diff --git a/crates/engine/src/cmd/sketch_constraints_tests.rs b/crates/engine/src/cmd/sketch_constraints_tests.rs
index 8ebf5714..ebfbde03 100644
--- a/crates/engine/src/cmd/sketch_constraints_tests.rs
+++ b/crates/engine/src/cmd/sketch_constraints_tests.rs
@@ -187,3 +187,16 @@ fn drawing_from_a_line_midpoint_keeps_it_there() {
     let b = sk.resolve_point(&format!("{l}.end")).and_then(|i| sk.point(i)).unwrap();
     assert!(st.dist((a + b) * 0.5) < 1e-7, "{st:?}");
 }
+
+#[test]
+fn drawing_onto_a_curve_keeps_the_point_on_it() {
+    let mut s = new_sketch();
+    let c = ids(&run(&mut s, "CircleCenterRadius", json!({"center": [0, 0], "radius": 10}))["curves"])[0].clone();
+    let r = run(&mut s, "DrawPolyline", json!({"points": [format!("on:{c}:7.2,7.1"), [30, 30]]}));
+    let n = ids(&r["curves"])[0].clone();
+    run(&mut s, "SketchDimension", json!({"entities": [c], "value": 16}));
+    let sk = sketch(&s);
+    let st = sk.resolve_point(&format!("{n}.start")).and_then(|i| sk.point(i)).unwrap();
+    let cc = sk.center(sk.curve_index(&c).unwrap()).unwrap();
+    assert!((st.dist(cc) - 8.0).abs() < 1e-7, "{st:?} {cc:?}");
+}
diff --git a/crates/engine/src/cmd/sketch_project.rs b/crates/engine/src/cmd/sketch_project.rs
index bbc09a21..73b11d89 100644
--- a/crates/engine/src/cmd/sketch_project.rs
+++ b/crates/engine/src/cmd/sketch_project.rs
@@ -347,7 +347,16 @@ pub(super) fn snap_points(s: &mut Session, p: &Value, cmd: &str) -> Result<Value
             }
         });
         let mid = v.as_str().and_then(|x| x.strip_prefix("mid:")).map(str::to_string);
-        if vertex.is_none() && edge.is_none() && mid.is_none() {
+        // "on:<curve>:x,y": a point on a curve of the sketch near x,y.
+        let on = v.as_str().and_then(|x| x.strip_prefix("on:")).and_then(|t| {
+            let (c, xy) = t.rsplit_once(':')?;
+            let n: Vec<f64> = xy.split(',').filter_map(|x| x.trim().parse().ok()).collect();
+            match n[..] {
+                [x, y] if x.is_finite() && y.is_finite() => Some((c.to_string(), solvecraft_geom::Vec2::new(x, y))),
+                _ => None,
+            }
+        });
+        if vertex.is_none() && edge.is_none() && mid.is_none() && on.is_none() {
             return Ok(());
         }
         if doc.is_none() {
@@ -358,7 +367,16 @@ pub(super) fn snap_points(s: &mut Session, p: &Value, cmd: &str) -> Result<Value
         }
         let Some((d, sk)) = doc.as_mut() else { return Ok(()) };
         let body = str_(v, "body").unwrap_or("").to_string();
-        let pid = if let Some(line) = mid {
+        let pid = if let Some((curve, near)) = on {
+            let c = sk.curve_index(&curve).ok_or_else(|| bad(cmd, format!("unknown curve `{curve}`")))?;
+            let at = match sk.shape(c) {
+                Some(sh) => sh.project(near),
+                None => near,
+            };
+            let pi = sk.add_point(at, None)?;
+            sk.add_constraint(ConstraintKind::PointOnCurve { p: pi, c }, None)?;
+            sk.points.get(pi).map(|q| q.id.clone()).unwrap_or_default()
+        } else if let Some(line) = mid {
             // "mid:<line>": a new point at the line's midpoint, kept there.
             let l = sk.curve_index(&line).ok_or_else(|| bad(cmd, format!("unknown curve `{line}`")))?;
             let Some(solvecraft_sketch::Shape::Line { a, b }) = sk.shape(l) else { return Err(bad(cmd, format!("`{line}` is not a line"))) };
diff --git a/crates/ui-egui/src/sketch_tools.rs b/crates/ui-egui/src/sketch_tools.rs
index c80d01a9..2c8e615c 100644
--- a/crates/ui-egui/src/sketch_tools.rs
+++ b/crates/ui-egui/src/sketch_tools.rs
@@ -71,6 +71,18 @@ pub fn refine_snap(
         }
         None => PREV.with(Cell::get),
     };
+    // On a curve (coincident with it).
+    if best.is_none() {
+        for (i, c) in sk.curves.iter().enumerate() {
+            let Some(sh) = sk.shape(i) else { continue };
+            let d = sh.dist(lp);
+            if d < tol * 0.8 && best.is_none_or(|x| d < x.0) {
+                let q = sh.project(lp);
+                best = Some((d, "On", q));
+                mid_of = Some(format!("{}:{},{}", c.id, q.x, q.y));
+            }
+        }
+    }
     if best.is_none()
         && let Some(prev) = prev
     {
@@ -86,7 +98,14 @@ pub fn refine_snap(
     }
     SNAP.with(|s| *s.borrow_mut() = best.map(|(_, l, p)| (l, p)));
     // A midpoint snap is a point argument the commands understand ("mid:<line>").
-    best.map(|(_, l, p)| (p, if l == "Mid" { mid_of.map(|m| format!("mid:{m}")) } else { None }))
+    best.map(|(_, l, p)| {
+        let arg = match l {
+            "Mid" => mid_of.map(|m| format!("mid:{m}")),
+            "On" => mid_of.map(|m| format!("on:{m}")),
+            _ => None,
+        };
+        (p, arg)
+    })
 }
 
 /// A click on a constraint glyph selects it (true: the click was used).