diff --git a/src/kernel/geometry/curves/circle.rs b/src/kernel/geometry/curves/circle.rs
index cfa96b4a5..6cad83bd4 100644
--- a/src/kernel/geometry/curves/circle.rs
+++ b/src/kernel/geometry/curves/circle.rs
@@ -35,13 +35,13 @@ impl Circle {
     /// Converts the provided point into curve coordinates between `0.`
     /// (inclusive) and `PI * 2.` (exclusive).
     ///
-    /// Ignores the radius, meaning points that are not on the circle will be
-    /// converted to the curve coordinate of their projection on the circle.
+    /// Projects the point onto the circle before computing curve coordinate,
+    /// ignoring the radius. This is done to make this method robust against
+    /// floating point accuracy issues.
     ///
-    /// This is done to make this method robust against floating point accuracy
-    /// issues. Callers are advised to be careful about the points they pass, as
-    /// the point not being on the circle, intended or not, will not result in
-    /// an error.
+    /// Callers are advised to be careful about the points they pass, as the
+    /// point not being on the curve, intentional or not, will not result in an
+    /// error.
     pub fn point_model_to_curve(&self, point: &Point<3>) -> Point<1> {
         let v = point - self.center;
         let atan = f64::atan2(v.y, v.x);
@@ -49,6 +49,24 @@ impl Circle {
         point![coord]
     }
 
+    /// Convert a point on the curve into model coordinates
+    pub fn point_curve_to_model(&self, point: &Point<1>) -> Point<3> {
+        self.center + self.vector_curve_to_model(&point.coords)
+    }
+
+    /// Convert a vector on the curve into model coordinates
+    pub fn vector_curve_to_model(&self, point: &Vector<1>) -> Vector<3> {
+        let radius = self.radius.magnitude();
+        let angle = point.x;
+
+        let (sin, cos) = angle.sin_cos();
+
+        let x = cos * radius;
+        let y = sin * radius;
+
+        vector![x, y, 0.]
+    }
+
     pub fn approx(&self, tolerance: f64, out: &mut Vec<Point<3>>) {
         let radius = self.radius.magnitude();
 
@@ -62,14 +80,7 @@ impl Circle {
 
         for i in 0..n {
             let angle = 2. * PI / n as f64 * i as f64;
-
-            let (sin, cos) = angle.sin_cos();
-
-            let x = cos * radius;
-            let y = sin * radius;
-
-            let point = self.center + vector![x, y, 0.];
-
+            let point = self.point_curve_to_model(&point![angle]);
             out.push(point);
         }
     }
@@ -93,7 +104,7 @@ mod tests {
     use super::Circle;
 
     #[test]
-    fn test_point_model_to_curve() {
+    fn point_model_to_curve() {
         let circle = Circle {
             center: point![1., 2., 3.],
             radius: vector![1., 0.],
@@ -118,7 +129,7 @@ mod tests {
     }
 
     #[test]
-    fn test_number_of_vertices() {
+    fn number_of_vertices() {
         verify_result(50., 100., 3);
         verify_result(10., 100., 7);
         verify_result(1., 100., 23);
diff --git a/src/kernel/geometry/curves/line.rs b/src/kernel/geometry/curves/line.rs
index c9d6fa45b..449ba2467 100644
--- a/src/kernel/geometry/curves/line.rs
+++ b/src/kernel/geometry/curves/line.rs
@@ -30,13 +30,12 @@ impl Line {
 
     /// Convert a point in model coordinates to curve coordinates
     ///
-    /// Ignores the distance of the point to the line, meaning points on the
-    /// line will be converted to the curve coordinates of their projection on
-    /// the line.
+    /// Projects the point onto the line before computing curve coordinate. This
+    /// is done to make this method robust against floating point accuracy
+    /// issues.
     ///
-    /// This is done to make this method robust against floating point accuracy
-    /// issues. Callers are advised to be careful about the points they pass, as
-    /// the point not being on the line, intended or not, will not result in an
+    /// Callers are advised to be careful about the points they pass, as the
+    /// point not being on the line, intentional or not, will never result in an
     /// error.
     pub fn point_model_to_curve(&self, point: &Point<3>) -> Point<1> {
         let p = point - self.origin;
@@ -46,6 +45,19 @@ impl Line {
 
         point![t]
     }
+
+    /// Convert a point on the curve into model coordinates
+    #[cfg(test)]
+    pub fn point_curve_to_model(&self, point: &Point<1>) -> Point<3> {
+        self.origin + self.vector_curve_to_model(&point.coords)
+    }
+
+    /// Convert a vector on the curve into model coordinates
+    #[cfg(test)]
+    pub fn vector_curve_to_model(&self, point: &Vector<1>) -> Vector<3> {
+        let t = point.x;
+        self.direction * t
+    }
 }
 
 impl AbsDiffEq for Line {
@@ -74,7 +86,7 @@ mod tests {
     use super::Line;
 
     #[test]
-    fn test_transform() {
+    fn transform() {
         let line = Line {
             origin: point![1., 0., 0.],
             direction: vector![0., 1., 0.],
@@ -96,7 +108,7 @@ mod tests {
     }
 
     #[test]
-    fn test_point_model_to_curve() {
+    fn point_model_to_curve() {
         let line = Line {
             origin: point![1., 0., 0.],
             direction: vector![2., 0., 0.],
@@ -108,7 +120,7 @@ mod tests {
         verify(line, 2.);
 
         fn verify(line: Line, t: f64) {
-            let point = line.origin + line.direction * t;
+            let point = line.point_curve_to_model(&point![t]);
             let t_result = line.point_model_to_curve(&point);
 
             assert_eq!(point![t], t_result);
diff --git a/src/kernel/geometry/curves/mod.rs b/src/kernel/geometry/curves/mod.rs
index 2afbfa1a8..8745429c6 100644
--- a/src/kernel/geometry/curves/mod.rs
+++ b/src/kernel/geometry/curves/mod.rs
@@ -36,8 +36,8 @@ impl Curve {
     #[must_use]
     pub fn transform(self, transform: &Isometry<f64>) -> Self {
         match self {
-            Self::Circle(circle) => Self::Circle(circle.transform(transform)),
-            Self::Line(line) => Self::Line(line.transform(transform)),
+            Self::Circle(curve) => Self::Circle(curve.transform(transform)),
+            Self::Line(curve) => Self::Line(curve.transform(transform)),
 
             #[cfg(test)]
             Self::Mock { .. } => todo!(),
@@ -46,18 +46,17 @@ impl Curve {
 
     /// Convert a point in model coordinates to curve coordinates
     ///
-    /// Whether the point is actually on the curve or not will be ignored. The
-    /// curve coordinates of the projection of the point on the curve will be
-    /// returned.
-    ///
+    /// Projects the point onto the curve before computing curve coordinate.
     /// This is done to make this method robust against floating point accuracy
-    /// issues. Callers are advised to be careful about the points they pass, as
-    /// the point not being on the curve, intended or not, will not result in an
-    /// error.
+    /// issues.
+    ///
+    /// Callers are advised to be careful about the points they pass, as the
+    /// point not being on the curve, intentional or not, will never result in
+    /// an error.
     pub fn point_model_to_curve(&self, point: &Point<3>) -> Point<1> {
         match self {
-            Self::Circle(circle) => circle.point_model_to_curve(point),
-            Self::Line(line) => line.point_model_to_curve(point),
+            Self::Circle(curve) => curve.point_model_to_curve(point),
+            Self::Line(curve) => curve.point_model_to_curve(point),
 
             #[cfg(test)]
             Self::Mock { coords, .. } => coords.borrow_mut().remove(0),