diff --git a/crates/fj-kernel/src/algorithms/approx/cycle.rs b/crates/fj-kernel/src/algorithms/approx/cycle.rs
index b819de3e35..02c4043111 100644
--- a/crates/fj-kernel/src/algorithms/approx/cycle.rs
+++ b/crates/fj-kernel/src/algorithms/approx/cycle.rs
@@ -2,17 +2,15 @@
 //!
 //! See [`CycleApprox`].
 
-use std::ops::Deref;
-
 use fj_math::Segment;
 
-use crate::objects::Cycle;
+use crate::objects::{Cycle, Surface};
 
 use super::{
     curve::CurveCache, edge::HalfEdgeApprox, Approx, ApproxPoint, Tolerance,
 };
 
-impl Approx for &Cycle {
+impl Approx for (&Cycle, &Surface) {
     type Approximation = CycleApprox;
     type Cache = CurveCache;
 
@@ -21,13 +19,13 @@ impl Approx for &Cycle {
         tolerance: impl Into<Tolerance>,
         cache: &mut Self::Cache,
     ) -> Self::Approximation {
+        let (cycle, surface) = self;
         let tolerance = tolerance.into();
 
-        let half_edges = self
+        let half_edges = cycle
             .half_edges()
             .map(|half_edge| {
-                (half_edge, self.surface().deref())
-                    .approx_with_cache(tolerance, cache)
+                (half_edge, surface).approx_with_cache(tolerance, cache)
             })
             .collect();
 
diff --git a/crates/fj-kernel/src/algorithms/approx/face.rs b/crates/fj-kernel/src/algorithms/approx/face.rs
index 6ee5c70807..aa99ea0a96 100644
--- a/crates/fj-kernel/src/algorithms/approx/face.rs
+++ b/crates/fj-kernel/src/algorithms/approx/face.rs
@@ -2,7 +2,7 @@
 //!
 //! See [`FaceApprox`].
 
-use std::collections::BTreeSet;
+use std::{collections::BTreeSet, ops::Deref};
 
 use fj_interop::mesh::Color;
 
@@ -87,11 +87,13 @@ 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.exterior().approx_with_cache(tolerance, cache);
+        let exterior = (self.exterior().deref(), self.surface().deref())
+            .approx_with_cache(tolerance, cache);
 
         let mut interiors = BTreeSet::new();
         for cycle in self.interiors() {
-            let cycle = cycle.approx_with_cache(tolerance, cache);
+            let cycle = (cycle.deref(), self.surface().deref())
+                .approx_with_cache(tolerance, cache);
             interiors.insert(cycle);
         }
 
diff --git a/crates/fj-kernel/src/algorithms/intersect/curve_face.rs b/crates/fj-kernel/src/algorithms/intersect/curve_face.rs
index 5e5048d891..de42b5e606 100644
--- a/crates/fj-kernel/src/algorithms/intersect/curve_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/curve_face.rs
@@ -183,7 +183,10 @@ mod tests {
         ];
 
         let face = {
-            let mut face = PartialFace::default();
+            let mut face = PartialFace {
+                surface: surface.clone(),
+                ..Default::default()
+            };
             face.exterior.write().surface = surface;
             face.exterior
                 .write()
diff --git a/crates/fj-kernel/src/algorithms/intersect/face_face.rs b/crates/fj-kernel/src/algorithms/intersect/face_face.rs
index fcf233da83..904f626303 100644
--- a/crates/fj-kernel/src/algorithms/intersect/face_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/face_face.rs
@@ -93,7 +93,10 @@ mod tests {
             services.objects.surfaces.xz_plane(),
         ]
         .map(|surface| {
-            let mut face = PartialFace::default();
+            let mut face = PartialFace {
+                surface: Partial::from(surface.clone()),
+                ..Default::default()
+            };
             face.exterior.write().surface = Partial::from(surface);
             face.exterior.write().update_as_polygon_from_points(points);
 
@@ -122,7 +125,10 @@ mod tests {
             services.objects.surfaces.xz_plane(),
         ];
         let [a, b] = surfaces.clone().map(|surface| {
-            let mut face = PartialFace::default();
+            let mut face = PartialFace {
+                surface: Partial::from(surface.clone()),
+                ..Default::default()
+            };
             face.exterior.write().surface = Partial::from(surface);
             face.exterior.write().update_as_polygon_from_points(points);
 
diff --git a/crates/fj-kernel/src/algorithms/intersect/face_point.rs b/crates/fj-kernel/src/algorithms/intersect/face_point.rs
index 5fcdcb72a3..a44ca8c078 100644
--- a/crates/fj-kernel/src/algorithms/intersect/face_point.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/face_point.rs
@@ -148,9 +148,13 @@ mod tests {
     fn point_is_outside_face() {
         let mut services = Services::new();
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.xy_plane());
+        let surface = Partial::from(services.objects.surfaces.xy_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [1., 1.],
@@ -169,9 +173,13 @@ mod tests {
     fn ray_hits_vertex_while_passing_outside() {
         let mut services = Services::new();
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.xy_plane());
+        let surface = Partial::from(services.objects.surfaces.xy_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [2., 1.],
@@ -193,9 +201,13 @@ mod tests {
     fn ray_hits_vertex_at_cycle_seam() {
         let mut services = Services::new();
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.xy_plane());
+        let surface = Partial::from(services.objects.surfaces.xy_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [4., 2.],
             [0., 4.],
@@ -217,9 +229,13 @@ mod tests {
     fn ray_hits_vertex_while_staying_inside() {
         let mut services = Services::new();
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.xy_plane());
+        let surface = Partial::from(services.objects.surfaces.xy_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [2., 1.],
@@ -242,9 +258,13 @@ mod tests {
     fn ray_hits_parallel_edge_and_leaves_face_at_vertex() {
         let mut services = Services::new();
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.xy_plane());
+        let surface = Partial::from(services.objects.surfaces.xy_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [2., 1.],
@@ -267,9 +287,13 @@ mod tests {
     fn ray_hits_parallel_edge_and_does_not_leave_face_there() {
         let mut services = Services::new();
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.xy_plane());
+        let surface = Partial::from(services.objects.surfaces.xy_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [2., 1.],
@@ -293,9 +317,13 @@ mod tests {
     fn point_is_coincident_with_edge() {
         let mut services = Services::new();
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.xy_plane());
+        let surface = Partial::from(services.objects.surfaces.xy_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [2., 0.],
@@ -326,9 +354,13 @@ mod tests {
     fn point_is_coincident_with_vertex() {
         let mut services = Services::new();
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.xy_plane());
+        let surface = Partial::from(services.objects.surfaces.xy_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [0., 0.],
             [1., 0.],
diff --git a/crates/fj-kernel/src/algorithms/intersect/ray_face.rs b/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
index 336ffb5be9..44269fbdf3 100644
--- a/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
@@ -163,9 +163,13 @@ mod tests {
 
         let ray = HorizontalRayToTheRight::from([0., 0., 0.]);
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.yz_plane());
+        let surface = Partial::from(services.objects.surfaces.yz_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
@@ -186,9 +190,13 @@ mod tests {
 
         let ray = HorizontalRayToTheRight::from([0., 0., 0.]);
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.yz_plane());
+        let surface = Partial::from(services.objects.surfaces.yz_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
@@ -212,9 +220,13 @@ mod tests {
 
         let ray = HorizontalRayToTheRight::from([0., 0., 0.]);
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.yz_plane());
+        let surface = Partial::from(services.objects.surfaces.yz_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
@@ -235,9 +247,13 @@ mod tests {
 
         let ray = HorizontalRayToTheRight::from([0., 0., 0.]);
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.yz_plane());
+        let surface = Partial::from(services.objects.surfaces.yz_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
@@ -269,9 +285,13 @@ mod tests {
 
         let ray = HorizontalRayToTheRight::from([0., 0., 0.]);
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.yz_plane());
+        let surface = Partial::from(services.objects.surfaces.yz_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
@@ -303,9 +323,13 @@ mod tests {
 
         let ray = HorizontalRayToTheRight::from([0., 0., 0.]);
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.xy_plane());
+        let surface = Partial::from(services.objects.surfaces.xy_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
@@ -328,9 +352,13 @@ mod tests {
 
         let ray = HorizontalRayToTheRight::from([0., 0., 0.]);
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.xy_plane());
+        let surface = Partial::from(services.objects.surfaces.xy_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior.write().update_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
diff --git a/crates/fj-kernel/src/algorithms/reverse/face.rs b/crates/fj-kernel/src/algorithms/reverse/face.rs
index caed07c558..3330ee8223 100644
--- a/crates/fj-kernel/src/algorithms/reverse/face.rs
+++ b/crates/fj-kernel/src/algorithms/reverse/face.rs
@@ -1,7 +1,6 @@
 use crate::{
     insert::Insert,
     objects::{Face, Objects},
-    partial::{FullToPartialCache, Partial, PartialFace, PartialObject},
     services::Service,
     storage::Handle,
 };
@@ -10,24 +9,13 @@ use super::Reverse;
 
 impl Reverse for Handle<Face> {
     fn reverse(self, objects: &mut Service<Objects>) -> Self {
-        let mut cache = FullToPartialCache::default();
-
-        let exterior = Partial::from_full(
-            self.exterior().clone().reverse(objects),
-            &mut cache,
-        );
+        let exterior = self.exterior().clone().reverse(objects);
         let interiors = self
             .interiors()
-            .map(|cycle| {
-                Partial::from_full(cycle.clone().reverse(objects), &mut cache)
-            })
+            .map(|cycle| cycle.clone().reverse(objects))
             .collect::<Vec<_>>();
 
-        let face = PartialFace {
-            exterior,
-            interiors,
-            color: Some(self.color()),
-        };
-        face.build(objects).insert(objects)
+        Face::new(self.surface().clone(), exterior, interiors, self.color())
+            .insert(objects)
     }
 }
diff --git a/crates/fj-kernel/src/algorithms/sweep/edge.rs b/crates/fj-kernel/src/algorithms/sweep/edge.rs
index 70a4c1cf1f..ca25066c7b 100644
--- a/crates/fj-kernel/src/algorithms/sweep/edge.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/edge.rs
@@ -33,10 +33,14 @@ impl Sweep for (Handle<HalfEdge>, &Surface, Color) {
         // A face (and everything in it) is defined on a surface. A surface can
         // be created by sweeping a curve, so let's sweep the curve of the edge
         // we're sweeping.
-        face.exterior.write().surface = Partial::from(
-            (edge.curve().clone(), surface)
-                .sweep_with_cache(path, cache, objects),
-        );
+        {
+            let surface = Partial::from(
+                (edge.curve().clone(), surface)
+                    .sweep_with_cache(path, cache, objects),
+            );
+            face.surface = surface.clone();
+            face.exterior.write().surface = surface;
+        }
 
         // Now we're ready to create the edges.
         let mut edge_bottom = face.exterior.write().add_half_edge();
@@ -266,7 +270,7 @@ mod tests {
             };
 
             let mut cycle = PartialCycle {
-                surface: Partial::from(surface),
+                surface: Partial::from(surface.clone()),
                 ..Default::default()
             };
             cycle.half_edges.extend(
@@ -274,6 +278,7 @@ mod tests {
             );
 
             let face = PartialFace {
+                surface: Partial::from(surface),
                 exterior: Partial::from_partial(cycle),
                 ..Default::default()
             };
diff --git a/crates/fj-kernel/src/algorithms/sweep/face.rs b/crates/fj-kernel/src/algorithms/sweep/face.rs
index 34a7cfb07a..51731cbc5e 100644
--- a/crates/fj-kernel/src/algorithms/sweep/face.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/face.rs
@@ -53,12 +53,13 @@ impl Sweep for Handle<Face> {
         };
         faces.push(bottom_face.clone());
 
+        let top_surface =
+            bottom_face.surface().clone().translate(path, objects);
         let mut top_face = PartialFace {
+            surface: Partial::from(top_surface.clone()),
             color: Some(self.color()),
             ..PartialFace::default()
         };
-        let top_surface =
-            bottom_face.surface().clone().translate(path, objects);
 
         for (i, cycle) in bottom_face.all_cycles().cloned().enumerate() {
             let cycle = cycle.reverse(objects);
@@ -84,7 +85,9 @@ impl Sweep for Handle<Face> {
 
             top_cycle.write().surface = Partial::from(top_surface.clone());
 
-            top_cycle.write().connect_to_closed_edges(top_edges);
+            top_cycle
+                .write()
+                .connect_to_closed_edges(top_edges, &top_surface.geometry());
 
             for half_edge in &mut top_cycle.write().half_edges {
                 for (_, surface_vertex) in &mut half_edge.write().vertices {
@@ -161,6 +164,7 @@ mod tests {
             let mut sketch = PartialSketch::default();
 
             let mut face = sketch.add_face();
+            face.write().surface = Partial::from(surface.clone());
             face.write().exterior.write().surface =
                 Partial::from(surface.clone());
             face.write()
@@ -175,23 +179,37 @@ mod tests {
             .insert(&mut services.objects)
             .sweep(UP, &mut services.objects);
 
-        let mut bottom = PartialFace::default();
-        bottom.exterior.write().surface = Partial::from(surface.clone());
-        bottom
-            .exterior
-            .write()
-            .update_as_polygon_from_points(TRIANGLE);
-        let bottom = bottom
-            .build(&mut services.objects)
-            .insert(&mut services.objects)
-            .reverse(&mut services.objects);
-        let mut top = PartialFace::default();
-        top.exterior.write().surface =
-            Partial::from(surface.clone().translate(UP, &mut services.objects));
-        top.exterior.write().update_as_polygon_from_points(TRIANGLE);
-        let top = top
-            .build(&mut services.objects)
-            .insert(&mut services.objects);
+        let bottom = {
+            let mut bottom = PartialFace {
+                surface: Partial::from(surface.clone()),
+                ..Default::default()
+            };
+
+            bottom.exterior.write().surface = Partial::from(surface.clone());
+            bottom
+                .exterior
+                .write()
+                .update_as_polygon_from_points(TRIANGLE);
+
+            bottom
+                .build(&mut services.objects)
+                .insert(&mut services.objects)
+                .reverse(&mut services.objects)
+        };
+        let top = {
+            let surface = surface.clone().translate(UP, &mut services.objects);
+
+            let mut top = PartialFace {
+                surface: Partial::from(surface.clone()),
+                ..Default::default()
+            };
+
+            top.exterior.write().surface = Partial::from(surface);
+            top.exterior.write().update_as_polygon_from_points(TRIANGLE);
+
+            top.build(&mut services.objects)
+                .insert(&mut services.objects)
+        };
 
         assert!(solid.find_face(&bottom).is_some());
         assert!(solid.find_face(&top).is_some());
@@ -227,6 +245,7 @@ mod tests {
             let mut sketch = PartialSketch::default();
 
             let mut face = sketch.add_face();
+            face.write().surface = Partial::from(surface.clone());
             face.write().exterior.write().surface =
                 Partial::from(surface.clone());
             face.write()
@@ -241,24 +260,38 @@ mod tests {
             .insert(&mut services.objects)
             .sweep(DOWN, &mut services.objects);
 
-        let mut bottom = PartialFace::default();
-        bottom.exterior.write().surface = Partial::from(
-            surface.clone().translate(DOWN, &mut services.objects),
-        );
-        bottom
-            .exterior
-            .write()
-            .update_as_polygon_from_points(TRIANGLE);
-        let bottom = bottom
-            .build(&mut services.objects)
-            .insert(&mut services.objects)
-            .reverse(&mut services.objects);
-        let mut top = PartialFace::default();
-        top.exterior.write().surface = Partial::from(surface.clone());
-        top.exterior.write().update_as_polygon_from_points(TRIANGLE);
-        let top = top
-            .build(&mut services.objects)
-            .insert(&mut services.objects);
+        let bottom = {
+            let surface =
+                surface.clone().translate(DOWN, &mut services.objects);
+
+            let mut bottom = PartialFace {
+                surface: Partial::from(surface.clone()),
+                ..Default::default()
+            };
+
+            bottom.exterior.write().surface = Partial::from(surface);
+            bottom
+                .exterior
+                .write()
+                .update_as_polygon_from_points(TRIANGLE);
+
+            bottom
+                .build(&mut services.objects)
+                .insert(&mut services.objects)
+                .reverse(&mut services.objects)
+        };
+        let top = {
+            let mut top = PartialFace {
+                surface: Partial::from(surface.clone()),
+                ..Default::default()
+            };
+
+            top.exterior.write().surface = Partial::from(surface.clone());
+            top.exterior.write().update_as_polygon_from_points(TRIANGLE);
+
+            top.build(&mut services.objects)
+                .insert(&mut services.objects)
+        };
 
         assert!(solid.find_face(&bottom).is_some());
         assert!(solid.find_face(&top).is_some());
diff --git a/crates/fj-kernel/src/algorithms/transform/face.rs b/crates/fj-kernel/src/algorithms/transform/face.rs
index 86a9f2b8ec..fb4a589069 100644
--- a/crates/fj-kernel/src/algorithms/transform/face.rs
+++ b/crates/fj-kernel/src/algorithms/transform/face.rs
@@ -17,6 +17,10 @@ impl TransformObject for Face {
         // Color does not need to be transformed.
         let color = self.color();
 
+        let surface = self
+            .surface()
+            .clone()
+            .transform_with_cache(transform, objects, cache);
         let exterior = self
             .exterior()
             .clone()
@@ -25,7 +29,7 @@ impl TransformObject for Face {
             interior.transform_with_cache(transform, objects, cache)
         });
 
-        Self::new(exterior, interiors, color)
+        Self::new(surface, exterior, interiors, color)
     }
 }
 
diff --git a/crates/fj-kernel/src/algorithms/triangulate/mod.rs b/crates/fj-kernel/src/algorithms/triangulate/mod.rs
index ec841dd1a7..87d29de38b 100644
--- a/crates/fj-kernel/src/algorithms/triangulate/mod.rs
+++ b/crates/fj-kernel/src/algorithms/triangulate/mod.rs
@@ -96,9 +96,13 @@ mod tests {
         let c = [2., 2.];
         let d = [0., 1.];
 
-        let mut face = PartialFace::default();
-        face.exterior.write().surface =
-            Partial::from(services.objects.surfaces.xy_plane());
+        let surface = Partial::from(services.objects.surfaces.xy_plane());
+
+        let mut face = PartialFace {
+            surface: surface.clone(),
+            ..Default::default()
+        };
+        face.exterior.write().surface = surface;
         face.exterior
             .write()
             .update_as_polygon_from_points([a, b, c, d]);
@@ -136,7 +140,10 @@ mod tests {
         let h = [3., 1.];
 
         let surface = services.objects.surfaces.xy_plane();
-        let mut face = PartialFace::default();
+        let mut face = PartialFace {
+            surface: Partial::from(surface.clone()),
+            ..Default::default()
+        };
         face.exterior.write().surface = Partial::from(surface.clone());
         face.exterior
             .write()
@@ -200,7 +207,10 @@ mod tests {
         let e = [0.0, 1.0];
 
         let surface = services.objects.surfaces.xy_plane();
-        let mut face = PartialFace::default();
+        let mut face = PartialFace {
+            surface: Partial::from(surface.clone()),
+            ..Default::default()
+        };
         face.exterior.write().surface = Partial::from(surface.clone());
         face.exterior
             .write()
diff --git a/crates/fj-kernel/src/builder/cycle.rs b/crates/fj-kernel/src/builder/cycle.rs
index 3673b0ee82..c20ed62561 100644
--- a/crates/fj-kernel/src/builder/cycle.rs
+++ b/crates/fj-kernel/src/builder/cycle.rs
@@ -1,10 +1,7 @@
-use std::collections::VecDeque;
-
-use fj_interop::ext::ArrayExt;
 use fj_math::Point;
 
 use crate::{
-    builder::SurfaceBuilder,
+    geometry::surface::SurfaceGeometry,
     objects::HalfEdge,
     partial::{Partial, PartialCycle},
 };
@@ -64,35 +61,6 @@ pub trait CycleBuilder {
     /// Will update each half-edge in the cycle to be a line segment.
     fn update_as_polygon(&mut self);
 
-    /// Update cycle as a triangle, from global (3D) points
-    ///
-    /// Uses the three points to infer a plane that is used as the surface.
-    ///
-    /// # Implementation Note
-    ///
-    /// This method is probably just temporary, and will be generalized into a
-    /// "update as polygon from global points" method sooner or later. For now,
-    /// I didn't want to deal with the question of how to infer the surface, and
-    /// how to handle points that don't fit that surface.
-    fn update_as_triangle_from_global_points(
-        &mut self,
-        points: [impl Into<Point<3>>; 3],
-    ) -> [Partial<HalfEdge>; 3];
-
-    /// Connect the cycle to the provided half-edges
-    ///
-    /// Assumes that the provided half-edges, once translated into local
-    /// equivalents of this cycle, will not form a cycle themselves.
-    ///
-    /// Returns the local equivalents of the provided half-edges and, as the
-    /// last entry, an additional half-edge that closes the cycle.
-    fn connect_to_open_edges<O>(
-        &mut self,
-        edges: O,
-    ) -> O::SizePlusOne<Partial<HalfEdge>>
-    where
-        O: ObjectArgument<Partial<HalfEdge>>;
-
     /// Connect the cycles to the provided half-edges
     ///
     /// Assumes that the provided half-edges, once translated into local
@@ -102,6 +70,7 @@ pub trait CycleBuilder {
     fn connect_to_closed_edges<O>(
         &mut self,
         edges: O,
+        surface: &SurfaceGeometry,
     ) -> O::SameSize<Partial<HalfEdge>>
     where
         O: ObjectArgument<Partial<HalfEdge>>;
@@ -204,65 +173,17 @@ impl CycleBuilder for PartialCycle {
         }
     }
 
-    fn update_as_triangle_from_global_points(
-        &mut self,
-        points_global: [impl Into<Point<3>>; 3],
-    ) -> [Partial<HalfEdge>; 3] {
-        let points_global = points_global.map(Into::into);
-
-        let (points_surface, _) = self
-            .surface
-            .write()
-            .update_as_plane_from_points(points_global);
-
-        let half_edges = self.update_as_polygon_from_points(points_surface);
-
-        for (mut half_edge, point) in half_edges.clone().zip_ext(points_global)
-        {
-            let [vertex, _] = &mut half_edge.write().vertices;
-            vertex.1.write().global_form.write().position = Some(point);
-        }
-
-        half_edges
-    }
-
-    fn connect_to_open_edges<O>(
-        &mut self,
-        edges: O,
-    ) -> O::SizePlusOne<Partial<HalfEdge>>
-    where
-        O: ObjectArgument<Partial<HalfEdge>>,
-    {
-        // We need to create the additional half-edge last, but at the same time
-        // need to provide it to the `map_plus_one` method first. Really no
-        // choice but to create them all in one go, as we do here.
-        let mut half_edges = VecDeque::new();
-        for _ in 0..edges.num_objects() {
-            half_edges.push_back(self.add_half_edge());
-        }
-        let additional_half_edge = self.add_half_edge();
-
-        edges.map_plus_one(additional_half_edge, |other| {
-            let mut this = half_edges.pop_front().expect(
-                "Pushed correct number of half-edges; should be able to pop",
-            );
-            this.write()
-                .update_from_other_edge(&other, &self.surface.read().geometry);
-            this
-        })
-    }
-
     fn connect_to_closed_edges<O>(
         &mut self,
         edges: O,
+        surface: &SurfaceGeometry,
     ) -> O::SameSize<Partial<HalfEdge>>
     where
         O: ObjectArgument<Partial<HalfEdge>>,
     {
         edges.map(|other| {
             let mut this = self.add_half_edge();
-            this.write()
-                .update_from_other_edge(&other, &self.surface.read().geometry);
+            this.write().update_from_other_edge(&other, surface);
             this
         })
     }
diff --git a/crates/fj-kernel/src/builder/edge.rs b/crates/fj-kernel/src/builder/edge.rs
index c4f5b74022..2807164c28 100644
--- a/crates/fj-kernel/src/builder/edge.rs
+++ b/crates/fj-kernel/src/builder/edge.rs
@@ -58,7 +58,7 @@ pub trait HalfEdgeBuilder {
     fn update_from_other_edge(
         &mut self,
         other: &Partial<HalfEdge>,
-        surface: &Option<SurfaceGeometry>,
+        surface: &SurfaceGeometry,
     );
 }
 
@@ -222,7 +222,7 @@ impl HalfEdgeBuilder for PartialHalfEdge {
     fn update_from_other_edge(
         &mut self,
         other: &Partial<HalfEdge>,
-        surface: &Option<SurfaceGeometry>,
+        surface: &SurfaceGeometry,
     ) {
         let global_curve = other.read().curve.read().global_form.clone();
         self.curve.write().global_form = global_curve.clone();
@@ -230,103 +230,78 @@ impl HalfEdgeBuilder for PartialHalfEdge {
 
         self.curve.write().path =
             other.read().curve.read().path.as_ref().and_then(|path| {
-                match surface {
-                    Some(surface) => {
-                        // We have information about the other edge's surface
-                        // available. We need to use that to interpret what the
-                        // other edge's curve path means for our curve path.
-                        match surface.u {
-                            GlobalPath::Circle(circle) => {
-                                // The other surface is curved. We're entering
-                                // some dodgy territory here, as only some edge
-                                // cases can be represented using our current
-                                // curve/surface representation.
-                                match path {
-                                    MaybeSurfacePath::Defined(
-                                        SurfacePath::Line(_),
-                                    )
-                                    | MaybeSurfacePath::UndefinedLine => {
-                                        // We're dealing with a line on a
-                                        // rounded surface.
-                                        //
-                                        // Based on the current uses of this
-                                        // method, we can make some assumptions:
-                                        //
-                                        // 1. The line is parallel to the u-axis
-                                        //    of the other surface.
-                                        // 2. The surface that *our* edge is in
-                                        //    is a plane that is parallel to the
-                                        //    the plane of the circle that
-                                        //    defines the curvature of the other
-                                        //    surface.
-                                        //
-                                        // These assumptions are necessary
-                                        // preconditions for the following code
-                                        // to work. But unfortunately, I see no
-                                        // way to check those preconditions
-                                        // here, as neither the other line nor
-                                        // our surface is necessarily defined
-                                        // yet.
-                                        //
-                                        // Handling this case anyway feels like
-                                        // a grave sin, but I don't know what
-                                        // else to do. If you tracked some
-                                        // extremely subtle and annoying bug
-                                        // back to this code, I apologize.
-                                        //
-                                        // I hope that I'll come up with a
-                                        // better curve/surface representation
-                                        // before this becomes a problem.
-                                        Some(
-                                            MaybeSurfacePath::UndefinedCircle {
-                                                radius: circle.radius(),
-                                            },
-                                        )
-                                    }
-                                    _ => {
-                                        // The other edge is a line segment in a
-                                        // curved surface. No idea how to deal
-                                        // with this.
-                                        todo!(
-                                            "Can't connect edge to circle on \
-                                            curved surface"
-                                        )
-                                    }
-                                }
+                // We have information about the other edge's surface available.
+                // We need to use that to interpret what the other edge's curve
+                // path means for our curve path.
+                match surface.u {
+                    GlobalPath::Circle(circle) => {
+                        // The other surface is curved. We're entering some
+                        // dodgy territory here, as only some edge cases can be
+                        // represented using our current curve/surface
+                        // representation.
+                        match path {
+                            MaybeSurfacePath::Defined(SurfacePath::Line(_))
+                            | MaybeSurfacePath::UndefinedLine => {
+                                // We're dealing with a line on a rounded
+                                // surface.
+                                //
+                                // Based on the current uses of this method, we
+                                // can make some assumptions:
+                                //
+                                // 1. The line is parallel to the u-axis of the
+                                //    other surface.
+                                // 2. The surface that *our* edge is in is a
+                                //    plane that is parallel to the the plane of
+                                //    the circle that defines the curvature of
+                                //    the other surface.
+                                //
+                                // These assumptions are necessary preconditions
+                                // for the following code to work. But
+                                // unfortunately, I see no way to check those
+                                // preconditions here, as neither the other line
+                                // nor our surface is necessarily defined yet.
+                                //
+                                // Handling this case anyway feels like a grave
+                                // sin, but I don't know what else to do. If you
+                                // tracked some extremely subtle and annoying
+                                // bug back to this code, I apologize.
+                                //
+                                // I hope that I'll come up with a better curve/
+                                // surface representation before this becomes a
+                                // problem.
+                                Some(MaybeSurfacePath::UndefinedCircle {
+                                    radius: circle.radius(),
+                                })
                             }
-                            GlobalPath::Line(_) => {
-                                // The other edge is defined on a plane.
-                                match path {
-                                    MaybeSurfacePath::Defined(
-                                        SurfacePath::Line(_),
-                                    )
-                                    | MaybeSurfacePath::UndefinedLine => {
-                                        // The other edge is a line segment on
-                                        // a plane. That means our edge must be
-                                        // a line segment too.
-                                        Some(MaybeSurfacePath::UndefinedLine)
-                                    }
-                                    _ => {
-                                        // The other edge is a circle or arc on
-                                        // a plane. I'm actually not sure what
-                                        // that means for our edge. We might be
-                                        // able to represent it somehow, but
-                                        // let's leave that as an exercise for
-                                        // later.
-                                        todo!(
-                                            "Can't connect edge to circle on \
-                                            plane"
-                                        )
-                                    }
-                                }
+                            _ => {
+                                // The other edge is a line segment in a curved
+                                // surface. No idea how to deal with this.
+                                todo!(
+                                    "Can't connect edge to circle on curved \
+                                    surface"
+                                )
                             }
                         }
                     }
-                    None => {
-                        // We know nothing about the surface the other edge is
-                        // on. This means we can't infer anything about our
-                        // curve from the other curve.
-                        None
+                    GlobalPath::Line(_) => {
+                        // The other edge is defined on a plane.
+                        match path {
+                            MaybeSurfacePath::Defined(SurfacePath::Line(_))
+                            | MaybeSurfacePath::UndefinedLine => {
+                                // The other edge is a line segment on a plane.
+                                // That means our edge must be a line segment
+                                // too.
+                                Some(MaybeSurfacePath::UndefinedLine)
+                            }
+                            _ => {
+                                // The other edge is a circle or arc on a plane.
+                                // I'm actually not sure what that means for our
+                                // edge. We might be able to represent it
+                                // somehow, but let's leave that as an exercise
+                                // for later.
+                                todo!("Can't connect edge to circle on plane")
+                            }
+                        }
                     }
                 }
             });
diff --git a/crates/fj-kernel/src/objects/full/face.rs b/crates/fj-kernel/src/objects/full/face.rs
index 95e9d6b87a..f43c318fed 100644
--- a/crates/fj-kernel/src/objects/full/face.rs
+++ b/crates/fj-kernel/src/objects/full/face.rs
@@ -34,6 +34,7 @@ use crate::{
 /// [`Shell`]: crate::objects::Shell
 #[derive(Clone, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
 pub struct Face {
+    surface: Handle<Surface>,
     exterior: Handle<Cycle>,
     interiors: Vec<Handle<Cycle>>,
     color: Color,
@@ -42,6 +43,7 @@ pub struct Face {
 impl Face {
     /// Construct an instance of `Face`
     pub fn new(
+        surface: Handle<Surface>,
         exterior: Handle<Cycle>,
         interiors: impl IntoIterator<Item = Handle<Cycle>>,
         color: Color,
@@ -49,6 +51,7 @@ impl Face {
         let interiors = interiors.into_iter().collect();
 
         Self {
+            surface,
             exterior,
             interiors,
             color,
@@ -57,7 +60,7 @@ impl Face {
 
     /// Access the surface of the face
     pub fn surface(&self) -> &Handle<Surface> {
-        self.exterior().surface()
+        &self.surface
     }
 
     /// Access the cycle that bounds the face on the outside
diff --git a/crates/fj-kernel/src/partial/objects/face.rs b/crates/fj-kernel/src/partial/objects/face.rs
index b39e04ff0a..2d4d5d0c43 100644
--- a/crates/fj-kernel/src/partial/objects/face.rs
+++ b/crates/fj-kernel/src/partial/objects/face.rs
@@ -1,7 +1,7 @@
 use fj_interop::mesh::Color;
 
 use crate::{
-    objects::{Cycle, Face, Objects},
+    objects::{Cycle, Face, Objects, Surface},
     partial::{FullToPartialCache, Partial, PartialObject},
     services::Service,
 };
@@ -9,6 +9,9 @@ use crate::{
 /// A partial [`Face`]
 #[derive(Clone, Debug, Default)]
 pub struct PartialFace {
+    /// The surface that the face is defined in
+    pub surface: Partial<Surface>,
+
     /// The cycle that bounds the face on the outside
     pub exterior: Partial<Cycle>,
 
@@ -26,6 +29,7 @@ impl PartialObject for PartialFace {
 
     fn from_full(face: &Self::Full, cache: &mut FullToPartialCache) -> Self {
         Self {
+            surface: Partial::from_full(face.surface().clone(), cache),
             exterior: Partial::from_full(face.exterior().clone(), cache),
             interiors: face
                 .interiors()
@@ -36,11 +40,12 @@ impl PartialObject for PartialFace {
     }
 
     fn build(self, objects: &mut Service<Objects>) -> Self::Full {
+        let surface = self.surface.build(objects);
         let exterior = self.exterior.build(objects);
         let interiors =
             self.interiors.into_iter().map(|cycle| cycle.build(objects));
         let color = self.color.unwrap_or_default();
 
-        Face::new(exterior, interiors, color)
+        Face::new(surface, exterior, interiors, color)
     }
 }
diff --git a/crates/fj-kernel/src/validate/face.rs b/crates/fj-kernel/src/validate/face.rs
index 2fa6a00223..ff500cffa8 100644
--- a/crates/fj-kernel/src/validate/face.rs
+++ b/crates/fj-kernel/src/validate/face.rs
@@ -115,7 +115,10 @@ mod tests {
         let surface = services.objects.surfaces.xy_plane();
 
         let valid = {
-            let mut face = PartialFace::default();
+            let mut face = PartialFace {
+                surface: Partial::from(surface.clone()),
+                ..Default::default()
+            };
             face.exterior.write().surface = Partial::from(surface);
             face.exterior.write().update_as_polygon_from_points([
                 [0., 0.],
@@ -141,7 +144,12 @@ mod tests {
                 .insert(&mut services.objects);
 
             let interiors = [cycle];
-            Face::new(valid.exterior().clone(), interiors, valid.color())
+            Face::new(
+                valid.surface().clone(),
+                valid.exterior().clone(),
+                interiors,
+                valid.color(),
+            )
         };
 
         valid.validate_and_return_first_error()?;
@@ -157,7 +165,10 @@ mod tests {
         let surface = services.objects.surfaces.xy_plane();
 
         let valid = {
-            let mut face = PartialFace::default();
+            let mut face = PartialFace {
+                surface: Partial::from(surface.clone()),
+                ..Default::default()
+            };
             face.exterior.write().surface = Partial::from(surface);
             face.exterior.write().update_as_polygon_from_points([
                 [0., 0.],
@@ -178,7 +189,12 @@ mod tests {
                 .map(|cycle| cycle.reverse(&mut services.objects))
                 .collect::<Vec<_>>();
 
-            Face::new(valid.exterior().clone(), interiors, valid.color())
+            Face::new(
+                valid.surface().clone(),
+                valid.exterior().clone(),
+                interiors,
+                valid.color(),
+            )
         };
 
         valid.validate_and_return_first_error()?;
diff --git a/crates/fj-operations/src/difference_2d.rs b/crates/fj-operations/src/difference_2d.rs
index 898b07e212..6f9fa37377 100644
--- a/crates/fj-operations/src/difference_2d.rs
+++ b/crates/fj-operations/src/difference_2d.rs
@@ -82,6 +82,7 @@ impl Shape for fj::Difference2d {
             );
 
             let face = PartialFace {
+                surface: Partial::from(surface.clone()),
                 exterior: Partial::from(exterior),
                 interiors,
                 color: Some(Color(self.color())),
diff --git a/crates/fj-operations/src/sketch.rs b/crates/fj-operations/src/sketch.rs
index f9e402aeb6..83dec0e289 100644
--- a/crates/fj-operations/src/sketch.rs
+++ b/crates/fj-operations/src/sketch.rs
@@ -37,7 +37,7 @@ impl Shape for fj::Sketch {
                 };
                 let exterior = {
                     let mut cycle = PartialCycle {
-                        surface,
+                        surface: surface.clone(),
                         ..Default::default()
                     };
                     cycle.half_edges.push(half_edge);
@@ -45,6 +45,7 @@ impl Shape for fj::Sketch {
                 };
 
                 PartialFace {
+                    surface,
                     exterior,
                     color: Some(Color(self.color())),
                     ..Default::default()
@@ -59,7 +60,7 @@ impl Shape for fj::Sketch {
 
                 let exterior = {
                     let mut cycle = PartialCycle {
-                        surface: Partial::from(surface),
+                        surface: Partial::from(surface.clone()),
                         ..Default::default()
                     };
                     let mut line_segments = vec![];
@@ -98,6 +99,7 @@ impl Shape for fj::Sketch {
                 };
 
                 PartialFace {
+                    surface: Partial::from(surface),
                     exterior,
                     color: Some(Color(self.color())),
                     ..Default::default()