed tessellation's colours, line attributes, texture and behaviours, through
+    /// its added schema fields; returns what its faces' styles are made from.
+    fn compressed_attributes(&mut self, faces: usize) -> Result<FaceAttributes, PrcError> {
+        let r = &mut self.r;
+        if r.bit()? {
+            arrays::bool_array(r, faces)?; // is_point_color_on_face
+            arrays::character_array(r, 8, None, false)?; // point_color_array
+        }
+        // is_multiple_line_attribute_on_face
+        let multi = if r.bit()? { arrays::bool_array(r, faces)? } else { Vec::new() };
+        let line_attributes = arrays::short_array(r, 16)?;
+        if !r.bit()? {
+            self.compressed_texture_parameter()?;
+            if !self.r.bit()? {
+                arrays::bool_array(&mut self.r, faces)?; // face_has_texture
+            }
+        }
+        let behaviours = if self.r.bit()? { arrays::character_array(&mut self.r, 8, None, false)? } else { Vec::new() };
+        self.schema.skip_added_fields(TESS_3D_COMPRESSED, &mut self.r)?;
+        Ok(FaceAttributes { multi, line_attributes, behaviours })
     }
 
     /// `CompressedTextureParameter` [WD 7.8.9.8-7.8.9.9; pdf-issues #729, #749], read past.
@@ -387,6 +423,71 @@ impl Ctx<'_, '_> {
     }
 }
 
+/// Budget elements one face range (`Range<usize>`) takes.
+const RANGE_ELEMENTS: usize = std::mem::size_of::<std::ops::Range<usize>>().div_ceil(8);
+
+/// What a compressed tessellation's faces are styled from [WD 7.8.9.5]: with no face flagged
+/// `is_multiple_line_attribute_on_face` (`multi`), `line_attributes` holds one style index + 1
+/// per face (0 keeps the owner's graphics) and `behaviours` one behaviour byte per face.
+struct FaceAttributes {
+    multi: Vec<bool>,
+    line_attributes: Vec<i32>,
+    behaviours: Vec<i32>,
+}
+
+impl FaceAttributes {
+    /// Whether the faces give their triangles styles. The one-per-triangle layout of a flagged
+    /// face is not established, so such a tessellation keeps its owner's graphics, as does one
+    /// whose attributes and behaviours are all 0.
+    fn styled(&self, faces: usize) -> bool {
+        let behaviours = self.behaviours.len() == faces && self.behaviours.iter().any(|&b| b != 0);
+        !self.multi.iter().any(|&b| b) && self.line_attributes.len() == faces && (behaviours || self.line_attributes.iter().any(|&a| a != 0))
+    }
+
+    /// Per triangle, the style its face gives it; call only when [`Self::styled`].
+    fn styles(&self, face_of: &[u32], faces: usize) -> Vec<TriangleStyle> {
+        let behaviour = |f: usize| {
+            if self.behaviours.len() == faces { self.behaviours.get(f).and_then(|&b| u16::try_from(b).ok()).unwrap_or(0) } else { 0 }
+        };
+        face_of
+            .iter()
+            .map(|&f| {
+                let f = f as usize;
+                TriangleStyle { style: self.line_attributes.get(f).and_then(|&a| u32::try_from(a).ok()).unwrap_or(0), behaviour: behaviour(f) }
+            })
+            .collect()
+    }
+}
+
+/// A rebuilt compressed mesh with its triangles grouped by face, in walk order within each,
+/// so face `i` is `triangles[faces[i].clone()]` as for a `TESS_3D`. `faces` is charged to the
+/// element budget by the caller.
+fn by_face(m: compressed::Rebuilt, face_of: &[u32], faces: usize, styles: &[TriangleStyle]) -> TriangleMesh {
+    let mut order: Vec<usize> = (0..m.triangles.len()).collect();
+    order.sort_by_key(|&i| face_of.get(i).copied().unwrap_or(0));
+    let mut mesh = TriangleMesh { positions: m.positions, normals_recalculated: true, ..TriangleMesh::default() };
+    mesh.faces = Vec::with_capacity(faces);
+    for &i in &order {
+        let f = face_of.get(i).map_or(0, |&f| f as usize);
+        while mesh.faces.len() <= f {
+            let at = mesh.triangles.len();
+            mesh.faces.push(at..at);
+        }
+        if let (Some(&t), Some(r)) = (m.triangles.get(i), mesh.faces.last_mut()) {
+            mesh.triangles.push(t);
+            r.end = mesh.triangles.len();
+        }
+        if let Some(&s) = styles.get(i) {
+            mesh.triangle_styles.push(s);
+        }
+    }
+    while mesh.faces.len() < faces {
+        let at = mesh.triangles.len();
+        mesh.faces.push(at..at);
+    }
+    mesh
+}
+
 /// The normal array and each triangle's corner normals, or both empty when any slot fails to
 /// name a normal: a mesh shades from its stored normals wholly or not at all.
 fn stored_normals(coords: &[f64], slots: &[[u32; 3]], triangles: usize) -> (Vec<[f64; 3]>, Vec<[u32; 3]>) {

crates/prc/src/tess/compressed.rs

@@ -0,0 +1,353 @@
+//! Triangle reconstruction for a `TESS_3D_Compressed` (173) whose normals are recalculated
+//! [WD 7.8.9, 7.8.9.2, 7.8.9.7].
+//!
+//! The WD names the arrays but not the traversal that wrote them; the rules below are the ones
+//! the arrays fit when every slot, reference and point is consumed exactly. They are inferred
+//! from compressed tessellations and are not WD text.
+//!
+//! Traversal. A component starts with a seed triangle of three slots; every later triangle
+//! takes one slot, its apex. A slot flagged in `point_is_reference` takes the next
+//! `point_reference_array` value, a vertex number; any other slot makes the next vertex from
+//! the next `point_array` triple. Triangle `[A B C]`, entered across `[A B]`, reads
+//! `edge_status & 3`: bit 1 continues across `[C B]`, bit 0 across `[A C]`; with both,
+//! `[C B]` is taken and `[A C]` pushed. With neither, pushed edges are popped, skipping any
+//! already shared by two triangles; an empty stack starts the next component. A triangle
+//! whose continuation edge is already shared by two triangles, while the other edge is open,
+//! continues across the open one.
+//!
+//! Geometry, in units of `tolerance`:
+//! - seed: `V0 = origin + d`, `V1 = V0 + d`, `V2 = mid(V0, V1) + d`;
+//! - apex across `[P Q]` with the triangle's third vertex `W` [WD 7.8.9.2]: with
+//!   `O = (P + Q) * 0.5`, `L` and `H` the lower- and higher-numbered of `P`, `Q`,
+//!   `X = unitize(L - H)`, `Z = unitize((W - O) x X)` and `Y = unitize(Z x X)`, the apex is
+//!   `((O + d.x X) - d.y Y) - d.z Z`.
+//!
+//! Arithmetic. The encoder frames each apex on the points it already approximated [WD 7.8.9],
+//! so the decoder repeats the encoder's floating point: a last-bit difference compounds from
+//! apex to apex. `unitize` therefore matches the WD's `PrcPt::Unitize` pseudocode [WD 12.3] as
+//! read, whose length is not exact, and the operations keep the order above.
+//!
+//! Fold. A triangle folded against its parent swaps bits 1 and 0 for its own continuation.
+//! Without stored normals to orient it, a new apex lying in its parent's plane on the parent's
+//! side (`d.z == 0`, `d.y > 0`) is folded. A component that does not fit is rewound and walked
+//! again with the fold inverted at its second triangle, where an apex that is a reference
+//! leaves no other trace of its fold.
+//!
+//! Only the three-statuses-per-triangle `edge_status` form is reconstructed (the first `T`
+//! are read); the one-per-triangle form walks differently.
+
+use std::collections::HashMap;
+
+type V = [f64; 3];
+
+/// The arrays of one `TESS_3D_Compressed` the walk reads.
+pub(super) struct Arrays<'a> {
+    pub(super) tolerance: f64,
+    pub(super) origin: V,
+    pub(super) points: &'a [i64],
+    pub(super) edge_status: &'a [i32],
+    pub(super) triangles: usize,
+    pub(super) is_reference: &'a [bool],
+    pub(super) references: &'a [u32],
+    /// Budget elements the walk may hold at once (see [`Walk::held`]).
+    pub(super) limit: usize,
+}
+
+/// The rebuilt vertices and triangles, in walk order.
+#[derive(Debug)]
+pub(super) struct Rebuilt {
+    pub(super) positions: Vec<V>,
+    pub(super) triangles: Vec<[u32; 3]>,
+}
+
+/// Rebuilds the mesh, or says why the arrays do not fit the traversal.
+pub(super) fn rebuild(a: &Arrays<'_>) -> Result<Rebuilt, &'static str> {
+    if a.edge_status.len() != a.triangles.saturating_mul(3) {
+        return Err("the one-status-per-triangle edge form is not reconstructed");
+    }
+    let mut w = Walk::new(a.triangles, a.limit);
+    w.walk(a)?;
+    if w.slot != a.is_reference.len() || w.ri != a.references.len() || w.pi != a.points.len() {
+        return Err("the decoded triangles do not use up the stored arrays");
+    }
+    Ok(Rebuilt { positions: w.pos, triangles: w.tris })
+}
+
+fn add(a: V, b: V) -> V {
+    [a[0] + b[0], a[1] + b[1], a[2] + b[2]]
+}
+fn sub(a: V, b: V) -> V {
+    [a[0] - b[0], a[1] - b[1], a[2] - b[2]]
+}
+fn scale(a: V, s: f64) -> V {
+    a.map(|c| c * s)
+}
+fn dot(a: V, b: V) -> f64 {
+    a[0] * b[0] + a[1] * b[1] + a[2] * b[2]
+}
+fn cross(a: V, b: V) -> V {
+    [a[1] * b[2] - a[2] * b[1], a[2] * b[0] - a[0] * b[2], a[0] * b[1] - a[1] * b[0]]
+}
+fn key(a: u32, b: u32) -> (u32, u32) {
+    if a < b { (a, b) } else { (b, a) }
+}
+
+/// `PrcPt::Length` [WD 12.3]: the squared length summed in `f64` and rounded to `f32`, then
+/// Newton's square root in `f64`, seeded with that `f32` with its exponent halved. The printed
+/// loop never advances its iterate; it is read as iterating until two successive values are
+/// equal, `None` (the WD's -1) when 100 steps do not settle.
+fn wd_length(a: V) -> Option<f64> {
+    let squared = (a[0] * a[0] + a[1] * a[1] + a[2] * a[2]) as f32;
+    let bits = squared.to_bits();
+    let exponent = (bits >> 23) & 0xff;
+    let seed = if exponent > 127 { f32::from_bits((bits & !(0xff << 23)) | (((exponent - 127) / 2 + 127) << 23)) } else { squared };
+    let squared = f64::from(squared);
+    let mut x = f64::from(seed);
+    for _ in 0..100 {
+        let next = 0.5 * (x + squared / x);
+        if next == x {
+            return Some(next);
+        }
+        x = next;
+    }
+    None
+}
+
+/// `PrcPt::Unitize` [WD 12.3]: each component divided by [`wd_length`]; `None` below
+/// `FLT_EPSILON`.
+fn unitize(a: V) -> Option<V> {
+    let l = wd_length(a).filter(|&l| l >= f64::from(f32::EPSILON))?;
+    Some([a[0] / l, a[1] / l, a[2] / l])
+}
+
+/// The `Z` of `PrcPt::MakeOrthoRep` [WD 12.3] on `X`: `unitize(X x (0, 1, 0))`, or
+/// `X x (1, 0, 0)` when that is null, from `X` unitized again as the pseudocode does.
+fn make_ortho_rep_z(x: V) -> Option<V> {
+    let x = unitize(x)?;
+    unitize(cross(x, [0.0, 1.0, 0.0])).or_else(|| unitize(cross(x, [1.0, 0.0, 0.0])))
+}
+
+/// The apex across `[lo hi]` (`lo` the lower-numbered vertex) opposite `w`, `d` already scaled
+/// by the tolerance [WD 7.8.9.2]. A null `Z` or `Y` takes `MakeOrthoRep(X)`'s `Z` as the WD
+/// prescribes, with `Y = unitize(Z x X)` from the frame's own `X` (not the copy `MakeOrthoRep`
+/// unitized again, whose last bits differ), turned a half-turn about `X` when the residue of
+/// `(W - O) x X` points against that `Z`. A null `X`, an edge the WD's non-degeneracy rule
+/// excludes, is used as it is.
+fn apex(lo: V, hi: V, w: V, d: V) -> V {
+    let o = scale(add(lo, hi), 0.5);
+    let x = sub(lo, hi);
+    let x = unitize(x).unwrap_or(x);
+    let residue = cross(sub(w, o), x);
+    let (y, z) = unitize(residue)
+        .and_then(|z| Some((unitize(cross(z, x))?, z)))
+        .or_else(|| {
+            let z = make_ortho_rep_z(x)?;
+            let y = unitize(cross(z, x))?;
+            Some(if dot(residue, z) < 0.0 { (scale(y, -1.0), scale(z, -1.0)) } else { (y, z) })
+        })
+        .unwrap_or_default();
+    sub(sub(add(o, scale(x, d[0])), scale(y, d[1])), scale(z, d[2]))
+}
+
+/// Traversal state; `log` records every edge-count increment so a component can be rewound.
+/// Everything it holds is checked against [`Arrays::limit`] after each triangle.
+#[derive(Default)]
+struct Walk {
+    pos: Vec<V>,
+    tris: Vec<[u32; 3]>,
+    edges: HashMap<(u32, u32), u8>,
+    log: Vec<(u32, u32)>,
+    stack: Vec<(u32, u32, u32)>,
+    next: Option<(u32, u32, u32)>,
+    slot: usize,
+    ri: usize,
+    pi: usize,
+}
+
+/// Where a component starts, to rewind to.
+#[derive(Clone, Copy)]
+struct Mark {
+    tri: usize,
+    pos: usize,
+    log: usize,
+    slot: usize,
+    ri: usize,
+    pi: usize,
+}
+
+impl Walk {
+    /// Reserves room for `t` triangles, no more than `limit` elements' worth (two each).
+    fn new(t: usize, limit: usize) -> Self {
+        Walk { tris: Vec::with_capacity(t.min(limit / 2).min(1 << 20)), ..Walk::default() }
+    }
+
+    fn mark(&self, tri: usize) -> Mark {
+        Mark { tri, pos: self.pos.len(), log: self.log.len(), slot: self.slot, ri: self.ri, pi: self.pi }
+    }
+
+    fn rewind(&mut self, m: Mark) {
+        for k in self.log.drain(m.log..) {
+            if let Some(n) = self.edges.get_mut(&k) {
+                *n = n.saturating_sub(1);
+            }
+        }
+        self.pos.truncate(m.pos);
+        self.tris.truncate(m.tri);
+        self.stack.clear();
+        self.next = None;
+        (self.slot, self.ri, self.pi) = (m.slot, m.ri, m.pi);
+    }
+
+    /// One slot: `Some(v)` a reference to vertex `v`, `None` a new point.
+    fn take(&mut self, a: &Arrays<'_>) -> Result<Option<u32>, &'static str> {
+        let flagged = *a.is_reference.get(self.slot).ok_or("slots run out")?;
+        self.slot = self.slot.saturating_add(1);
+        if !flagged {
+            return Ok(None);
+        }
+        let v = *a.references.get(self.ri).ok_or("references run out")?;
+        self.ri = self.ri.saturating_add(1);
+        self.get(v)?;
+        Ok(Some(v))
+    }
+
+    fn point(&mut self, a: &Arrays<'_>) -> Result<V, &'static str> {
+        let Some(&[x, y, z]) = a.points.get(self.pi..self.pi.saturating_add(3)) else {
+            return Err("points run out");
+        };
+        self.pi = self.pi.saturating_add(3);
+        let tol = a.tolerance;
+        Ok([x as f64 * tol, y as f64 * tol, z as f64 * tol])
+    }
+
+    fn get(&self, v: u32) -> Result<V, &'static str> {
+        self.pos.get(v as usize).copied().ok_or("a triangle refers to a vertex not yet decoded")
+    }
+
+    fn push(&mut self, p: V) -> Result<u32, &'static str> {
+        let id = u32::try_from(self.pos.len()).map_err(|_| "too many vertices")?;
+        self.pos.push(p);
+        Ok(id)
+    }
+
+    /// The next triangle and its default fold: the seed, or the apex across the edge to
+    /// continue from.
+    fn triangle(&mut self, a: &Arrays<'_>) -> Result<([u32; 3], bool), &'static str> {
+        if let Some((p, q, w)) = self.next.take() {
+            if let Some(r) = self.take(a)? {
+                return Ok(([p, q, r], false));
+            }
+            let (pp, qq, ww) = (self.get(p)?, self.get(q)?, self.get(w)?);
+            let (lo, hi) = if p < q { (pp, qq) } else { (qq, pp) };
+            let fold = matches!(a.points.get(self.pi..self.pi.saturating_add(3)), Some(&[_, dy, 0]) if dy > 0);
+            let d = self.point(a)?;
+            let r = self.push(apex(lo, hi, ww, d))?;
+            return Ok(([p, q, r], fold));
+        }
+        let mut v = [0u32; 3];
+        for k in 0..3 {
+            let id = match self.take(a)? {
+                Some(r) => r,
+                None => {
+                    let d = self.point(a)?;
+                    let p = match k {
+                        0 => add(a.origin, d),
+                        1 => add(self.get(v[0])?, d),
+                        _ => add(scale(add(self.get(v[0])?, self.get(v[1])?), 0.5), d),
+                    };
+                    self.push(p)?
+                }
+            };
+            if let Some(s) = v.get_mut(k) {
+                *s = id;
+            }
+        }
+        Ok((v, false))
+    }
+
+    /// Budget elements (8 bytes each) the walk's arrays hold, by length: growth can add up to
+    /// as much again in spare capacity.
+    fn held(&self) -> usize {
+        let pos = self.pos.len().saturating_mul(3);
+        let tris = self.tris.len().saturating_mul(2);
+        let edges = self.edges.len().saturating_mul(2);
+        let stack = self.stack.len().saturating_mul(2);
+        pos.saturating_add(tris).saturating_add(edges).saturating_add(self.log.len()).saturating_add(stack)
+    }
+
+    fn closed(&self, e: (u32, u32, u32)) -> bool {
+        self.edges.get(&key(e.0, e.1)).copied().unwrap_or(0) >= 2
+    }
+
+    /// Adds one triangle; `flip` inverts its fold.
+    fn step(&mut self, a: &Arrays<'_>, status: i32, flip: bool) -> Result<(), &'static str> {
+        let ([ta, tb, tc], fold) = self.triangle(a)?;
+        for (x, y) in [(ta, tb), (tb, tc), (tc, ta)] {
+            let n = self.edges.entry(key(x, y)).or_insert(0);
+            *n = n.saturating_add(1);
+            let shared = *n;
+            self.log.push(key(x, y));
+            if shared > 2 {
+                return Err("an edge is shared by more than two triangles");
+            }
+        }
+        self.tris.push([ta, tb, tc]);
+        if self.held() > a.limit {
+            return Err("the rebuild needs more than the decoded-element allowance left");
+        }
+        let (mut left, mut right) = ((tc, tb, ta), (ta, tc, tb));
+        if fold != flip {
+            std::mem::swap(&mut left, &mut right);
+        }
+        let (taken, other) = if status & 2 != 0 { (left, right) } else { (right, left) };
+        if status & 3 != 0 && self.closed(taken) && !self.closed(other) {
+            std::mem::swap(&mut left, &mut right);
+        }
+        match (status & 2 != 0, status & 1 != 0) {
+            (true, true) => {
+                self.next = Some(left);
+                self.stack.push(right);
+            }
+            (true, false) => self.next = Some(left),
+            (false, true) => self.next = Some(right),
+            (false, false) => {}
+        }
+        while self.next.is_none() {
+            let Some(e) = self.stack.pop() else { break };
+            if !self.closed(e) {
+                self.next = Some(e);
+            }
+        }
+        Ok(())
+    }
+
+    /// Walks every component. One that fails is rewound and walked once more with the fold
+    /// inverted at its second triangle, so the walk is at most twice the triangle count.
+    fn walk(&mut self, a: &Arrays<'_>) -> Result<(), &'static str> {
+        let mut start = self.mark(0);
+        let mut flip_at = None;
+        let mut i = 0;
+        while i < a.triangles {
+            if self.next.is_none() && !(flip_at.is_some() && i == start.tri) {
+                start = self.mark(i);
+                flip_at = None;
+            }
+            let status = a.edge_status.get(i).copied().unwrap_or(0);
+            match self.step(a, status, flip_at == Some(i)) {
+                Ok(()) => i = i.saturating_add(1),
+                Err(e) => {
+                    if flip_at.is_some() || i == start.tri {
+                        return Err(e);
+                    }
+                    self.rewind(start);
+                    flip_at = Some(start.tri.saturating_add(1));
+                    i = start.tri;
+                }
+            }
+        }
+        Ok(())
+    }
+}
+
+#[cfg(test)]
+mod tests;

crates/prc/src/tess/compressed/tests.rs

@@ -0,0 +1,299 @@
+#![allow(clippy::unwrap_used, clippy::expect_used, clippy::panic)]
+
+use super::*;
+
+const S: f64 = std::f64::consts::SQRT_2;
+
+fn run(points: &[i64], status: &[i32], t: usize, is_ref: &[bool], refs: &[u32]) -> Result<Rebuilt, &'static str> {
+    rebuild(&Arrays {
+        tolerance: 0.5,
+        origin: [10.0, 0.0, 0.0],
+        points,
+        edge_status: status,
+        triangles: t,
+        is_reference: is_ref,
+        references: refs,
+        limit: usize::MAX,
+    })
+}
+
+fn close(a: V, b: V) -> bool {
+    (0..3).all(|i| (a[i] - b[i]).abs() < 1e-9)
+}
+
+/// Seed `[0 1 2]`, then the apex across `[2 1]`: `X` runs from vertex 2 to vertex 1, and
+/// each of `d`'s three components lands on its own axis with its own sign. Tolerance 0.5
+/// halves every step.
+#[test]
+fn a_seed_and_one_apex_rebuild() {
+    let pts = [0, 0, 0, 4, 0, 0, -2, 4, 0, 2, -6, 2];
+    let m = run(&pts, &[2, 0, 0, 0, 0, 0], 2, &[false; 4], &[]).unwrap();
+    assert_eq!(m.triangles, [[0, 1, 2], [2, 1, 3]]);
+    let p: Vec<V> = m.positions.iter().map(|v| [v[0] - 10.0, v[1], v[2]]).collect();
+    assert!(close(p[0], [0.0, 0.0, 0.0]) && close(p[1], [2.0, 0.0, 0.0]));
+    assert!(close(p[2], [0.0, 2.0, 0.0]), "{:?}", p[2]);
+    assert!(close(p[3], [1.0 + 2.0 * S, 1.0 + S, -1.0]), "{:?}", p[3]);
+}
+
+/// The walk stops once its arrays would hold more than `limit` elements: the seed-and-apex
+/// pair ends holding four vertices (12), two triangles (4), five edges (10) and six logged
+/// edges (6), 32 in all.
+#[test]
+fn a_walk_past_its_limit_stops() {
+    let pts = [0, 0, 0, 4, 0, 0, -2, 4, 0, 2, -6, 2];
+    let with = |limit| {
+        rebuild(&Arrays {
+            tolerance: 0.5,
+            origin: [0.0; 3],
+            points: &pts,
+            edge_status: &[2, 0, 0, 0, 0, 0],
+            triangles: 2,
+            is_reference: &[false; 4],
+            references: &[],
+            limit,
+        })
+    };
+    assert_eq!(with(32).unwrap().triangles.len(), 2);
+    assert_eq!(with(31).unwrap_err(), "the rebuild needs more than the decoded-element allowance left");
+}
+
+/// The triangles reserved up front fit the limit too.
+#[test]
+fn the_walk_reserves_within_its_limit() {
+    assert!(Walk::new(1 << 20, 10).tris.capacity() < 1 << 20);
+    assert!(Walk::new(1 << 20, 10).tris.capacity() <= 5);
+    assert!(Walk::new(4, usize::MAX).tris.capacity() >= 4);
+}
+
+/// Bit 0 alone continues across `[A C]`, keeping `W = B`.
+#[test]
+fn the_right_edge_is_entered_from_a_to_c() {
+    let pts = [0, 0, 0, 4, 0, 0, -2, 4, 0, 0, -2, 0];
+    let m = run(&pts, &[1, 0, 0, 0, 0, 0], 2, &[false; 4], &[]).unwrap();
+    assert_eq!(m.triangles[1], [0, 2, 3]);
+}
+
+/// A reference slot reuses a vertex; a closed pushed edge is skipped and the next component
+/// starts from a fresh seed.
+#[test]
+fn references_and_components() {
+    let pts = [0; 9 + 9];
+    let is_ref = [false, false, false, true, false, false, false];
+    let m = run(&pts, &[3, 0, 0, 0, 0, 0, 0, 0, 0], 3, &is_ref, &[0]).unwrap();
+    assert_eq!(m.triangles, [[0, 1, 2], [2, 1, 0], [3, 4, 5]]);
+}
+
+/// A double-sided panel: four coplanar vertices, four triangles. The second triangle's apex
+/// folds back onto the seed's side, so its own continuation takes `[A C]` first; the last two
+/// apexes are the seed's first two vertices.
+#[test]
+fn a_folded_apex_swaps_its_continuation() {
+    let is_ref = [false, false, false, false, true, true];
+    let status = [3, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0];
+    let m = run(&[0, 0, 0, 4, 0, 0, -2, 4, 0, 1, 3, 0], &status, 4, &is_ref, &[0, 1]).unwrap();
+    assert_eq!(m.triangles, [[0, 1, 2], [2, 1, 3], [2, 3, 0], [0, 3, 1]]);
+    // The same panel unfolded is rescued by the component retry.
+    let m = run(&[0, 0, 0, 4, 0, 0, -2, 4, 0, 1, -3, 0], &status, 4, &is_ref, &[0, 1]).unwrap();
+    assert_eq!(m.triangles, [[0, 1, 2], [2, 1, 3], [2, 3, 0], [0, 3, 1]]);
+}
+
+/// The fourth apex `(1, 3, 0)` lies in its parent's plane on the parent's side, so its
+/// triangle folds and continues across `[A C]` to the reference.
+#[test]
+fn a_fold_past_the_retry_point_is_detected() {
+    let pts = [0, 0, 0, 4, 0, 0, -2, 4, 0, 0, -3, 1, 1, 3, 0];
+    let is_ref = [false, false, false, false, false, true];
+    let m = run(&pts, &[3, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0], 4, &is_ref, &[0]).unwrap();
+    assert_eq!(m.triangles[3], [3, 4, 0]);
+}
+
+/// A panel whose fold apex is a reference carries no fold signal: the unfolded walk closes an
+/// edge it still needs, so the component is rewound and walked again folded at its second
+/// triangle.
+#[test]
+fn a_reference_fold_is_found_by_retrying_the_component() {
+    let pts: Vec<i64> = (1..=30).collect();
+    let mut is_ref = vec![false; 9];
+    is_ref.extend([true, true, false, true, true, true]);
+    let mut status = vec![0; 21];
+    status[3..7].copy_from_slice(&[3, 3, 2, 0]);
+    let m = run(&pts, &status, 7, &is_ref, &[0, 3, 6, 0, 3]).unwrap();
+    assert_eq!(m.triangles[3..], [[0, 3, 9], [9, 3, 6], [9, 6, 0], [0, 6, 3]]);
+}
+
+/// A tetrahedron closed by two references: after `[3 1 0]` the continuation edge `[0 1]` is
+/// already shared twice, so the walk takes the open `[3 0]` instead.
+#[test]
+fn a_closed_continuation_edge_gives_way_to_the_open_one() {
+    let pts = [0, 0, 0, 4, 0, 0, -2, 4, 0, 0, 0, 4];
+    let mut status = [0; 12];
+    status[..3].copy_from_slice(&[2, 2, 2]);
+    let mut is_ref = [false; 6];
+    is_ref[4..].copy_from_slice(&[true, true]);
+    let m = run(&pts, &status, 4, &is_ref, &[0, 2]).unwrap();
+    assert_eq!(m.triangles, [[0, 1, 2], [2, 1, 3], [3, 1, 0], [3, 0, 2]]);
+}
+
+#[test]
+fn arrays_that_do_not_fit_are_refused() {
+    let pts = [0; 9];
+    let ok = [false; 3];
+    assert!(run(&pts, &[0, 0, 0], 1, &ok, &[]).is_ok());
+    assert!(run(&pts, &[0], 1, &ok, &[]).is_err(), "T form");
+    assert!(run(&[0; 12], &[0, 0, 0], 1, &ok, &[]).is_err(), "points left");
+    assert!(run(&[0; 6], &[0, 0, 0], 1, &ok, &[]).is_err(), "points short");
+    assert!(run(&pts, &[0, 0, 0], 1, &[false; 4], &[]).is_err(), "slots left");
+    assert!(run(&[0; 6], &[0, 0, 0], 1, &[false, false, true], &[7]).is_err(), "forward reference");
+    assert!(run(&[0; 9], &[0, 0, 0], 1, &[false, false, false, true], &[0]).is_err(), "references left");
+    // A third triangle on the seed's edge [0 1], each apex a reference.
+    let three = [false, false, false, true, true];
+    let shared = run(&[0; 9], &[2, 2, 0, 0, 0, 0, 0, 0, 0], 3, &three, &[0, 0]).map(|_| ());
+    assert_eq!(shared, Err("an edge is shared by more than two triangles"));
+}
+
+/// `PrcPt::Length` rounds the squared length to `f32` and takes Newton's root from a halved
+/// exponent, which can settle an ulp off the correctly rounded root; `Unitize` refuses
+/// lengths below `FLT_EPSILON` [WD 12.3].
+#[test]
+fn length_and_unitize_follow_the_wd_pseudocode() {
+    assert_eq!(wd_length([3.0, 4.0, 0.0]), Some(5.0));
+    let newton = f64::from_bits(S.to_bits() - 1);
+    assert_eq!(wd_length([1.0, 1.0, 0.0]), Some(newton));
+    let long = 1.0 + 2f64.powi(-30);
+    assert_eq!(wd_length([long, 0.0, 0.0]), Some(1.0));
+    assert_eq!(unitize([0.0, long, 0.0]), Some([0.0, long, 0.0]));
+    assert_eq!(unitize([1e-8, 0.0, 0.0]), None);
+    assert_eq!(unitize([0.0; 3]), None);
+}
+
+/// A `W` on the edge's line leaves `Z` null, so the frame comes from `MakeOrthoRep(X)`: `Z`
+/// from `X x (0, 1, 0)`, or from `X x (1, 0, 0)` when `X` is along `Y` [WD 7.8.9.2, 12.3].
+#[test]
+fn a_null_z_takes_the_ortho_rep_of_x() {
+    let d = [1.0, 2.0, 3.0];
+    assert_eq!(apex([0.0; 3], [-2.0, 0.0, 0.0], [5.0, 0.0, 0.0], d), [0.0, -2.0, -3.0]);
+    assert_eq!(apex([0.0; 3], [0.0, -2.0, 0.0], [0.0, 5.0, 0.0], d), [-2.0, 0.0, 3.0]);
+}
+
+/// A degenerate frame is turned a half-turn about `X` when the sub-epsilon residue of
+/// `(W - O) x X` points against `MakeOrthoRep`'s `Z`, and kept when it points along it.
+#[test]
+fn a_degenerate_frame_follows_its_residue() {
+    let (lo, hi, d) = ([0.0; 3], [-2.0, 0.0, 0.0], [1.0, 2.0, 3.0]);
+    assert_eq!(apex(lo, hi, [5.0, 1e-9, 0.0], d), [0.0, 2.0, 3.0]);
+    assert_eq!(apex(lo, hi, [5.0, -1e-9, 0.0], d), [0.0, -2.0, -3.0]);
+}
+
+/// A degenerate frame's `Y` comes from the frame's own `X`, whose length `unitize` left off by
+/// a few 1e-8, not from the copy `MakeOrthoRep` unitized again: the two differ by that much,
+/// and the walk would carry the difference into later apexes.
+#[test]
+fn a_degenerate_frame_takes_y_from_its_own_x() {
+    let (lo, hi) = ([0.1, 0.2, 0.3], [0.7, -0.4, 0.9]);
+    let o = scale(add(lo, hi), 0.5);
+    let x = unitize(sub(lo, hi)).unwrap();
+    let w = add(o, scale(x, 0.37));
+    let d = [1.0, 2.0, 3.0];
+    let z = make_ortho_rep_z(x).unwrap();
+    let p = apex(lo, hi, w, d);
+    let own = unitize(cross(z, x)).unwrap();
+    let again = unitize(cross(z, unitize(x).unwrap())).unwrap();
+    assert_ne!(own, again, "the probe must tell the two Ys apart");
+    let sign = if dot(cross(sub(w, o), x), z) < 0.0 { -1.0 } else { 1.0 };
+    let expect = |y: V| sub(sub(add(o, scale(x, d[0])), scale(y, sign * d[1])), scale(z, sign * d[2]));
+    assert_eq!(p, expect(own));
+    assert_ne!(p, expect(again));
+}
+
+/// A strip of thin, non-planar triangles compressed by an encoder written out from the WD:
+/// each point is expressed in the frame of the working mesh, rounded to the tolerance and
+/// reinjected [WD 7.8.9, 7.8.9.2], with `PrcPt::Length` and `Unitize` transcribed from the
+/// pseudocode [WD 12.3] as read. The decoder must land exactly on that working mesh, which
+/// keeps every vertex within the rounding bound of its source.
+#[test]
+fn a_strip_decodes_onto_the_encoders_working_mesh() {
+    fn length(v: V) -> f64 {
+        let f = (v[0] * v[0] + v[1] * v[1] + v[2] * v[2]) as f32;
+        let mut bits = f.to_bits();
+        let exponent = (bits >> 23) & 0xff;
+        if exponent > 127 {
+            bits = (bits & 0x807f_ffff) | (((exponent - 127) / 2 + 127) << 23);
+        }
+        let (squared, mut x0) = (f64::from(f), f64::from(f32::from_bits(bits)));
+        for _ in 0..100 {
+            let xi = 0.5 * (x0 + squared / x0);
+            if xi == x0 {
+                return xi;
+            }
+            x0 = xi;
+        }
+        -1.0
+    }
+    fn unitized(v: V) -> V {
+        let l = length(v);
+        assert!(l >= f64::from(f32::EPSILON), "null axis");
+        [v[0] / l, v[1] / l, v[2] / l]
+    }
+    let (tol, n) = (1e-4, 48);
+    let mut state = 0x9e37_79b9_7f4a_7c15_u64;
+    let mut jitter = |scale: f64| {
+        state = state.wrapping_mul(6_364_136_223_846_793_005).wrapping_add(1_442_695_040_888_963_407);
+        ((state >> 11) as f64 / (1u64 << 53) as f64 - 0.5) * scale
+    };
+    // Two rails 4 apart, 0.05 between a rail's points, 0.02 of relief.
+    let source: Vec<V> = (0..n)
+        .map(|i| {
+            let (along, rail) = (i as f64 * 0.025, (i % 2) as f64 * 4.0);
+            [along + jitter(0.01), rail + jitter(0.01), jitter(0.02)]
+        })
+        .collect();
+    let round = |v: f64| (v / tol).round() as i64;
+    let (mut points, mut work) = (Vec::new(), Vec::<V>::new());
+    for (i, target) in source.iter().take(3).enumerate() {
+        let base = match i {
+            0 => [0.0; 3],
+            1 => work[0],
+            _ => scale(add(work[0], work[1]), 0.5),
+        };
+        let d = [0, 1, 2].map(|c| round(target[c] - base[c]));
+        points.extend(d);
+        work.push(add(base, d.map(|c| c as f64 * tol)));
+    }
+    // Triangle k enters across vertices k and k + 1, opposite k - 1, and continues right on
+    // odd k, left on even k, unless it folds.
+    let mut status = vec![0; 3 * (n - 2)];
+    status[0] = 2;
+    for k in 1..n - 2 {
+        let (lo, hi, w) = (work[k], work[k + 1], work[k - 1]);
+        let o = scale(add(hi, lo), 0.5);
+        let x = unitized(sub(lo, hi));
+        let z = unitized(cross(sub(w, o), x));
+        let y = unitized(cross(z, x));
+        let r = sub(source[k + 2], o);
+        let d = [round(dot(r, x)), round(-dot(r, y)), round(-dot(r, z))];
+        points.extend(d);
+        let [dx, dy, dz] = d.map(|c| c as f64 * tol);
+        work.push(sub(sub(add(o, scale(x, dx)), scale(y, dy)), scale(z, dz)));
+        if k < n - 3 {
+            let fold = d[2] == 0 && d[1] > 0;
+            status[k] = if (k % 2 == 1) != fold { 1 } else { 2 };
+        }
+    }
+    let m = rebuild(&Arrays {
+        tolerance: tol,
+        origin: [0.0; 3],
+        points: &points,
+        edge_status: &status,
+        triangles: n - 2,
+        is_reference: &vec![false; n],
+        references: &[],
+        limit: usize::MAX,
+    })
+    .unwrap();
+    assert_eq!(m.triangles[1..4], [[2, 1, 3], [2, 3, 4], [4, 3, 5]]);
+    assert_eq!(m.positions, work);
+    for (p, s) in m.positions.iter().zip(&source) {
+        let off = dot(sub(*p, *s), sub(*p, *s)).sqrt() / tol;
+        assert!(off < 0.87, "{off} tolerances off");
+    }
+}

crates/prc/src/tess/tests.rs

@@ -529,12 +529,281 @@ fn a_compressed_mesh_is_read_to_its_end() {
             let schema = Schema::default();
             let mut ctx = Ctx::new(BitReader::new(&bytes), &schema, v);
             let t = ctx.file_structure_tessellation().unwrap();
-            assert!(matches!(t[..], [Tessellation::Compressed { triangles: 1, .. }, Tessellation::Mesh(_)]), "full={full} v={v}: {t:?}");
             assert_eq!(ctx.r.position(), w.len(), "full={full} v={v}");
+            assert!(matches!(t[1], Tessellation::Mesh(_)), "full={full} v={v}: {t:?}");
+            // Stored normals and the one-status-per-triangle form are not rebuilt; the
+            // recalculated 3T seed is.
+            if full || !three_t {
+                assert!(matches!(t[0], Tessellation::Compressed { triangles: 1, .. }), "full={full} v={v}: {t:?}");
+            } else {
+                let m = mesh(&t[0]);
+                assert_eq!((m.triangles.as_slice(), m.faces.as_slice()), ([[0, 1, 2]].as_slice(), std::slice::from_ref(&(0..1))), "v={v}");
+                assert!(m.normals_recalculated && m.normals.is_empty());
+                assert_eq!(m.triangle_styles, [TriangleStyle { style: 1, behaviour: 0 }]);
+                // Each point component is 1, in units of the 0.001 tolerance, from the origin 0.
+                let want = [0.001, 0.002, 0.0025];
+                assert!(m.positions.iter().zip(want).all(|(p, c)| p.iter().all(|x| (x - c).abs() < 1e-12)), "{:?}", m.positions);
+            }
         }
     }
 }
 
+/// A `TESS_3D_Compressed` (authoring version 8137, origin 0) whose normals are recalculated,
+/// with `points` in units of `tolerance`, the 3T `status`, each triangle's face, the slots'
+/// reference flags, the references and one line attribute per face; every other optional
+/// branch absent.
+struct Packed<'a> {
+    tolerance: f64,
+    points: &'a [i64],
+    status: &'a [u8],
+    face_of: &'a [u32],
+    /// The width of each face index; 8 unless an index needs more.
+    face_bits: u32,
+    is_ref: &'a [bool],
+    refs: &'a [u32],
+    line_attributes: &'a [u16],
+}
+
+/// A `CompressedIndiceArray` [WD 11.9] of `bits`-wide deltas, `implicit` with no compressed
+/// flag.
+fn indices(w: &mut W, v: &[u32], implicit: bool, bits: u32) {
+    if !implicit {
+        w.bit(false);
+    }
+    w.uint(u32::try_from(v.len()).unwrap());
+    for i in 0..v.len() {
+        w.put(if i == 0 { u64::from(bits) } else { 0 }, 8);
+    }
+    let mut prev = 0i64;
+    for &x in v {
+        let d = i64::from(x) - prev;
+        w.bit(d < 0).put(d.unsigned_abs(), bits - 1);
+        prev = i64::from(x);
+    }
+}
+
+fn packed(w: &mut W, c: &Packed<'_>) {
+    w.uint(173).bit(false).bit(true).double(c.tolerance);
+    w.put(0, 32).put(0, 32).put(0, 32);
+    // point_array: 8-bit sign-magnitude integers.
+    w.bit(false).uint(u32::try_from(c.points.len()).unwrap());
+    for _ in c.points {
+        w.put(8, 8);
+    }
+    for &p in c.points {
+        w.bit(p < 0).put(p.unsigned_abs(), 7);
+    }
+    w.bit(false).uint(u32::try_from(c.status.len()).unwrap());
+    for &e in c.status {
+        w.put(u64::from(e), 8);
+    }
+    indices(w, c.face_of, false, c.face_bits);
+    w.uint(u32::try_from(c.is_ref.len()).unwrap());
+    for &b in c.is_ref {
+        w.bit(b);
+    }
+    assert!(c.refs.len() <= 3, "more than three references are Huffman-coded");
+    indices(w, c.refs, true, 8);
+    // must_recalculate_normals, normal_is_reversed, crease angle, flags; no colours, no
+    // multiple line attributes.
+    w.bit(true);
+    for _ in c.face_of {
+        w.bit(false);
+    }
+    w.double(0.5).put(0, 8).bit(false).bit(false);
+    w.bit(false).uint(u32::try_from(c.line_attributes.len()).unwrap());
+    for &a in c.line_attributes {
+        w.put(u64::from(a & 0xff), 8).put(u64::from(a >> 8), 8);
+    }
+    // No texture, no behaviours.
+    w.bit(true).bit(false);
+}
+
+/// A double-sided panel of two faces packed in walk order, its fold apex and its last two
+/// apexes references: rebuilt into a mesh grouped by face, each triangle styled by its face.
+#[test]
+fn a_recalculated_compressed_mesh_is_rebuilt_by_face() {
+    let mut b = W::default();
+    packed(
+        &mut b,
+        &Packed {
+            tolerance: 0.5,
+            points: &[0, 0, 0, 4, 0, 0, -2, 4, 0, 1, 3, 0],
+            status: &[3, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+            face_of: &[1, 0, 1, 0],
+            face_bits: 8,
+            is_ref: &[false, false, false, false, true, true],
+            refs: &[0, 1],
+            line_attributes: &[3, 0],
+        },
+    );
+    let w = section(1, &b);
+    let bytes = w.bytes();
+    let schema = Schema::default();
+    let mut ctx = Ctx::new(BitReader::new(&bytes), &schema, 8137);
+    let t = ctx.file_structure_tessellation().unwrap();
+    assert_eq!(ctx.r.position(), w.len());
+    let m = mesh(&t[0]);
+    // Walk order [0 1 2] [2 1 3] [2 3 0] [0 3 1]; faces 1, 0, 1, 0.
+    assert_eq!(m.triangles, [[2, 1, 3], [0, 3, 1], [0, 1, 2], [2, 3, 0]]);
+    assert_eq!(m.faces, [0..2, 2..4]);
+    assert_eq!(m.positions[..3], [[0.0, 0.0, 0.0], [2.0, 0.0, 0.0], [0.0, 2.0, 0.0]]);
+    let style = |style| TriangleStyle { style, behaviour: 0 };
+    assert_eq!(m.triangle_styles, [style(3), style(3), style(0), style(0)]);
+    assert!(m.normals_recalculated);
+}
+
+/// Arrays the walk cannot fit stay a [`Tessellation::Compressed`], read to their end.
+#[test]
+fn a_compressed_mesh_that_does_not_fit_is_left_compressed() {
+    let mut b = W::default();
+    // One point too many for a lone seed.
+    let c = Packed {
+        tolerance: 0.5,
+        points: &[0; 12],
+        status: &[0, 0, 0],
+        face_of: &[0],
+        face_bits: 8,
+        is_ref: &[false; 3],
+        refs: &[],
+        line_attributes: &[0],
+    };
+    packed(&mut b, &c);
+    let w = section(1, &b);
+    let t = decode(&w, &Schema::default(), 8137).unwrap();
+    assert!(matches!(t[..], [Tessellation::Compressed { triangles: 1, .. }]), "{t:?}");
+}
+
+/// A face index near `u32::MAX` claims four billion faces: once the triangles rebuild, they are
+/// charged before any face range is made, so the decode stops at the element allowance instead
+/// of allocating them.
+#[test]
+fn a_huge_face_index_is_charged_not_allocated() {
+    let mut b = W::default();
+    let c = Packed {
+        tolerance: 0.5,
+        points: &[0; 9],
+        status: &[0, 0, 0],
+        face_of: &[u32::MAX - 1],
+        face_bits: 33,
+        is_ref: &[false; 3],
+        refs: &[],
+        line_attributes: &[],
+    };
+    packed(&mut b, &c);
+    let r = decode(&section(1, &b), &Schema::default(), 8137);
+    assert!(matches!(r, Err(PrcError::TooManyElements { .. })), "{r:?}");
+}
+
+/// The same huge face index on arrays the walk cannot fit charges nothing for triangles or
+/// faces: the tessellation stays compressed and the file reads on.
+#[test]
+fn arrays_that_do_not_fit_spend_no_triangles_or_faces() {
+    let mut b = W::default();
+    let c = Packed {
+        tolerance: 0.5,
+        points: &[0; 12],
+        status: &[0, 0, 0],
+        face_of: &[u32::MAX - 1],
+        face_bits: 33,
+        is_ref: &[false; 3],
+        refs: &[],
+        line_attributes: &[],
+    };
+    packed(&mut b, &c);
+    let t = decode(&section(1, &b), &Schema::default(), 8137).unwrap();
+    assert!(matches!(t[..], [Tessellation::Compressed { triangles: 1, .. }]), "{t:?}");
+}
+
+/// The two-face panel of [`a_recalculated_compressed_mesh_is_rebuilt_by_face`], its second
+/// reference `last_ref`.
+fn panel(last_ref: u32) -> W {
+    let mut b = W::default();
+    packed(
+        &mut b,
+        &Packed {
+            tolerance: 0.5,
+            points: &[0, 0, 0, 4, 0, 0, -2, 4, 0, 1, 3, 0],
+            status: &[3, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+            face_of: &[1, 0, 1, 0],
+            face_bits: 8,
+            is_ref: &[false, false, false, false, true, true],
+            refs: &[0, last_ref],
+            line_attributes: &[3, 0],
+        },
+    );
+    section(1, &b)
+}
+
+/// A rebuilt mesh is charged its triangles, each with its style, and each face range at its
+/// size; the same arrays left compressed (a reference to a vertex not yet made) are charged
+/// neither.
+#[test]
+fn a_rebuilt_compressed_mesh_is_charged_its_triangles_and_faces() {
+    let fits = panel(1);
+    let t = decode(&fits, &Schema::default(), 8137).unwrap();
+    assert_eq!(mesh(&t[0]).triangles.len(), 4);
+    let left = panel(9);
+    let t = decode(&left, &Schema::default(), 8137).unwrap();
+    assert!(matches!(t[..], [Tessellation::Compressed { triangles: 4, .. }]), "{t:?}");
+    assert_eq!(spent(&fits) - spent(&left), 4 * triangle_elements(false, 0, true) + 2 * RANGE_ELEMENTS);
+    assert_eq!(triangle_elements(false, 0, true), 3);
+    #[cfg(target_pointer_width = "64")]
+    assert_eq!(RANGE_ELEMENTS, 2);
+}
+
+/// A rebuild whose walk would outgrow the allowance left keeps the arrays compressed, even
+/// when the rebuilt mesh alone would have fit, and the stream reads on.
+#[test]
+fn a_compressed_rebuild_is_bounded_by_the_allowance_left() {
+    let fits = panel(1);
+    let bytes = fits.bytes();
+    let schema = Schema::default();
+    let read = |n| Ctx::new(BitReader::with_elements(&bytes, n), &schema, 8137).file_structure_tessellation();
+    let arrays = spent(&panel(9));
+    let mesh_alone = spent(&fits) - arrays;
+    let t = read(arrays + mesh_alone).unwrap();
+    assert!(matches!(t[..], [Tessellation::Compressed { triangles: 4, .. }]), "{t:?}");
+    assert!(matches!(read(crate::MAX_DECODED_ELEMENTS).unwrap()[..], [Tessellation::Mesh(_)]));
+}
+
+/// A face's nonzero behaviour styles its triangles even when its line attribute is 0 (the
+/// owner's graphics); all zero, the faces give no styles.
+#[test]
+fn a_compressed_face_behaviour_alone_styles_its_triangles() {
+    let attributes = |behaviours: Vec<i32>| FaceAttributes { multi: Vec::new(), line_attributes: vec![0], behaviours };
+    let a = attributes(vec![0x30]);
+    assert!(a.styled(1));
+    assert_eq!(a.styles(&[0, 0], 1), [TriangleStyle { style: 0, behaviour: 0x30 }; 2]);
+    assert!(!attributes(vec![0]).styled(1));
+    assert!(!attributes(Vec::new()).styled(1));
+}
+
+fn rebuilt(triangles: usize) -> compressed::Rebuilt {
+    compressed::Rebuilt { positions: vec![[0.0; 3]; 3], triangles: (0..triangles).map(|i| [0, 1, u32::try_from(i).unwrap()]).collect() }
+}
+
+/// Triangles are grouped by face in walk order within each, faces no triangle names (between
+/// and after) get empty ranges, and each style moves with its triangle.
+#[test]
+fn rebuilt_triangles_are_grouped_by_face_with_their_styles() {
+    let face_of = [2, 0, 2, 1, 0];
+    let m = by_face(rebuilt(5), &face_of, 5, &[]);
+    let apex: Vec<u32> = m.triangles.iter().map(|t| t[2]).collect();
+    assert_eq!(apex, [1, 4, 3, 0, 2]);
+    assert_eq!(m.faces, [0..2, 2..3, 3..5, 5..5, 5..5]);
+    assert!(m.triangle_styles.is_empty());
+    assert!(m.normals_recalculated);
+
+    let style = |style| TriangleStyle { style, behaviour: u16::try_from(style).unwrap() };
+    let styles: Vec<TriangleStyle> = face_of.iter().map(|&f| style(f + 1)).collect();
+    let m = by_face(rebuilt(5), &[2, 0, 2, 3, 0], 4, &styles);
+    let apex: Vec<u32> = m.triangles.iter().map(|t| t[2]).collect();
+    assert_eq!(apex, [1, 4, 0, 2, 3]);
+    assert_eq!(m.faces, [0..2, 2..2, 2..4, 4..5]);
+    assert_eq!(m.triangle_styles, [style(1), style(1), style(3), style(3), style(2)]);
+}
+
 #[test]
 fn refusals_and_damage_are_errors() {
     // Edge statuses neither T nor 3T long.