diff --git a/crates/fj-core/src/algorithms/approx/curve.rs b/crates/fj-core/src/algorithms/approx/curve.rs
index 43e7ddd15..837a24fe5 100644
--- a/crates/fj-core/src/algorithms/approx/curve.rs
+++ b/crates/fj-core/src/algorithms/approx/curve.rs
@@ -10,13 +10,13 @@ use crate::{
         SurfacePath,
     },
     storage::Handle,
-    topology::Curve,
+    topology::{Curve, Surface},
     Core,
 };
 
 use super::{Approx, ApproxPoint, Tolerance};
 
-impl Approx for (&Handle<Curve>, &HalfEdgeGeometry, &SurfaceGeometry) {
+impl Approx for (&Handle<Curve>, &HalfEdgeGeometry, &Handle<Surface>) {
     type Approximation = CurveApprox;
     type Cache = CurveApproxCache;
 
@@ -33,7 +33,7 @@ impl Approx for (&Handle<Curve>, &HalfEdgeGeometry, &SurfaceGeometry) {
             None => {
                 let approx = approx_curve(
                     &half_edge.path,
-                    surface,
+                    &core.layers.geometry.of_surface(surface),
                     half_edge.boundary,
                     tolerance,
                     core,
@@ -183,12 +183,9 @@ mod tests {
 
     use crate::{
         algorithms::approx::{Approx, ApproxPoint},
-        geometry::{
-            CurveBoundary, GlobalPath, HalfEdgeGeometry, SurfaceGeometry,
-            SurfacePath,
-        },
-        operations::insert::Insert,
-        topology::Curve,
+        geometry::{CurveBoundary, GlobalPath, HalfEdgeGeometry, SurfacePath},
+        operations::{build::BuildSurface, insert::Insert},
+        topology::{Curve, Surface},
         Core,
     };
 
@@ -201,7 +198,7 @@ mod tests {
             SurfacePath::line_from_points([[1., 1.], [2., 1.]]);
         let boundary = CurveBoundary::from(boundary);
         let half_edge = HalfEdgeGeometry { path, boundary };
-        let surface = core.layers.geometry.xz_plane();
+        let surface = core.layers.topology.surfaces.xz_plane();
 
         let tolerance = 1.;
         let approx =
@@ -219,10 +216,11 @@ mod tests {
             SurfacePath::line_from_points([[1., 1.], [2., 1.]]);
         let boundary = CurveBoundary::from(boundary);
         let half_edge = HalfEdgeGeometry { path, boundary };
-        let surface = SurfaceGeometry {
-            u: GlobalPath::circle_from_radius(1.),
-            v: [0., 0., 1.].into(),
-        };
+        let surface = Surface::from_uv(
+            GlobalPath::circle_from_radius(1.),
+            [0., 0., 1.],
+            &mut core,
+        );
 
         let tolerance = 1.;
         let approx =
@@ -243,10 +241,7 @@ mod tests {
         ]);
         let boundary = CurveBoundary::from([[0.], [TAU]]);
         let half_edge = HalfEdgeGeometry { path, boundary };
-        let surface = SurfaceGeometry {
-            u: global_path,
-            v: [0., 0., 1.].into(),
-        };
+        let surface = Surface::from_uv(global_path, [0., 0., 1.], &mut core);
 
         let tolerance = 1.;
         let approx =
@@ -257,8 +252,11 @@ mod tests {
             .into_iter()
             .map(|(point_local, _)| {
                 let point_surface = path.point_from_path_coords(point_local);
-                let point_global =
-                    surface.point_from_surface_coords(point_surface);
+                let point_global = core
+                    .layers
+                    .geometry
+                    .of_surface(&surface)
+                    .point_from_surface_coords(point_surface);
                 ApproxPoint::new(point_local, point_global)
             })
             .collect::<Vec<_>>();
@@ -273,7 +271,7 @@ mod tests {
         let path = SurfacePath::circle_from_center_and_radius([0., 0.], 1.);
         let boundary = CurveBoundary::from([[0.], [TAU]]);
         let half_edge = HalfEdgeGeometry { path, boundary };
-        let surface = core.layers.geometry.xz_plane();
+        let surface = core.layers.topology.surfaces.xz_plane();
 
         let tolerance = 1.;
         let approx =
@@ -284,8 +282,11 @@ mod tests {
             .into_iter()
             .map(|(point_local, _)| {
                 let point_surface = path.point_from_path_coords(point_local);
-                let point_global =
-                    surface.point_from_surface_coords(point_surface);
+                let point_global = core
+                    .layers
+                    .geometry
+                    .of_surface(&surface)
+                    .point_from_surface_coords(point_surface);
                 ApproxPoint::new(point_local, point_global)
             })
             .collect::<Vec<_>>();
diff --git a/crates/fj-core/src/algorithms/approx/cycle.rs b/crates/fj-core/src/algorithms/approx/cycle.rs
index f2b473afb..fd4311231 100644
--- a/crates/fj-core/src/algorithms/approx/cycle.rs
+++ b/crates/fj-core/src/algorithms/approx/cycle.rs
@@ -4,14 +4,18 @@
 
 use fj_math::Segment;
 
-use crate::{geometry::SurfaceGeometry, topology::Cycle, Core};
+use crate::{
+    storage::Handle,
+    topology::{Cycle, Surface},
+    Core,
+};
 
 use super::{
     edge::{HalfEdgeApprox, HalfEdgeApproxCache},
     Approx, ApproxPoint, Tolerance,
 };
 
-impl Approx for (&Cycle, &SurfaceGeometry) {
+impl Approx for (&Cycle, &Handle<Surface>) {
     type Approximation = CycleApprox;
     type Cache = HalfEdgeApproxCache;
 
diff --git a/crates/fj-core/src/algorithms/approx/edge.rs b/crates/fj-core/src/algorithms/approx/edge.rs
index 0484ea7b9..aa6f73f2a 100644
--- a/crates/fj-core/src/algorithms/approx/edge.rs
+++ b/crates/fj-core/src/algorithms/approx/edge.rs
@@ -6,7 +6,9 @@
 //! the caller doesn't have to deal with duplicate vertices.
 
 use crate::{
-    geometry::SurfaceGeometry, storage::Handle, topology::HalfEdge, Core,
+    storage::Handle,
+    topology::{HalfEdge, Surface},
+    Core,
 };
 
 use super::{
@@ -14,7 +16,7 @@ use super::{
     Tolerance,
 };
 
-impl Approx for (&Handle<HalfEdge>, &SurfaceGeometry) {
+impl Approx for (&Handle<HalfEdge>, &Handle<Surface>) {
     type Approximation = HalfEdgeApprox;
     type Cache = HalfEdgeApproxCache;
 
@@ -36,7 +38,10 @@ impl Approx for (&Handle<HalfEdge>, &SurfaceGeometry) {
             match cache.start_position.get(half_edge.start_vertex()) {
                 Some(position) => position,
                 None => {
-                    let position_global = surface
+                    let position_global = core
+                        .layers
+                        .geometry
+                        .of_surface(surface)
                         .point_from_surface_coords(start_position_surface);
                     cache.start_position.insert(
                         half_edge.start_vertex().clone(),
diff --git a/crates/fj-core/src/algorithms/approx/face.rs b/crates/fj-core/src/algorithms/approx/face.rs
index 8deb299e7..be81fbc9f 100644
--- a/crates/fj-core/src/algorithms/approx/face.rs
+++ b/crates/fj-core/src/algorithms/approx/face.rs
@@ -90,18 +90,12 @@ impl Approx for &Face {
         // would need to provide its own approximation, as the edges that bound
         // it have nothing to do with its curvature.
 
-        let exterior = (
-            self.region().exterior().deref(),
-            &core.layers.geometry.of_surface(self.surface()),
-        )
+        let exterior = (self.region().exterior().deref(), self.surface())
             .approx_with_cache(tolerance, cache, core);
 
         let mut interiors = BTreeSet::new();
         for cycle in self.region().interiors() {
-            let cycle = (
-                cycle.deref(),
-                &core.layers.geometry.of_surface(self.surface()),
-            )
+            let cycle = (cycle.deref(), self.surface())
                 .approx_with_cache(tolerance, cache, core);
             interiors.insert(cycle);
         }