diff --git a/crates/geom/src/curve.rs b/crates/geom/src/curve.rs
index c6be8da..9d45453 100644
--- a/crates/geom/src/curve.rs
+++ b/crates/geom/src/curve.rs
@@ -256,9 +256,22 @@ impl Ellipse {
         }
     }
     pub fn bounds(&self) -> Bounds2 {
-        let mut pts = Vec::new();
-        self.tessellate(self.major.len() * 1e-4, &mut pts);
-        Bounds2::from_points(pts)
+        // Sampled bounds can cut off the true extrema, especially on a small arc or
+        // a large drawing. The derivative of each coordinate gives two exact
+        // candidate parameters (opposite sides of the ellipse).
+        let minor = self.minor();
+        let mut bounds = Bounds2::from_points([self.at_param(self.start), self.at_param(self.end)]);
+        for angle in [
+            minor.x.atan2(self.major.x),
+            minor.x.atan2(self.major.x) + PI,
+            minor.y.atan2(self.major.y),
+            minor.y.atan2(self.major.y) + PI,
+        ] {
+            if angle_in_sweep(angle, self.start, self.end) {
+                bounds.add(self.at_param(angle));
+            }
+        }
+        bounds
     }
     /// Parameter of the point on the ellipse nearest the direction of `p` (approximate).
     pub fn param_of(&self, p: Vec2) -> f64 {
diff --git a/crates/geom/src/tests.rs b/crates/geom/src/tests.rs
index d6f1320..a9f5daf 100644
--- a/crates/geom/src/tests.rs
+++ b/crates/geom/src/tests.rs
@@ -206,3 +206,33 @@ fn circle_three_points_accepts_small_valid_triangles() {
     assert!(Circle::from_3_points(Vec2::ZERO, Vec2::new(r, 0.0), Vec2::new(2.0 * r, 0.0)).is_none());
     assert!(Circle::from_3_points(Vec2::ZERO, Vec2::X, Vec2::new(f64::NAN, 0.0)).is_none());
 }
+
+#[test]
+fn ellipse_bounds_find_analytic_extrema() {
+    // A tilted ellipse's extrema generally fall between tessellated samples.
+    let center = Vec2::new(7.0, -11.0);
+    let e = Ellipse::full(center, Vec2::new(3.0, 4.0), 0.5);
+    let bounds = e.bounds();
+    let dx = 13.0_f64.sqrt(); // hypot(major.x, minor.x)
+    let dy = 18.25_f64.sqrt(); // hypot(major.y, minor.y)
+    assert!((bounds.min.x - (center.x - dx)).abs() < 1e-12);
+    assert!((bounds.max.x - (center.x + dx)).abs() < 1e-12);
+    assert!((bounds.min.y - (center.y - dy)).abs() < 1e-12);
+    assert!((bounds.max.y - (center.y + dy)).abs() < 1e-12);
+}
+
+#[test]
+fn elliptical_arc_bounds_exclude_extrema_outside_the_sweep() {
+    let e = Ellipse {
+        center: Vec2::ZERO,
+        major: Vec2::new(3.0, 4.0),
+        ratio: 0.5,
+        start: 0.0,
+        end: PI / 4.0,
+    };
+    let bounds = e.bounds();
+    assert!((bounds.max.x - 3.0).abs() < 1e-12);
+    assert!((bounds.min.x - e.at_param(e.end).x).abs() < 1e-12);
+    assert!((bounds.min.y - e.at_param(e.end).y).abs() < 1e-12);
+    assert!((bounds.max.y - 18.25_f64.sqrt()).abs() < 1e-12);
+}