diff --git a/crates/geom/src/curve.rs b/crates/geom/src/curve.rs
index 04f665f88..972d0f8e8 100644
--- a/crates/geom/src/curve.rs
+++ b/crates/geom/src/curve.rs
@@ -484,6 +484,9 @@ pub struct Polyline {
     pub closed: bool,
 }
 
+/// Sampled points one [`Polyline::tessellate`] call produces before arcs become chords.
+pub const MAX_TESSELLATED_POINTS: usize = 4_000_000;
+
 impl Polyline {
     pub fn from_points(pts: &[Vec2], closed: bool) -> Self {
         Polyline { vertices: pts.iter().map(|p| PolyVertex::new(*p)).collect(), closed }
@@ -516,10 +519,23 @@ impl Polyline {
         b
     }
     pub fn tessellate(&self, tol: f64) -> Vec<Vec2> {
+        let mut budget = MAX_TESSELLATED_POINTS;
+        self.tessellate_within(tol, &mut budget)
+    }
+    /// [`Polyline::tessellate`] spending at most `budget` sampled points (shared by the caller
+    /// across many polylines): once it is spent, the remaining arcs are drawn as chords, so the
+    /// output stays proportional to the input vertices. Each arc can sample thousands of points,
+    /// so a compact polyline could otherwise expand into gigabytes (CWE-400).
+    pub fn tessellate_within(&self, tol: f64, budget: &mut usize) -> Vec<Vec2> {
         let mut out: Vec<Vec2> = Vec::new();
         for s in self.segments() {
             let mut pts = Vec::new();
-            s.tessellate(tol, &mut pts);
+            if *budget > 0 {
+                s.tessellate(tol, &mut pts);
+                *budget = budget.saturating_sub(pts.len());
+            } else {
+                pts.extend([s.start(), s.end()]);
+            }
             let skip = usize::from(out.last().is_some_and(|l| pts.first().is_some_and(|f| l.near(*f, 1e-9))));
             out.extend(pts.into_iter().skip(skip));
         }
diff --git a/crates/geom/src/lib.rs b/crates/geom/src/lib.rs
index e62f478fb..d482141fd 100644
--- a/crates/geom/src/lib.rs
+++ b/crates/geom/src/lib.rs
@@ -20,7 +20,8 @@ mod vec;
 
 pub use bounds::Bounds2;
 pub use curve::{
-    Arc, Circle, Ellipse, Line, PolyVertex, Polyline, Segment, arc_to_bulge, bulge_to_arc, point_in_polygon, polygon_centroid, shoelace,
+    Arc, Circle, Ellipse, Line, MAX_TESSELLATED_POINTS, PolyVertex, Polyline, Segment, arc_to_bulge, bulge_to_arc, point_in_polygon,
+    polygon_centroid, shoelace,
 };
 pub use fillet::{CornerFit, TanCurve, tangent_circles};
 pub use intersect::{circle_circle, intersect_ext, intersect_segments, line_circle, line_line, line_line_infinite};
diff --git a/crates/geom/src/tests.rs b/crates/geom/src/tests.rs
index 915a1071e..57ffcd663 100644
--- a/crates/geom/src/tests.rs
+++ b/crates/geom/src/tests.rs
@@ -221,6 +221,18 @@ fn segment_offset() {
     assert!(s.offset(6.0).is_none());
 }
 
+/// A polyline of many fine arcs samples at most the point budget, then draws chords.
+#[test]
+fn polyline_tessellation_is_bounded() {
+    let vertices: Vec<PolyVertex> = (0..3000).map(|i| PolyVertex { bulge: 1.0, ..PolyVertex::new(Vec2::new(f64::from(i) * 1e6, 0.0)) }).collect();
+    let pl = Polyline { vertices, closed: false };
+    let pts = pl.tessellate(1e-9);
+    assert!(pts.len() <= MAX_TESSELLATED_POINTS + 2 * 3000 + 4096, "{}", pts.len());
+    assert!(pts.len() > 3000);
+    let mut budget = 10;
+    assert!(pl.tessellate_within(1e-9, &mut budget).len() <= 4096 + 2 * 3000);
+}
+
 #[test]
 fn arc_segments_bounded() {
     assert_eq!(arc_segments(0.0, 1.0, 0.1), 1);
diff --git a/crates/render/src/hatch.rs b/crates/render/src/hatch.rs
index 1bfeb1b6c..ecd78fdfe 100644
--- a/crates/render/src/hatch.rs
+++ b/crates/render/src/hatch.rs
@@ -128,7 +128,10 @@ const MAX_DEDUP_COMPARISONS: u64 = 50_000_000;
 /// within islands alternate), 1 outer keeps the outermost loops and their first islands, 2
 /// ignore keeps only the outermost loops. Nesting depth = how many other loops enclose a loop.
 pub fn fill_loops(h: &Hatch, tol: f64) -> Vec<Vec<Vec2>> {
-    let mut loops: Vec<Vec<Vec2>> = h.loops.iter().map(|l| Polyline { vertices: l.vertices.clone(), closed: true }.tessellate(tol)).collect();
+    // One point budget for all the loops: many short arcs across loops must not add up unbounded.
+    let mut budget = cadcraft_geom::MAX_TESSELLATED_POINTS;
+    let mut loops: Vec<Vec<Vec2>> =
+        h.loops.iter().map(|l| Polyline { vertices: l.vertices.clone(), closed: true }.tessellate_within(tol, &mut budget)).collect();
     dedup_polygons(&mut loops);
     let max_depth = match h.style {
         1 => 1,