diff --git a/geometry/src/outline.rs b/geometry/src/outline.rs
index f46b2be4..2cb65a13 100644
--- a/geometry/src/outline.rs
+++ b/geometry/src/outline.rs
@@ -385,12 +385,15 @@ impl Contour {
         loop {
             let mut sweep_vector = end_vector.rev_rotate_by(vector);
             let last = sweep_vector.0.x() >= -EPSILON && sweep_vector.0.y() >= -EPSILON;
+
+            let mut segment;
             if !last {
                 sweep_vector = UnitVector(Point2DF::new(0.0, 1.0));
+                segment = Segment::quarter_circle_arc();
+            } else {
+                segment = Segment::arc_from_cos(sweep_vector.0.x());
             }
 
-            // TODO(pcwalton): Cache 90 degree arc segments so we aren't calling this all the time.
-            let mut segment = Segment::arc_from_cos(sweep_vector.0.x());
             let rotation =
                 Transform2DF::from_rotation_vector(sweep_vector.halve_angle().rotate_by(vector));
             segment = segment.transform(&rotation.post_mul(&transform));
@@ -413,23 +416,15 @@ impl Contour {
     }
 
     pub fn push_ellipse(&mut self, transform: &Transform2DF) {
-        let rotations = [
-            Transform2DF::default(),
-            Transform2DF::from_rotation_vector(UnitVector(Point2DF::new( 0.0,  1.0))),
-            Transform2DF::from_rotation_vector(UnitVector(Point2DF::new(-1.0,  0.0))),
-            Transform2DF::from_rotation_vector(UnitVector(Point2DF::new( 0.0, -1.0))),
-        ];
-
-        let base_segment = Segment::arc_from_cos(0.0);
-
-        for (rotation_index, rotation) in rotations.iter().enumerate() {
-            let segment = base_segment.transform(&rotation.post_mul(&transform));
-            if rotation_index == 0 {
-                self.push_full_segment(&segment, true);
-            } else {
-                self.push_segment(segment, true);
-            }
-        }
+        let segment = Segment::quarter_circle_arc();
+        let mut rotation;
+        self.push_full_segment(&segment.transform(transform), true);
+        rotation = Transform2DF::from_rotation_vector(UnitVector(Point2DF::new( 0.0,  1.0)));
+        self.push_segment(segment.transform(&rotation.post_mul(&transform)), true);
+        rotation = Transform2DF::from_rotation_vector(UnitVector(Point2DF::new(-1.0,  0.0)));
+        self.push_segment(segment.transform(&rotation.post_mul(&transform)), true);
+        rotation = Transform2DF::from_rotation_vector(UnitVector(Point2DF::new( 0.0, -1.0)));
+        self.push_segment(segment.transform(&rotation.post_mul(&transform)), true);
     }
 
     #[inline]
diff --git a/geometry/src/segment.rs b/geometry/src/segment.rs
index 4ef5620e..d71ee473 100644
--- a/geometry/src/segment.rs
+++ b/geometry/src/segment.rs
@@ -94,6 +94,15 @@ impl Segment {
         const K: f32 = 4.0 / 3.0 * (SQRT_2 - 1.0);
     }
 
+    #[inline]
+    pub fn quarter_circle_arc() -> Segment {
+        let p0 = Point2DF::splat(SQRT_2 * 0.5);
+        let p1 = Point2DF::new(-SQRT_2 / 6.0 + 4.0 / 3.0, 7.0 * SQRT_2 / 6.0 - 4.0 / 3.0);
+        let flip = Point2DF::new(1.0, -1.0);
+        let (p2, p3) = (p1.scale_xy(flip), p0.scale_xy(flip));
+        Segment::cubic(&LineSegmentF::new(p3, p0), &LineSegmentF::new(p2, p1))
+    }
+
     #[inline]
     pub fn as_line_segment(&self) -> LineSegmentF {
         debug_assert!(self.is_line());
diff --git a/geometry/src/stroke.rs b/geometry/src/stroke.rs
index fe70d67d..a3acee91 100644
--- a/geometry/src/stroke.rs
+++ b/geometry/src/stroke.rs
@@ -122,6 +122,7 @@ impl<'a> OutlineStrokeToFill<'a> {
 
         match self.style.line_cap {
             LineCap::Butt => unreachable!(),
+
             LineCap::Square => {
                 let offset = gradient.scale(width * 0.5);
 
@@ -133,6 +134,7 @@ impl<'a> OutlineStrokeToFill<'a> {
                 contour.push_endpoint(p3);
                 contour.push_endpoint(p4);
             }
+
             LineCap::Round => {
                 let scale = Point2DF::splat(width * 0.5);
                 let offset = gradient.yx().scale_xy(Point2DF::new(-1.0, 1.0));