diff --git a/crates/fj-kernel/src/algorithms/intersect/face_face.rs b/crates/fj-kernel/src/algorithms/intersect/face_face.rs
index 920cd76a7..7e6a23b80 100644
--- a/crates/fj-kernel/src/algorithms/intersect/face_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/face_face.rs
@@ -36,9 +36,7 @@ impl FaceFaceIntersection {
 
         let intersection_curves =
             match SurfaceSurfaceIntersection::compute(surfaces, objects) {
-                Some(intersection) => {
-                    intersection.intersection_curves.map(|curve| curve.path())
-                }
+                Some(intersection) => intersection.intersection_curves,
                 None => return None,
             };
 
diff --git a/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs b/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs
index 9406e92d4..4b71b2e1a 100644
--- a/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs
@@ -2,8 +2,7 @@ use fj_math::{Line, Plane, Point, Scalar};
 
 use crate::{
     geometry::path::{GlobalPath, SurfacePath},
-    insert::Insert,
-    objects::{Curve, Objects, Surface},
+    objects::{Objects, Surface},
     services::Service,
     storage::Handle,
 };
@@ -12,14 +11,14 @@ use crate::{
 #[derive(Clone, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
 pub struct SurfaceSurfaceIntersection {
     /// The intersection curves
-    pub intersection_curves: [Handle<Curve>; 2],
+    pub intersection_curves: [SurfacePath; 2],
 }
 
 impl SurfaceSurfaceIntersection {
     /// Compute the intersection between two surfaces
     pub fn compute(
         surfaces: [Handle<Surface>; 2],
-        objects: &mut Service<Objects>,
+        _: &mut Service<Objects>,
     ) -> Option<Self> {
         // Algorithm from Real-Time Collision Detection by Christer Ericson. See
         // section 5.4.4, Intersection of Two Planes.
@@ -53,10 +52,8 @@ impl SurfaceSurfaceIntersection {
 
         let line = Line::from_origin_and_direction(origin, direction);
 
-        let curves = planes.map(|plane| {
-            let path = SurfacePath::Line(plane.project_line(&line));
-            Curve::new(path).insert(objects)
-        });
+        let curves =
+            planes.map(|plane| SurfacePath::Line(plane.project_line(&line)));
 
         Some(Self {
             intersection_curves: curves,
@@ -83,10 +80,7 @@ mod tests {
     use pretty_assertions::assert_eq;
 
     use crate::{
-        algorithms::transform::TransformObject,
-        builder::CurveBuilder,
-        insert::Insert,
-        partial::{PartialCurve, PartialObject},
+        algorithms::transform::TransformObject, geometry::path::SurfacePath,
         services::Services,
     };
 
@@ -114,17 +108,8 @@ mod tests {
             None,
         );
 
-        let mut expected_xy = PartialCurve::default();
-        expected_xy.update_as_u_axis();
-        let expected_xy = expected_xy
-            .build(&mut services.objects)
-            .insert(&mut services.objects);
-
-        let mut expected_xz = PartialCurve::default();
-        expected_xz.update_as_u_axis();
-        let expected_xz = expected_xz
-            .build(&mut services.objects)
-            .insert(&mut services.objects);
+        let expected_xy = SurfacePath::u_axis();
+        let expected_xz = SurfacePath::u_axis();
 
         assert_eq!(
             SurfaceSurfaceIntersection::compute(