diff --git a/crates/fj-kernel/src/algorithms/triangulation/mod.rs b/crates/fj-kernel/src/algorithms/triangulation/mod.rs
index aa207dc99..54b079612 100644
--- a/crates/fj-kernel/src/algorithms/triangulation/mod.rs
+++ b/crates/fj-kernel/src/algorithms/triangulation/mod.rs
@@ -32,7 +32,7 @@ pub fn triangulate(
                     .map(|vertex| {
                         // Can't panic, unless the approximation wrongfully
                         // generates points that are not in the surface.
-                        surface.point_model_to_surface(vertex)
+                        surface.convert_point_to_surface_coords(vertex)
                     })
                     .collect();
                 let face_as_polygon = Polygon::new(surface)
@@ -40,7 +40,9 @@ pub fn triangulate(
                         |point| {
                             // Can't panic, unless the approximation wrongfully
                             // generates points that are not in the surface.
-                            surface.point_model_to_surface(point).native()
+                            surface
+                                .convert_point_to_surface_coords(point)
+                                .native()
                         },
                     ))
                     .with_interiors(approx.interiors.into_iter().map(
@@ -49,7 +51,9 @@ pub fn triangulate(
                                 // Can't panic, unless the approximation
                                 // wrongfully generates points that are not in
                                 // the surface.
-                                surface.point_model_to_surface(point).native()
+                                surface
+                                    .convert_point_to_surface_coords(point)
+                                    .native()
                             })
                         },
                     ));
diff --git a/crates/fj-kernel/src/algorithms/triangulation/polygon.rs b/crates/fj-kernel/src/algorithms/triangulation/polygon.rs
index 5ab3f37d3..68fce75cb 100644
--- a/crates/fj-kernel/src/algorithms/triangulation/polygon.rs
+++ b/crates/fj-kernel/src/algorithms/triangulation/polygon.rs
@@ -140,7 +140,7 @@ impl Polygon {
         };
 
         let mut check = TriangleEdgeCheck::new(
-            self.surface.point_surface_to_model(&ray.origin),
+            self.surface.convert_point_from_surface_coords(&ray.origin),
         );
 
         let mut num_hits = 0;
@@ -201,7 +201,8 @@ impl Polygon {
 
                     let edge =
                         Segment::from_points(edge.points().map(|point| {
-                            self.surface.point_surface_to_model(&point)
+                            self.surface
+                                .convert_point_from_surface_coords(&point)
                         }));
                     check.hits.push(edge);
                 }
diff --git a/crates/fj-kernel/src/geometry/surfaces/mod.rs b/crates/fj-kernel/src/geometry/surfaces/mod.rs
index 0c88f1adc..ebcca1430 100644
--- a/crates/fj-kernel/src/geometry/surfaces/mod.rs
+++ b/crates/fj-kernel/src/geometry/surfaces/mod.rs
@@ -59,13 +59,13 @@ impl Surface {
     }
 
     /// Convert a point in model coordinates to surface coordinates
-    pub fn point_model_to_surface(
+    pub fn convert_point_to_surface_coords(
         &self,
         point_3d: Point<3>,
     ) -> geometry::Point<2> {
         let point_2d = match self {
             Self::SweptCurve(surface) => {
-                surface.point_model_to_surface(&point_3d)
+                surface.convert_point_to_surface_coords(&point_3d)
             }
         };
 
@@ -73,17 +73,25 @@ impl Surface {
     }
 
     /// Convert a point in surface coordinates to model coordinates
-    pub fn point_surface_to_model(&self, point: &Point<2>) -> Point<3> {
+    pub fn convert_point_from_surface_coords(
+        &self,
+        point: &Point<2>,
+    ) -> Point<3> {
         match self {
-            Self::SweptCurve(surface) => surface.point_surface_to_model(point),
+            Self::SweptCurve(surface) => {
+                surface.convert_point_from_surface_coords(point)
+            }
         }
     }
 
     /// Convert a vector in surface coordinates to model coordinates
-    pub fn vector_surface_to_model(&self, vector: &Vector<2>) -> Vector<3> {
+    pub fn convert_vector_from_surface_coords(
+        &self,
+        vector: &Vector<2>,
+    ) -> Vector<3> {
         match self {
             Self::SweptCurve(surface) => {
-                surface.vector_surface_to_model(vector)
+                surface.convert_vector_from_surface_coords(vector)
             }
         }
     }
diff --git a/crates/fj-kernel/src/geometry/surfaces/swept.rs b/crates/fj-kernel/src/geometry/surfaces/swept.rs
index c8d2b3513..ce9d5081e 100644
--- a/crates/fj-kernel/src/geometry/surfaces/swept.rs
+++ b/crates/fj-kernel/src/geometry/surfaces/swept.rs
@@ -1,4 +1,4 @@
-use fj_math::{Point, Transform, Vector};
+use fj_math::{Line, Point, Transform, Vector};
 
 use crate::geometry::Curve;
 
@@ -16,8 +16,6 @@ impl SweptCurve {
     /// Construct a plane from 3 points
     #[cfg(test)]
     pub fn plane_from_points([a, b, c]: [Point<3>; 3]) -> Self {
-        use fj_math::Line;
-
         let curve = Curve::Line(Line::from_points([a, b]));
         let path = c - a;
 
@@ -40,22 +38,36 @@ impl SweptCurve {
     }
 
     /// Convert a point in model coordinates to surface coordinates
-    pub fn point_model_to_surface(&self, point: &Point<3>) -> Point<2> {
+    pub fn convert_point_to_surface_coords(
+        &self,
+        point: &Point<3>,
+    ) -> Point<2> {
         let u = self.curve.point_model_to_curve(point).t;
-        let v = (point - self.curve.origin())
-            .scalar_projection_onto(&self.path)
-            / self.path.magnitude();
+        let v = {
+            let line = Line {
+                origin: self.curve.origin(),
+                direction: self.path,
+            };
+
+            line.convert_point_to_line_coords(point).t
+        };
 
         Point::from([u, v])
     }
 
     /// Convert a point in surface coordinates to model coordinates
-    pub fn point_surface_to_model(&self, point: &Point<2>) -> Point<3> {
+    pub fn convert_point_from_surface_coords(
+        &self,
+        point: &Point<2>,
+    ) -> Point<3> {
         self.curve.point_curve_to_model(&point.to_t()) + self.path * point.v
     }
 
     /// Convert a vector in surface coordinates to model coordinates
-    pub fn vector_surface_to_model(&self, vector: &Vector<2>) -> Vector<3> {
+    pub fn convert_vector_from_surface_coords(
+        &self,
+        vector: &Vector<2>,
+    ) -> Vector<3> {
         self.curve.vector_curve_to_model(&vector.to_t()) + self.path * vector.v
     }
 }
@@ -70,13 +82,13 @@ mod tests {
     use super::SweptCurve;
 
     #[test]
-    fn point_model_to_surface() {
+    fn convert_point_to_surface_coords() {
         let swept = SweptCurve {
             curve: Curve::Line(Line {
                 origin: Point::from([1., 0., 0.]),
                 direction: Vector::from([0., 2., 0.]),
             }),
-            path: Vector::from([0., 0., 2.]),
+            path: Vector::from([0., 0., 3.]),
         };
 
         verify(&swept, Point::from([-1., -1.]));
@@ -85,15 +97,15 @@ mod tests {
         verify(&swept, Point::from([2., 3.]));
 
         fn verify(swept: &SweptCurve, surface_point: Point<2>) {
-            let point = swept.point_surface_to_model(&surface_point);
-            let result = swept.point_model_to_surface(&point);
+            let point = swept.convert_point_from_surface_coords(&surface_point);
+            let result = swept.convert_point_to_surface_coords(&point);
 
             assert_eq!(result, surface_point);
         }
     }
 
     #[test]
-    fn point_surface_to_model() {
+    fn convert_point_from_surface_coords() {
         let swept = SweptCurve {
             curve: Curve::Line(Line {
                 origin: Point::from([1., 0., 0.]),
@@ -103,13 +115,13 @@ mod tests {
         };
 
         assert_eq!(
-            swept.point_surface_to_model(&Point::from([2., 4.])),
+            swept.convert_point_from_surface_coords(&Point::from([2., 4.])),
             Point::from([1., 4., 8.]),
         );
     }
 
     #[test]
-    fn vector_surface_to_model() {
+    fn convert_vector_from_surface_coords() {
         let swept = SweptCurve {
             curve: Curve::Line(Line {
                 origin: Point::from([1., 0., 0.]),
@@ -119,7 +131,7 @@ mod tests {
         };
 
         assert_eq!(
-            swept.vector_surface_to_model(&Vector::from([2., 4.])),
+            swept.convert_vector_from_surface_coords(&Vector::from([2., 4.])),
             Vector::from([0., 4., 8.]),
         );
     }
diff --git a/crates/fj-operations/src/sketch.rs b/crates/fj-operations/src/sketch.rs
index b822bd99b..4f8306196 100644
--- a/crates/fj-operations/src/sketch.rs
+++ b/crates/fj-operations/src/sketch.rs
@@ -15,7 +15,7 @@ impl ToShape for fj::Sketch {
             .to_points()
             .into_iter()
             .map(Point::from)
-            .map(|point| surface.point_surface_to_model(&point));
+            .map(|point| surface.convert_point_from_surface_coords(&point));
 
         Face::builder(surface, &mut shape)
             .with_exterior_polygon(points)