diff --git a/crates/engine/src/cmd/anchors.rs b/crates/engine/src/cmd/anchors.rs
index 4ccbeec7e..0968fb588 100644
--- a/crates/engine/src/cmd/anchors.rs
+++ b/crates/engine/src/cmd/anchors.rs
@@ -291,33 +291,61 @@ fn convert_points(s: &mut Session, p: &Value) -> Result<Value> {
     })
 }
 
+/// Most samples of a drag that Erase / Smooth look at.
+const MAX_DRAG_SAMPLES: usize = 4096;
+
 /// Erase / Smooth: the stretch of one subpath touched by the drag.
 fn erase_or_smooth(s: &mut Session, p: &Value, smooth: bool) -> Result<Value> {
     let cmd = if smooth { "path.smooth" } else { "path.erase" };
     let id = super::id_param(p, "id").ok_or_else(|| bad(cmd, "missing id"))?;
     let tol = p.get("tolerance").and_then(Value::as_f64).unwrap_or(4.0);
-    let pts: Vec<Point> = p
+    let mut pts: Vec<Point> = p
         .get("points")
         .and_then(Value::as_array)
         .map(|a| a.iter().filter_map(|v| Some(Point::new(v.get(0)?.as_f64()?, v.get(1)?.as_f64()?))).collect())
         .unwrap_or_default();
+    // Each sample is a search of every segment: repeated samples add nothing, and a drag has no use
+    // for more than MAX_DRAG_SAMPLES (evenly thinned, the first and last kept). Whether the drag
+    // touched the path is decided on every distinct sample, though: see below.
+    let mut seen = std::collections::HashSet::new();
+    pts.retain(|q| seen.insert((q.x.to_bits(), q.y.to_bits())));
+    let thinned: Vec<Point> = if pts.len() > MAX_DRAG_SAMPLES {
+        let (n, last) = (pts.len(), pts.last().copied());
+        (0..MAX_DRAG_SAMPLES).filter_map(|i| pts.get((i as u64 * n as u64 / MAX_DRAG_SAMPLES as u64) as usize).copied()).chain(last).collect()
+    } else {
+        Vec::new()
+    };
     s.edit(|d, sel| {
         path_editable(d, id, cmd)?;
         let inv = to_inner(d, id).ok_or(designcraft_doc::DocError::NoItem(id))?;
         let scale = inv.determinant().abs().sqrt().max(1e-9);
         let it = d.item_mut(id).ok_or(designcraft_doc::DocError::NoItem(id))?;
         // Where the drag touched the path, per subpath.
-        let mut hits: Vec<(usize, usize, f64)> = Vec::new();
-        for q in &pts {
-            if let Some((si, seg, t, _, dist)) = it.path.nearest(inv * *q)
-                && dist <= tol * scale
-            {
-                hits.push((si, seg, t));
+        let hits_of = |set: &[Point]| -> Vec<(usize, usize, f64)> {
+            let mut hits = Vec::new();
+            for q in set {
+                if let Some((si, seg, t, _, dist)) = it.path.nearest(inv * *q)
+                    && dist <= tol * scale
+                {
+                    hits.push((si, seg, t));
+                }
+            }
+            hits
+        };
+        // A long drag is searched thinned; only when that finds nothing are all its samples, so a
+        // touch the thinning skipped over still counts and a long drag is never refused for it.
+        let mut hits = hits_of(if thinned.is_empty() { &pts } else { &thinned });
+        if hits.is_empty() && !thinned.is_empty() {
+            hits = hits_of(&pts);
+        }
+        // The subpath with the most hits (the last of equals), counted in one pass.
+        let mut counts = vec![0usize; it.path.subpaths.len()];
+        for h in &hits {
+            if let Some(n) = counts.get_mut(h.0) {
+                *n += 1;
             }
         }
-        let Some(si) =
-            (0..it.path.subpaths.len()).max_by_key(|i| hits.iter().filter(|h| h.0 == *i).count()).filter(|i| hits.iter().any(|h| h.0 == *i))
-        else {
+        let Some((si, _)) = counts.iter().copied().enumerate().filter(|(_, n)| *n > 0).max_by_key(|(_, n)| *n) else {
             return Err(bad(cmd, "the drag didn't touch the path"));
         };
         let on: Vec<(usize, f64)> = hits.iter().filter(|h| h.0 == si).map(|h| (h.1, h.2)).collect();
@@ -663,6 +691,31 @@ mod tests {
         assert!(s.execute("path.erase", &json!({"id": id, "points": [[100, 600]]})).is_err());
     }
 
+    /// A touch only thinning would skip still counts: a long drag is not refused as missing the path.
+    #[test]
+    fn a_touch_the_thinning_skips_still_counts() {
+        let mut s = Session::new();
+        s.execute("file.new", &json!({})).unwrap();
+        let pts: Vec<Value> = (0..=20).map(|i| json!({"p": [100.0 + i as f64 * 20.0, 300.0]})).collect();
+        let id = s.execute("path.create", &json!({"anchors": pts})).unwrap()["id"].as_u64().unwrap();
+        // 6,000 distinct samples far from the path but one, at an index thinning to 4,096 drops.
+        let drag: Vec<Value> = (0..6_000).map(|i| if i == 3 { json!([200.0, 300.0]) } else { json!([200.0 + f64::from(i) * 0.01, 900.0]) }).collect();
+        assert!(s.execute("path.erase", &json!({"id": id, "points": drag})).is_ok());
+    }
+
+    /// A drag of many repeated and many distinct samples is thinned and still erases where it went.
+    #[test]
+    fn a_drag_with_many_samples_still_erases() {
+        let mut s = Session::new();
+        s.execute("file.new", &json!({})).unwrap();
+        let pts: Vec<Value> = (0..=20).map(|i| json!({"p": [100.0 + i as f64 * 20.0, if i % 2 == 0 { 300.0 } else { 306.0 }]})).collect();
+        let id = s.execute("path.create", &json!({"anchors": pts})).unwrap()["id"].as_u64().unwrap();
+        // 60,000 samples along the middle of the path, each point repeated.
+        let drag: Vec<Value> = (0..60_000).map(|i| json!([200.0 + f64::from(i % 6000) * 0.025, 303.0])).collect();
+        let r = s.execute("path.erase", &json!({"id": id, "points": drag})).unwrap();
+        assert_eq!(r["subpaths"], 2, "a gap in the middle");
+    }
+
     /// The Direct Selection's selected points stay the same points when the path changes under
     /// them: a point added before them, one of them deleted, the path reversed.
     #[test]