diff --git a/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs b/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs
index 969f5736f..0f643519e 100644
--- a/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs
@@ -15,7 +15,7 @@ pub struct SurfaceSurfaceIntersection {
 
 impl SurfaceSurfaceIntersection {
     /// Compute the intersection between two surfaces
-    pub fn compute(surfaces: [&Surface; 2], store: &Stores) -> Option<Self> {
+    pub fn compute(surfaces: [&Surface; 2], stores: &Stores) -> Option<Self> {
         // Algorithm from Real-Time Collision Detection by Christer Ericson. See
         // section 5.4.4, Intersection of Two Planes.
         //
@@ -52,14 +52,14 @@ impl SurfaceSurfaceIntersection {
         let line = Line::from_origin_and_direction(origin, direction);
 
         let curves = planes_parametric.map(|(surface, plane)| {
-            let local = project_line_into_plane(&line, &plane);
-            let global = store.global_curves.insert(GlobalCurve::from_path(
-                GlobalPath::Line(Line::from_origin_and_direction(
-                    origin, direction,
+            let path = project_line_into_plane(&line, &plane);
+            let global_form = stores.global_curves.insert(
+                GlobalCurve::from_path(GlobalPath::Line(
+                    Line::from_origin_and_direction(origin, direction),
                 )),
-            ));
+            );
 
-            Curve::new(surface, local, global)
+            Curve::new(surface, path, global_form)
         });
 
         Some(Self {
diff --git a/crates/fj-kernel/src/algorithms/validate/coherence.rs b/crates/fj-kernel/src/algorithms/validate/coherence.rs
index 91850c7c9..dc9b4581c 100644
--- a/crates/fj-kernel/src/algorithms/validate/coherence.rs
+++ b/crates/fj-kernel/src/algorithms/validate/coherence.rs
@@ -48,11 +48,11 @@ pub fn validate_vertex(
     // lies on the curve.
 
     let local = vertex.position();
+    let local_as_surface = vertex.curve().path().point_from_path_coords(local);
     let local_as_global = vertex
         .curve()
-        .global_form()
-        .path()
-        .point_from_path_coords(local);
+        .surface()
+        .point_from_surface_coords(local_as_surface);
     let global = vertex.global_form().position();
     let distance = (local_as_global - global).magnitude();
 
diff --git a/crates/fj-kernel/src/builder/curve.rs b/crates/fj-kernel/src/builder/curve.rs
index a9e0b3f3d..db9e5f5df 100644
--- a/crates/fj-kernel/src/builder/curve.rs
+++ b/crates/fj-kernel/src/builder/curve.rs
@@ -51,15 +51,15 @@ impl<'a> CurveBuilder<'a> {
     pub fn line_from_points(&self, points: [impl Into<Point<2>>; 2]) -> Curve {
         let points = points.map(Into::into);
 
-        let local = Line::from_points(points);
-        let global = self.stores.global_curves.insert(GlobalCurve::from_path(
-            GlobalPath::Line(Line::from_points(
+        let path = SurfacePath::Line(Line::from_points(points));
+        let global_form = self.stores.global_curves.insert(
+            GlobalCurve::from_path(GlobalPath::Line(Line::from_points(
                 points
                     .map(|point| self.surface.point_from_surface_coords(point)),
-            )),
-        ));
+            ))),
+        );
 
-        Curve::new(self.surface, SurfacePath::Line(local), global)
+        Curve::new(self.surface, path, global_form)
     }
 }
 
diff --git a/crates/fj-kernel/src/builder/edge.rs b/crates/fj-kernel/src/builder/edge.rs
index e5b074c9a..26df2eb7b 100644
--- a/crates/fj-kernel/src/builder/edge.rs
+++ b/crates/fj-kernel/src/builder/edge.rs
@@ -32,9 +32,8 @@ impl<'a> HalfEdgeBuilder<'a> {
             let [a_curve, b_curve] =
                 [Scalar::ZERO, Scalar::TAU].map(|coord| Point::from([coord]));
 
-            let global_vertex = GlobalVertex::from_position(
-                curve.global_form().path().point_from_path_coords(a_curve),
-            );
+            let global_vertex =
+                GlobalVertex::build().from_curve_and_position(&curve, a_curve);
 
             let surface_vertices = [a_curve, b_curve].map(|point_curve| {
                 let point_surface =
@@ -90,7 +89,7 @@ impl<'a> HalfEdgeBuilder<'a> {
 
         let curve = {
             let path = SurfacePath::Line(Line::from_points(points));
-            let curve_global = {
+            let global_form = {
                 let points = global_vertices
                     .map(|global_vertex| global_vertex.position());
                 self.stores.global_curves.insert(GlobalCurve::from_path(
@@ -98,7 +97,7 @@ impl<'a> HalfEdgeBuilder<'a> {
                 ))
             };
 
-            Curve::new(self.surface, path, curve_global)
+            Curve::new(self.surface, path, global_form)
         };
 
         let vertices = {