diff --git a/crates/geom/src/freehand.rs b/crates/geom/src/freehand.rs
index 53ddf9611..278898239 100644
--- a/crates/geom/src/freehand.rs
+++ b/crates/geom/src/freehand.rs
@@ -3,6 +3,10 @@
 use crate::Point;
 use crate::path::{Anchor, AnchorKind};
 
+/// Distance tests [`simplify`] may make. Worst-case input splits one point off at a time (quadratic);
+/// past this the spans still waiting keep only their end points, a coarser fit rather than a stall.
+const MAX_SIMPLIFY_STEPS: usize = 16 * 1024 * 1024;
+
 /// Ramer–Douglas–Peucker: the points that keep the polyline within `tol`.
 fn simplify(pts: &[Point], tol: f64) -> Vec<Point> {
     if pts.len() < 3 {
@@ -12,7 +16,12 @@ fn simplify(pts: &[Point], tol: f64) -> Vec<Point> {
     keep[0] = true;
     keep[pts.len() - 1] = true;
     let mut stack = vec![(0, pts.len() - 1)];
+    let mut steps = 0usize;
     while let Some((a, b)) = stack.pop() {
+        steps = steps.saturating_add(b - a);
+        if steps > MAX_SIMPLIFY_STEPS {
+            break;
+        }
         let (pa, pb) = (pts[a], pts[b]);
         let d = pb - pa;
         let len = d.hypot();
@@ -113,4 +122,13 @@ mod tests {
         assert!(c.iter().all(|x| x.kind == AnchorKind::Smooth));
         assert!((c[0].p - c[c.len() - 1].p).hypot() > 1.0);
     }
+
+    /// A convex curve splits one point at a time (quadratic): fitting stops at the step budget.
+    #[test]
+    fn unbalanced_input_stops_at_the_step_budget() {
+        let n = 100_000;
+        let curve: Vec<Point> = (0..n).map(|i| Point::new(f64::from(i), (f64::from(i) * 1e-3).powi(4))).collect();
+        let a = fit(&curve, 1e-9, false);
+        assert!(a.len() >= 2 && a.len() < n as usize);
+    }
 }
diff --git a/crates/geom/src/pathfinder.rs b/crates/geom/src/pathfinder.rs
index 75aa1afce..0b8237024 100644
--- a/crates/geom/src/pathfinder.rs
+++ b/crates/geom/src/pathfinder.rs
@@ -92,11 +92,31 @@ fn from_flo(paths: &[SimpleBezierPath]) -> PathData {
     PathData::new(subpaths)
 }
 
+/// Unions of more shapes than this combine in pairs; up to it each is added in turn (as always).
+const BALANCED_UNION_FROM: usize = 16;
+
 /// Combine `shapes` (back to front, all in one coordinate space). `None` when nothing is left.
 pub fn combine(op: Op, shapes: &[PathData]) -> Option<PathData> {
     let flo: Vec<Vec<SimpleBezierPath>> = shapes.iter().map(to_flo).filter(|p| !p.is_empty()).collect();
     let (first, rest) = flo.split_first()?;
     let union = |v: &[Vec<SimpleBezierPath>]| -> Vec<SimpleBezierPath> {
+        if v.len() > BALANCED_UNION_FROM {
+            // Pairs, then pairs of pairs: adding each shape to everything so far reprocesses the
+            // growing result every time (quadratic for many shapes).
+            let mut level: Vec<Vec<SimpleBezierPath>> = v.to_vec();
+            while level.len() > 1 {
+                level = level
+                    .chunks(2)
+                    .map(|c| match c {
+                        [a, b] if a.is_empty() => b.clone(),
+                        [a, b] if b.is_empty() => a.clone(),
+                        [a, b] => path_add(a, b, ACCURACY),
+                        _ => c.first().cloned().unwrap_or_default(),
+                    })
+                    .collect();
+            }
+            return level.pop().unwrap_or_default();
+        }
         let mut acc: Vec<SimpleBezierPath> = Vec::new();
         for p in v {
             acc = if acc.is_empty() { p.clone() } else { path_add(&acc, p, ACCURACY) };
@@ -147,6 +167,19 @@ mod tests {
         p.subpaths.iter().map(|s| s.area()).sum::<f64>().abs()
     }
 
+    #[test]
+    fn a_union_of_many_shapes_adds_up() {
+        // 40 squares in a row, each overlapping the next by 5: width 40 * 10 + 5, height 10.
+        let shapes: Vec<PathData> = (0..40).map(|i| shapes::rectangle(Rect::new(f64::from(i) * 5.0, 0.0, f64::from(i) * 5.0 + 10.0, 10.0))).collect();
+        let u = combine(Op::Add, &shapes).unwrap();
+        assert!((area(&u) - 205.0 * 10.0).abs() < 1.0, "{}", area(&u));
+        // Far-apart squares stay separate.
+        let apart: Vec<PathData> =
+            (0..40).map(|i| shapes::rectangle(Rect::new(f64::from(i) * 50.0, 0.0, f64::from(i) * 50.0 + 10.0, 10.0))).collect();
+        let u = combine(Op::Add, &apart).unwrap();
+        assert_eq!(u.subpaths.len(), 40);
+    }
+
     #[test]
     fn boolean_ops_on_overlapping_squares() {
         let a = shapes::rectangle(Rect::new(0.0, 0.0, 100.0, 100.0));
diff --git a/crates/geom/src/stroke_style.rs b/crates/geom/src/stroke_style.rs
index 26c201477..d3c25ecba 100644
--- a/crates/geom/src/stroke_style.rs
+++ b/crates/geom/src/stroke_style.rs
@@ -57,10 +57,18 @@ fn offset_subpath(bp: &BezPath, d: f64, tol: f64, out: &mut BezPath) {
 /// Bands (start, width) across a stroke, as fractions of the weight from its left edge.
 pub type Bands = [(f64, f64)];
 
+/// Most bands of a striped stroke that are drawn, and most path elements its fill may have:
+/// every band strokes the whole path again, so both multiply the path's size.
+pub const MAX_STRIPE_BANDS: usize = 64;
+const MAX_STRIPE_ELEMENTS: usize = 2_000_000;
+
 /// The filled area of a striped stroke of `weight` along `bp`.
 pub fn stripes(bp: &BezPath, weight: f64, bands: &Bands, tol: f64) -> BezPath {
     let mut out = BezPath::new();
-    for &(start, width) in bands {
+    for &(start, width) in bands.iter().take(MAX_STRIPE_BANDS) {
+        if out.elements().len() > MAX_STRIPE_ELEMENTS {
+            break;
+        }
         if width <= 0.0 {
             continue;
         }
@@ -139,6 +147,17 @@ pub fn hashed(bp: &BezPath, weight: f64, tol: f64) -> BezPath {
 #[cfg(test)]
 mod tests {
     use super::*;
+
+    #[test]
+    fn stripes_draw_at_most_the_band_cap() {
+        let mut line = BezPath::new();
+        line.move_to((0.0, 0.0));
+        line.line_to((100.0, 0.0));
+        let one = stripes(&line, 4.0, &[(0.0, 0.1)], 0.1).elements().len();
+        let many = vec![(0.0, 0.1); 100_000];
+        let all = stripes(&line, 4.0, &many, 0.1).elements().len();
+        assert_eq!(all, one * MAX_STRIPE_BANDS);
+    }
     use kurbo::Shape;
 
     fn line() -> BezPath {