diff --git a/crates/fj-core/src/geometry/curve.rs b/crates/fj-core/src/geometry/curve.rs
index 9b5100581..3d036ff68 100644
--- a/crates/fj-core/src/geometry/curve.rs
+++ b/crates/fj-core/src/geometry/curve.rs
@@ -2,7 +2,11 @@ use std::collections::BTreeMap;
 
 use fj_math::{Circle, Line, Point};
 
-use crate::{storage::Handle, topology::Surface};
+use crate::{
+    algorithms::approx::{PathApproxParams, Tolerance},
+    storage::Handle,
+    topology::Surface,
+};
 
 use super::Path;
 
@@ -68,18 +72,54 @@ pub struct LocalCurveGeom {
 pub trait CurveGeom2<const D: usize> {
     /// # Access the origin of the curve
     fn origin(&self) -> Point<D>;
+
+    /// # Compute a line segment to approximate the curve at this point
+    ///
+    /// If the curve requires no approximation (meaning it is a line), then per
+    /// convention, a degenerate line segment is returned, that collapses to the
+    /// provided point.
+    fn line_segment_at(
+        &self,
+        point: impl Into<Point<1>>,
+        tolerance: impl Into<Tolerance>,
+    ) -> [Point<D>; 2];
 }
 
 impl<const D: usize> CurveGeom2<D> for Circle<D> {
     fn origin(&self) -> Point<D> {
         self.center() + self.a()
     }
+
+    fn line_segment_at(
+        &self,
+        point: impl Into<Point<1>>,
+        tolerance: impl Into<Tolerance>,
+    ) -> [Point<D>; 2] {
+        let point = point.into();
+        let params = PathApproxParams::for_circle(self, tolerance);
+
+        [point.t - params.increment(), point.t + params.increment()]
+            .map(|point_circle| self.point_from_circle_coords([point_circle]))
+    }
 }
 
 impl<const D: usize> CurveGeom2<D> for Line<D> {
     fn origin(&self) -> Point<D> {
         self.origin()
     }
+
+    fn line_segment_at(
+        &self,
+        point: impl Into<Point<1>>,
+        _: impl Into<Tolerance>,
+    ) -> [Point<D>; 2] {
+        let point = point.into();
+
+        // Collapse line segment into a point, as per documentation.
+        let point = self.origin() + self.direction() * point.t;
+
+        [point, point]
+    }
 }
 
 // This implementation is temporary, to ease the transition towards a curve
@@ -91,4 +131,15 @@ impl<const D: usize> CurveGeom2<D> for Path<D> {
             Self::Line(line) => line.origin(),
         }
     }
+
+    fn line_segment_at(
+        &self,
+        point: impl Into<Point<1>>,
+        tolerance: impl Into<Tolerance>,
+    ) -> [Point<D>; 2] {
+        match self {
+            Self::Circle(circle) => circle.line_segment_at(point, tolerance),
+            Self::Line(line) => line.line_segment_at(point, tolerance),
+        }
+    }
 }
diff --git a/crates/fj-core/src/geometry/surface.rs b/crates/fj-core/src/geometry/surface.rs
index 40e5da5a5..7f564444b 100644
--- a/crates/fj-core/src/geometry/surface.rs
+++ b/crates/fj-core/src/geometry/surface.rs
@@ -2,7 +2,7 @@
 
 use fj_math::{Point, Scalar, Transform, Triangle, Vector};
 
-use crate::algorithms::approx::{PathApproxParams, Tolerance};
+use crate::algorithms::approx::Tolerance;
 
 use super::{CurveGeom2, Path};
 
@@ -67,23 +67,10 @@ impl SurfaceGeom {
 
         let line_segment = match &self.u {
             Path::Circle(circle) => {
-                let params = PathApproxParams::for_circle(circle, tolerance);
-
-                [
-                    point_surface.u - params.increment(),
-                    point_surface.u + params.increment(),
-                ]
-                .map(|point_circle| {
-                    circle.point_from_circle_coords([point_circle])
-                })
+                circle.line_segment_at([point_surface.u], tolerance)
             }
             Path::Line(line) => {
-                // We don't need to approximate a line. So instead of creating a
-                // line segment to represent the line at this point, we just
-                // need this single point.
-                let point = line.origin() + line.direction() * point_surface.u;
-
-                [point, point]
+                line.line_segment_at([point_surface.u], tolerance)
             }
         };