diff --git a/crates/pathops/src/edit.rs b/crates/pathops/src/edit.rs
index 338900041..80f0deffd 100644
--- a/crates/pathops/src/edit.rs
+++ b/crates/pathops/src/edit.rs
@@ -476,7 +476,8 @@ fn with_handle(a: Anchor, out: bool, h: Point) -> Anchor {
     Anchor { p: a.p, h_in, h_out, kind }
 }
 
-/// Ramer–Douglas–Peucker polyline reduction.
+/// Ramer–Douglas–Peucker polyline reduction. Each point dropped is within `tol` of the line segment
+/// that replaces it: a point beyond either end of the segment is measured to that end.
 fn rdp(pts: &[Point], tol: f64) -> Vec<Point> {
     if pts.len() < 3 {
         return pts.to_vec();
@@ -498,10 +499,10 @@ fn rdp(pts: &[Point], tol: f64) -> Vec<Point> {
         };
         let (pa, pb) = (pts[a], pts[b]);
         let chord = pb - pa;
-        let len = chord.hypot();
+        let len2 = chord.hypot2();
         let mut best = (0.0, a);
         for (i, &p) in pts.iter().enumerate().take(b).skip(a + 1) {
-            let d = if len < 1e-12 { p.distance(pa) } else { (p - pa).cross(chord).abs() / len };
+            let d = if len2 < 1e-24 { p.distance(pa) } else { p.distance(pa + chord * ((p - pa).dot(chord) / len2).clamp(0.0, 1.0)) };
             if d > best.0 {
                 best = (d, i);
             }
diff --git a/crates/pathops/src/fit.rs b/crates/pathops/src/fit.rs
index 4443e5b74..6b1f4eb0c 100644
--- a/crates/pathops/src/fit.rs
+++ b/crates/pathops/src/fit.rs
@@ -465,6 +465,23 @@ mod tests {
         }
     }
 
+    #[test]
+    fn a_path_that_turns_back_past_its_end_converted_to_straight_lines_keeps_its_tip() {
+        // A line to (100, 0), a smooth 0.2 pt U-turn and a line back to (50, 0.2): every point is
+        // within 0.4 pt of the line through the ends, but the tip is 50 pt past the end.
+        let k = 0.2 * 2.0 / 3.0;
+        let (a, b) = (Point::new(100.0, 0.0), Point::new(100.0, 0.2));
+        let anchors = vec![
+            Anchor::corner(Point::new(0.0, 0.0)),
+            Anchor::with_handles(a, a, Point::new(100.0 + k, 0.0)),
+            Anchor::with_handles(b, Point::new(100.0 + k, 0.2), b),
+            Anchor::corner(Point::new(50.0, 0.2)),
+        ];
+        let path = PathData::single(SubPath::new(anchors, false));
+        let out = crate::simplify_with(&path, &crate::SimplifyOptions { tolerance: 1.0, straight_lines: true, ..Default::default() });
+        assert_within(&cubics(&path), &cubics(&out), 1.0, "a U-turn past the end");
+    }
+
     #[test]
     fn long_and_short_segments_stay_within_the_tolerance() {
         for (r, tol) in [(50_000.0, 2.0), (50_000.0, 1.0 / 32.0), (8_000.0, 0.01), (500.0, 0.5)] {