diff --git a/fj-kernel/src/algorithms/triangulation/mod.rs b/fj-kernel/src/algorithms/triangulation/mod.rs
index bacb48f01..611380552 100644
--- a/fj-kernel/src/algorithms/triangulation/mod.rs
+++ b/fj-kernel/src/algorithms/triangulation/mod.rs
@@ -35,13 +35,23 @@ pub fn triangulate(
                     })
                     .collect();
                 let face_as_polygon = Polygon::new(surface)
-                    .with_exterior(approx.exterior.points)
-                    .with_interiors(
-                        approx
-                            .interiors
-                            .into_iter()
-                            .map(|interior| interior.points),
-                    );
+                    .with_exterior(approx.exterior.points.into_iter().map(
+                        |point| {
+                            // Can't panic, unless the approximation wrongfully
+                            // generates points that are not in the surface.
+                            surface.point_model_to_surface(point).native()
+                        },
+                    ))
+                    .with_interiors(approx.interiors.into_iter().map(
+                        |interior| {
+                            interior.points.into_iter().map(|point| {
+                                // Can't panic, unless the approximation
+                                // wrongfully generates points that are not in
+                                // the surface.
+                                surface.point_model_to_surface(point).native()
+                            })
+                        },
+                    ));
 
                 let mut triangles = delaunay(points);
                 triangles.retain(|triangle| {
diff --git a/fj-kernel/src/algorithms/triangulation/polygon.rs b/fj-kernel/src/algorithms/triangulation/polygon.rs
index c86ed3f15..7ee76fa53 100644
--- a/fj-kernel/src/algorithms/triangulation/polygon.rs
+++ b/fj-kernel/src/algorithms/triangulation/polygon.rs
@@ -22,13 +22,13 @@ impl Polygon {
         }
     }
 
-    pub fn with_exterior(self, exterior: impl Into<PolyChain<3>>) -> Self {
+    pub fn with_exterior(self, exterior: impl Into<PolyChain<2>>) -> Self {
         self.with_approx(exterior)
     }
 
     pub fn with_interiors(
         mut self,
-        interiors: impl IntoIterator<Item = impl Into<PolyChain<3>>>,
+        interiors: impl IntoIterator<Item = impl Into<PolyChain<2>>>,
     ) -> Self {
         for interior in interiors {
             self = self.with_approx(interior);
@@ -37,18 +37,14 @@ impl Polygon {
         self
     }
 
-    fn with_approx(mut self, approx: impl Into<PolyChain<3>>) -> Self {
+    fn with_approx(mut self, approx: impl Into<PolyChain<2>>) -> Self {
         for segment in approx.into().segments() {
             let segment = segment.points().map(|point| {
-                // Can't panic, unless the approximation wrongfully generates
-                // points that are not in the surface.
-                let point = self.surface.point_model_to_surface(point);
-
-                if point.native() > self.max {
-                    self.max = point.native();
+                if point > self.max {
+                    self.max = point;
                 }
 
-                point.native()
+                point
             });
 
             let edge = Segment::from(segment);