diff --git a/crates/fj-kernel/src/algorithms/approx/cycle.rs b/crates/fj-kernel/src/algorithms/approx/cycle.rs
index 9d398f2bc..b819de3e3 100644
--- a/crates/fj-kernel/src/algorithms/approx/cycle.rs
+++ b/crates/fj-kernel/src/algorithms/approx/cycle.rs
@@ -2,6 +2,8 @@
 //!
 //! See [`CycleApprox`].
 
+use std::ops::Deref;
+
 use fj_math::Segment;
 
 use crate::objects::Cycle;
@@ -23,7 +25,10 @@ impl Approx for &Cycle {
 
         let half_edges = self
             .half_edges()
-            .map(|half_edge| half_edge.approx_with_cache(tolerance, cache))
+            .map(|half_edge| {
+                (half_edge, self.surface().deref())
+                    .approx_with_cache(tolerance, cache)
+            })
             .collect();
 
         CycleApprox { half_edges }
diff --git a/crates/fj-kernel/src/algorithms/approx/edge.rs b/crates/fj-kernel/src/algorithms/approx/edge.rs
index 2d74e66d5..cefa0cee0 100644
--- a/crates/fj-kernel/src/algorithms/approx/edge.rs
+++ b/crates/fj-kernel/src/algorithms/approx/edge.rs
@@ -5,9 +5,10 @@
 //! approximations are usually used to build cycle approximations, and this way,
 //! the caller doesn't have to call with duplicate vertices.
 
-use std::ops::Deref;
-
-use crate::{objects::HalfEdge, storage::Handle};
+use crate::{
+    objects::{HalfEdge, Surface},
+    storage::Handle,
+};
 
 use super::{
     curve::{CurveApprox, CurveCache},
@@ -15,7 +16,7 @@ use super::{
     Approx, ApproxPoint, Tolerance,
 };
 
-impl Approx for &Handle<HalfEdge> {
+impl Approx for (&Handle<HalfEdge>, &Surface) {
     type Approximation = HalfEdgeApprox;
     type Cache = CurveCache;
 
@@ -24,15 +25,17 @@ impl Approx for &Handle<HalfEdge> {
         tolerance: impl Into<Tolerance>,
         cache: &mut Self::Cache,
     ) -> Self::Approximation {
-        let boundary = self.boundary();
+        let (half_edge, surface) = self;
+
+        let boundary = half_edge.boundary();
         let range = RangeOnPath { boundary };
 
         let first = ApproxPoint::new(
-            self.start_vertex().position(),
-            self.start_vertex().global_form().position(),
+            half_edge.start_vertex().position(),
+            half_edge.start_vertex().global_form().position(),
         )
-        .with_source((self.clone(), self.boundary()[0]));
-        let curve_approx = (self.curve(), self.surface().deref(), range)
+        .with_source((half_edge.clone(), half_edge.boundary()[0]));
+        let curve_approx = (half_edge.curve(), surface, range)
             .approx_with_cache(tolerance, cache);
 
         HalfEdgeApprox {
diff --git a/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs b/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
index d7ca55ac7..bec4fa99f 100644
--- a/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
@@ -76,7 +76,7 @@ mod tests {
 
     use crate::{
         builder::{CurveBuilder, HalfEdgeBuilder},
-        partial::{Partial, PartialCurve, PartialHalfEdge, PartialObject},
+        partial::{PartialCurve, PartialHalfEdge, PartialObject},
         services::Services,
     };
 
@@ -86,16 +86,14 @@ mod tests {
     fn compute_edge_in_front_of_curve_origin() {
         let mut services = Services::new();
 
-        let surface = Partial::from(services.objects.surfaces.xy_plane());
+        let surface = services.objects.surfaces.xy_plane();
         let mut curve = PartialCurve::default();
         curve.update_as_u_axis();
         let curve = curve.build(&mut services.objects);
         let half_edge = {
             let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                surface,
-                [[1., -1.], [1., 1.]],
-            );
+            half_edge.update_as_line_segment_from_points([[1., -1.], [1., 1.]]);
+            half_edge.infer_vertex_positions_if_necessary(&surface.geometry());
 
             half_edge.build(&mut services.objects)
         };
@@ -114,16 +112,15 @@ mod tests {
     fn compute_edge_behind_curve_origin() {
         let mut services = Services::new();
 
-        let surface = Partial::from(services.objects.surfaces.xy_plane());
+        let surface = services.objects.surfaces.xy_plane();
         let mut curve = PartialCurve::default();
         curve.update_as_u_axis();
         let curve = curve.build(&mut services.objects);
         let half_edge = {
             let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                surface,
-                [[-1., -1.], [-1., 1.]],
-            );
+            half_edge
+                .update_as_line_segment_from_points([[-1., -1.], [-1., 1.]]);
+            half_edge.infer_vertex_positions_if_necessary(&surface.geometry());
 
             half_edge.build(&mut services.objects)
         };
@@ -142,16 +139,15 @@ mod tests {
     fn compute_edge_parallel_to_curve() {
         let mut services = Services::new();
 
-        let surface = Partial::from(services.objects.surfaces.xy_plane());
+        let surface = services.objects.surfaces.xy_plane();
         let mut curve = PartialCurve::default();
         curve.update_as_u_axis();
         let curve = curve.build(&mut services.objects);
         let half_edge = {
             let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                surface,
-                [[-1., -1.], [1., -1.]],
-            );
+            half_edge
+                .update_as_line_segment_from_points([[-1., -1.], [1., -1.]]);
+            half_edge.infer_vertex_positions_if_necessary(&surface.geometry());
 
             half_edge.build(&mut services.objects)
         };
@@ -165,16 +161,14 @@ mod tests {
     fn compute_edge_on_curve() {
         let mut services = Services::new();
 
-        let surface = Partial::from(services.objects.surfaces.xy_plane());
+        let surface = services.objects.surfaces.xy_plane();
         let mut curve = PartialCurve::default();
         curve.update_as_u_axis();
         let curve = curve.build(&mut services.objects);
         let half_edge = {
             let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                surface,
-                [[-1., 0.], [1., 0.]],
-            );
+            half_edge.update_as_line_segment_from_points([[-1., 0.], [1., 0.]]);
+            half_edge.infer_vertex_positions_if_necessary(&surface.geometry());
 
             half_edge.build(&mut services.objects)
         };
diff --git a/crates/fj-kernel/src/algorithms/reverse/cycle.rs b/crates/fj-kernel/src/algorithms/reverse/cycle.rs
index 23f517430..fd08da75a 100644
--- a/crates/fj-kernel/src/algorithms/reverse/cycle.rs
+++ b/crates/fj-kernel/src/algorithms/reverse/cycle.rs
@@ -17,6 +17,6 @@ impl Reverse for Handle<Cycle> {
 
         edges.reverse();
 
-        Cycle::new(edges).insert(objects)
+        Cycle::new(self.surface().clone(), edges).insert(objects)
     }
 }
diff --git a/crates/fj-kernel/src/algorithms/reverse/edge.rs b/crates/fj-kernel/src/algorithms/reverse/edge.rs
index 721f77dc8..1a9023356 100644
--- a/crates/fj-kernel/src/algorithms/reverse/edge.rs
+++ b/crates/fj-kernel/src/algorithms/reverse/edge.rs
@@ -19,7 +19,6 @@ impl Reverse for Handle<HalfEdge> {
         };
 
         HalfEdge::new(
-            self.surface().clone(),
             self.curve().clone(),
             vertices,
             self.global_form().clone(),
diff --git a/crates/fj-kernel/src/algorithms/sweep/edge.rs b/crates/fj-kernel/src/algorithms/sweep/edge.rs
index 488aeb75e..70a4c1cf1 100644
--- a/crates/fj-kernel/src/algorithms/sweep/edge.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/edge.rs
@@ -1,12 +1,10 @@
-use std::ops::Deref;
-
 use fj_interop::{ext::ArrayExt, mesh::Color};
 use fj_math::{Point, Scalar, Vector};
 
 use crate::{
     builder::{CycleBuilder, HalfEdgeBuilder},
     insert::Insert,
-    objects::{Face, HalfEdge, Objects},
+    objects::{Face, HalfEdge, Objects, Surface},
     partial::{Partial, PartialFace, PartialObject},
     services::Service,
     storage::Handle,
@@ -14,7 +12,7 @@ use crate::{
 
 use super::{Sweep, SweepCache};
 
-impl Sweep for (Handle<HalfEdge>, Color) {
+impl Sweep for (Handle<HalfEdge>, &Surface, Color) {
     type Swept = (Handle<Face>, Handle<HalfEdge>);
 
     fn sweep_with_cache(
@@ -23,7 +21,7 @@ impl Sweep for (Handle<HalfEdge>, Color) {
         cache: &mut SweepCache,
         objects: &mut Service<Objects>,
     ) -> Self::Swept {
-        let (edge, color) = self;
+        let (edge, surface, color) = self;
         let path = path.into();
 
         // The result of sweeping an edge is a face. Let's create that.
@@ -36,7 +34,7 @@ impl Sweep for (Handle<HalfEdge>, Color) {
         // 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(), edge.surface().deref())
+            (edge.curve().clone(), surface)
                 .sweep_with_cache(path, cache, objects),
         );
 
@@ -150,6 +148,8 @@ impl Sweep for (Handle<HalfEdge>, Color) {
 
 #[cfg(test)]
 mod tests {
+    use std::ops::Deref;
+
     use fj_interop::{ext::ArrayExt, mesh::Color};
     use fj_math::Point;
     use pretty_assertions::assert_eq;
@@ -168,38 +168,33 @@ mod tests {
     fn sweep() {
         let mut services = Services::new();
 
+        let surface = services.objects.surfaces.xy_plane();
+
         let half_edge = {
             let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                services.objects.surfaces.xy_plane(),
-                [[0., 0.], [1., 0.]],
-            );
+            half_edge.update_as_line_segment_from_points([[0., 0.], [1., 0.]]);
+            half_edge.infer_vertex_positions_if_necessary(&surface.geometry());
 
             half_edge
                 .build(&mut services.objects)
                 .insert(&mut services.objects)
         };
 
-        let (face, _) = (half_edge, Color::default())
+        let (face, _) = (half_edge, surface.deref(), Color::default())
             .sweep([0., 0., 1.], &mut services.objects);
 
         let expected_face = {
-            let surface = Partial::from(services.objects.surfaces.xz_plane());
+            let surface = services.objects.surfaces.xz_plane();
 
             let bottom = {
                 let mut half_edge = PartialHalfEdge::default();
-                half_edge.update_as_line_segment_from_points(
-                    surface.clone(),
-                    [[0., 0.], [1., 0.]],
-                );
+                half_edge
+                    .update_as_line_segment_from_points([[0., 0.], [1., 0.]]);
 
                 half_edge
             };
             let side_up = {
-                let mut side_up = PartialHalfEdge {
-                    surface: surface.clone(),
-                    ..Default::default()
-                };
+                let mut side_up = PartialHalfEdge::default();
 
                 {
                     let [back, front] = side_up
@@ -219,10 +214,7 @@ mod tests {
                 side_up
             };
             let top = {
-                let mut top = PartialHalfEdge {
-                    surface: surface.clone(),
-                    ..Default::default()
-                };
+                let mut top = PartialHalfEdge::default();
 
                 {
                     let [(back, back_surface), (front, front_surface)] =
@@ -238,6 +230,7 @@ mod tests {
 
                 top.infer_global_form();
                 top.update_as_line_segment();
+                top.infer_vertex_positions_if_necessary(&surface.geometry());
 
                 Partial::from(
                     top.build(&mut services.objects)
@@ -247,10 +240,7 @@ mod tests {
                 .clone()
             };
             let side_down = {
-                let mut side_down = PartialHalfEdge {
-                    surface,
-                    ..Default::default()
-                };
+                let mut side_down = PartialHalfEdge::default();
 
                 let [(back, back_surface), (front, front_surface)] =
                     side_down.vertices.each_mut_ext();
@@ -263,6 +253,8 @@ mod tests {
 
                 side_down.infer_global_form();
                 side_down.update_as_line_segment();
+                side_down
+                    .infer_vertex_positions_if_necessary(&surface.geometry());
 
                 Partial::from(
                     side_down
@@ -273,7 +265,10 @@ mod tests {
                 .clone()
             };
 
-            let mut cycle = PartialCycle::default();
+            let mut cycle = PartialCycle {
+                surface: Partial::from(surface),
+                ..Default::default()
+            };
             cycle.half_edges.extend(
                 [bottom, side_up, top, side_down].map(Partial::from_partial),
             );
diff --git a/crates/fj-kernel/src/algorithms/sweep/face.rs b/crates/fj-kernel/src/algorithms/sweep/face.rs
index db79e37f2..34a7cfb07 100644
--- a/crates/fj-kernel/src/algorithms/sweep/face.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/face.rs
@@ -1,3 +1,5 @@
+use std::ops::Deref;
+
 use fj_interop::ext::ArrayExt;
 use fj_math::{Scalar, Vector};
 
@@ -70,8 +72,9 @@ impl Sweep for Handle<Face> {
             let mut original_edges = Vec::new();
             let mut top_edges = Vec::new();
             for half_edge in cycle.half_edges().cloned() {
-                let (face, top_edge) = (half_edge.clone(), self.color())
-                    .sweep_with_cache(path, cache, objects);
+                let (face, top_edge) =
+                    (half_edge.clone(), self.surface().deref(), self.color())
+                        .sweep_with_cache(path, cache, objects);
 
                 faces.push(face);
 
@@ -128,6 +131,8 @@ impl Sweep for Handle<Face> {
 
 #[cfg(test)]
 mod tests {
+    use std::ops::Deref;
+
     use fj_interop::{ext::SliceExt, mesh::Color};
 
     use crate::{
@@ -182,7 +187,7 @@ mod tests {
             .reverse(&mut services.objects);
         let mut top = PartialFace::default();
         top.exterior.write().surface =
-            Partial::from(surface.translate(UP, &mut services.objects));
+            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)
@@ -195,17 +200,16 @@ mod tests {
         let side_faces = triangle.array_windows_ext().map(|&[a, b]| {
             let half_edge = {
                 let mut half_edge = PartialHalfEdge::default();
-                half_edge.update_as_line_segment_from_points(
-                    services.objects.surfaces.xy_plane(),
-                    [a, b],
-                );
+                half_edge.update_as_line_segment_from_points([a, b]);
+                half_edge
+                    .infer_vertex_positions_if_necessary(&surface.geometry());
 
                 half_edge
                     .build(&mut services.objects)
                     .insert(&mut services.objects)
             };
-            let (face, _) =
-                (half_edge, Color::default()).sweep(UP, &mut services.objects);
+            let (face, _) = (half_edge, surface.deref(), Color::default())
+                .sweep(UP, &mut services.objects);
             face
         });
 
@@ -250,7 +254,7 @@ mod tests {
             .insert(&mut services.objects)
             .reverse(&mut services.objects);
         let mut top = PartialFace::default();
-        top.exterior.write().surface = Partial::from(surface);
+        top.exterior.write().surface = Partial::from(surface.clone());
         top.exterior.write().update_as_polygon_from_points(TRIANGLE);
         let top = top
             .build(&mut services.objects)
@@ -263,17 +267,16 @@ mod tests {
         let side_faces = triangle.array_windows_ext().map(|&[a, b]| {
             let half_edge = {
                 let mut half_edge = PartialHalfEdge::default();
-                half_edge.update_as_line_segment_from_points(
-                    services.objects.surfaces.xy_plane(),
-                    [a, b],
-                );
+                half_edge.update_as_line_segment_from_points([a, b]);
+                half_edge
+                    .infer_vertex_positions_if_necessary(&surface.geometry());
 
                 half_edge
                     .build(&mut services.objects)
                     .insert(&mut services.objects)
                     .reverse(&mut services.objects)
             };
-            let (face, _) = (half_edge, Color::default())
+            let (face, _) = (half_edge, surface.deref(), Color::default())
                 .sweep(DOWN, &mut services.objects);
             face
         });
diff --git a/crates/fj-kernel/src/algorithms/transform/cycle.rs b/crates/fj-kernel/src/algorithms/transform/cycle.rs
index 5fbc6eb32..e56793752 100644
--- a/crates/fj-kernel/src/algorithms/transform/cycle.rs
+++ b/crates/fj-kernel/src/algorithms/transform/cycle.rs
@@ -14,12 +14,16 @@ impl TransformObject for Cycle {
         objects: &mut Service<Objects>,
         cache: &mut TransformCache,
     ) -> Self {
+        let surface = self
+            .surface()
+            .clone()
+            .transform_with_cache(transform, objects, cache);
         let half_edges = self.half_edges().map(|half_edge| {
             half_edge
                 .clone()
                 .transform_with_cache(transform, objects, cache)
         });
 
-        Self::new(half_edges)
+        Self::new(surface, half_edges)
     }
 }
diff --git a/crates/fj-kernel/src/algorithms/transform/edge.rs b/crates/fj-kernel/src/algorithms/transform/edge.rs
index 796769f5e..4c5b7b242 100644
--- a/crates/fj-kernel/src/algorithms/transform/edge.rs
+++ b/crates/fj-kernel/src/algorithms/transform/edge.rs
@@ -15,10 +15,6 @@ impl TransformObject for HalfEdge {
         objects: &mut Service<Objects>,
         cache: &mut TransformCache,
     ) -> Self {
-        let surface = self
-            .surface()
-            .clone()
-            .transform_with_cache(transform, objects, cache);
         let curve = self
             .curve()
             .clone()
@@ -36,7 +32,7 @@ impl TransformObject for HalfEdge {
             .clone()
             .transform_with_cache(transform, objects, cache);
 
-        Self::new(surface, curve, boundary, global_form)
+        Self::new(curve, boundary, global_form)
     }
 }
 
diff --git a/crates/fj-kernel/src/builder/cycle.rs b/crates/fj-kernel/src/builder/cycle.rs
index 3e7202905..3673b0ee8 100644
--- a/crates/fj-kernel/src/builder/cycle.rs
+++ b/crates/fj-kernel/src/builder/cycle.rs
@@ -149,7 +149,6 @@ impl CycleBuilder for PartialCycle {
 
             let [_, vertex] = &mut new_half_edge.vertices;
             vertex.1 = shared_surface_vertex;
-            new_half_edge.surface = self.surface.clone();
             new_half_edge.infer_global_form();
         }
 
@@ -247,7 +246,8 @@ impl CycleBuilder for PartialCycle {
             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);
+            this.write()
+                .update_from_other_edge(&other, &self.surface.read().geometry);
             this
         })
     }
@@ -261,7 +261,8 @@ impl CycleBuilder for PartialCycle {
     {
         edges.map(|other| {
             let mut this = self.add_half_edge();
-            this.write().update_from_other_edge(&other);
+            this.write()
+                .update_from_other_edge(&other, &self.surface.read().geometry);
             this
         })
     }
diff --git a/crates/fj-kernel/src/builder/edge.rs b/crates/fj-kernel/src/builder/edge.rs
index 78f85e921..c4f5b7402 100644
--- a/crates/fj-kernel/src/builder/edge.rs
+++ b/crates/fj-kernel/src/builder/edge.rs
@@ -2,8 +2,11 @@ use fj_interop::ext::ArrayExt;
 use fj_math::{Point, Scalar};
 
 use crate::{
-    geometry::path::{GlobalPath, SurfacePath},
-    objects::{GlobalEdge, HalfEdge, Surface},
+    geometry::{
+        path::{GlobalPath, SurfacePath},
+        surface::SurfaceGeometry,
+    },
+    objects::{GlobalEdge, HalfEdge},
     partial::{MaybeSurfacePath, Partial, PartialGlobalEdge, PartialHalfEdge},
 };
 
@@ -25,7 +28,6 @@ pub trait HalfEdgeBuilder {
     /// Update partial half-edge to be a line segment, from the given points
     fn update_as_line_segment_from_points(
         &mut self,
-        surface: impl Into<Partial<Surface>>,
         points: [impl Into<Point<2>>; 2],
     );
 
@@ -38,6 +40,12 @@ pub trait HalfEdgeBuilder {
     /// it.
     fn infer_global_form(&mut self) -> Partial<GlobalEdge>;
 
+    /// Infer the vertex positions (surface and global), if not already set
+    fn infer_vertex_positions_if_necessary(
+        &mut self,
+        surface: &SurfaceGeometry,
+    );
+
     /// Update this edge from another
     ///
     /// Infers as much information about this edge from the other, under the
@@ -47,7 +55,11 @@ pub trait HalfEdgeBuilder {
     /// under various circumstances. As long as you're only dealing with lines
     /// and planes, you should be fine. Otherwise, please read the code of this
     /// method carefully, to make sure you don't run into trouble.
-    fn update_from_other_edge(&mut self, other: &Partial<HalfEdge>);
+    fn update_from_other_edge(
+        &mut self,
+        other: &Partial<HalfEdge>,
+        surface: &Option<SurfaceGeometry>,
+    );
 }
 
 impl HalfEdgeBuilder for PartialHalfEdge {
@@ -111,11 +123,8 @@ impl HalfEdgeBuilder for PartialHalfEdge {
 
     fn update_as_line_segment_from_points(
         &mut self,
-        surface: impl Into<Partial<Surface>>,
         points: [impl Into<Point<2>>; 2],
     ) {
-        self.surface = surface.into();
-
         for (vertex, point) in self.vertices.each_mut_ext().zip_ext(points) {
             let mut surface_form = vertex.1.write();
             surface_form.position = Some(point.into());
@@ -166,14 +175,62 @@ impl HalfEdgeBuilder for PartialHalfEdge {
         self.global_form.clone()
     }
 
-    fn update_from_other_edge(&mut self, other: &Partial<HalfEdge>) {
+    fn infer_vertex_positions_if_necessary(
+        &mut self,
+        surface: &SurfaceGeometry,
+    ) {
+        let path = self
+            .curve
+            .read()
+            .path
+            .expect("Can't infer vertex positions without curve");
+        let MaybeSurfacePath::Defined(path) = path else {
+            panic!("Can't infer vertex positions with undefined path");
+        };
+
+        for vertex in &mut self.vertices {
+            let position_curve = vertex
+                .0
+                .expect("Can't infer surface position without curve position");
+
+            let position_surface = vertex.1.read().position;
+
+            // Infer surface position, if not available.
+            let position_surface = match position_surface {
+                Some(position_surface) => position_surface,
+                None => {
+                    let position_surface =
+                        path.point_from_path_coords(position_curve);
+
+                    vertex.1.write().position = Some(position_surface);
+
+                    position_surface
+                }
+            };
+
+            // Infer global position, if not available.
+            let position_global = vertex.1.read().global_form.read().position;
+            if position_global.is_none() {
+                let position_global =
+                    surface.point_from_surface_coords(position_surface);
+                vertex.1.write().global_form.write().position =
+                    Some(position_global);
+            }
+        }
+    }
+
+    fn update_from_other_edge(
+        &mut self,
+        other: &Partial<HalfEdge>,
+        surface: &Option<SurfaceGeometry>,
+    ) {
         let global_curve = other.read().curve.read().global_form.clone();
         self.curve.write().global_form = global_curve.clone();
         self.global_form.write().curve = global_curve;
 
         self.curve.write().path =
             other.read().curve.read().path.as_ref().and_then(|path| {
-                match other.read().surface.read().geometry {
+                match surface {
                     Some(surface) => {
                         // We have information about the other edge's surface
                         // available. We need to use that to interpret what the
diff --git a/crates/fj-kernel/src/objects/full/cycle.rs b/crates/fj-kernel/src/objects/full/cycle.rs
index 66e56a200..5fdfb4a4a 100644
--- a/crates/fj-kernel/src/objects/full/cycle.rs
+++ b/crates/fj-kernel/src/objects/full/cycle.rs
@@ -12,6 +12,7 @@ use crate::{
 /// A cycle of connected half-edges
 #[derive(Clone, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
 pub struct Cycle {
+    surface: Handle<Surface>,
     half_edges: Vec<Handle<HalfEdge>>,
 }
 
@@ -21,7 +22,10 @@ impl Cycle {
     /// # Panics
     ///
     /// Panics, if `half_edges` does not yield at least one half-edge.
-    pub fn new(half_edges: impl IntoIterator<Item = Handle<HalfEdge>>) -> Self {
+    pub fn new(
+        surface: Handle<Surface>,
+        half_edges: impl IntoIterator<Item = Handle<HalfEdge>>,
+    ) -> Self {
         let half_edges = half_edges.into_iter().collect::<Vec<_>>();
 
         // This is not a validation check, and thus not part of the validation
@@ -33,18 +37,15 @@ impl Cycle {
             "Cycle must contain at least one half-edge"
         );
 
-        Self { half_edges }
+        Self {
+            surface,
+            half_edges,
+        }
     }
 
     /// Access the surface that the cycle is in
     pub fn surface(&self) -> &Handle<Surface> {
-        if let Some(half_edge) = self.half_edges.first() {
-            return half_edge.surface();
-        }
-
-        unreachable!(
-            "Cycle has no half-edges, which the constructor should prevent."
-        )
+        &self.surface
     }
 
     /// Access the half-edges that make up the cycle
diff --git a/crates/fj-kernel/src/objects/full/edge.rs b/crates/fj-kernel/src/objects/full/edge.rs
index f7a77b378..5288eec25 100644
--- a/crates/fj-kernel/src/objects/full/edge.rs
+++ b/crates/fj-kernel/src/objects/full/edge.rs
@@ -4,14 +4,13 @@ use fj_interop::ext::ArrayExt;
 use fj_math::Point;
 
 use crate::{
-    objects::{Curve, GlobalCurve, GlobalVertex, Surface, SurfaceVertex},
+    objects::{Curve, GlobalCurve, GlobalVertex, SurfaceVertex},
     storage::{Handle, HandleWrapper},
 };
 
 /// A half-edge
 #[derive(Clone, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
 pub struct HalfEdge {
-    surface: Handle<Surface>,
     curve: Handle<Curve>,
     boundary: [(Point<1>, Handle<SurfaceVertex>); 2],
     global_form: Handle<GlobalEdge>,
@@ -20,24 +19,17 @@ pub struct HalfEdge {
 impl HalfEdge {
     /// Create an instance of `HalfEdge`
     pub fn new(
-        surface: Handle<Surface>,
         curve: Handle<Curve>,
         boundary: [(Point<1>, Handle<SurfaceVertex>); 2],
         global_form: Handle<GlobalEdge>,
     ) -> Self {
         Self {
-            surface,
             curve,
             boundary,
             global_form,
         }
     }
 
-    /// Access the surface that the half-edge is defined in
-    pub fn surface(&self) -> &Handle<Surface> {
-        &self.surface
-    }
-
     /// Access the curve that defines the half-edge's geometry
     pub fn curve(&self) -> &Handle<Curve> {
         &self.curve
@@ -173,14 +165,15 @@ mod tests {
 
         let a_to_b = {
             let mut half_edge = PartialHalfEdge::default();
-            half_edge
-                .update_as_line_segment_from_points(surface.clone(), [a, b]);
+            half_edge.update_as_line_segment_from_points([a, b]);
+            half_edge.infer_vertex_positions_if_necessary(&surface.geometry());
 
             half_edge.build(&mut services.objects)
         };
         let b_to_a = {
             let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(surface, [b, a]);
+            half_edge.update_as_line_segment_from_points([b, a]);
+            half_edge.infer_vertex_positions_if_necessary(&surface.geometry());
 
             half_edge.build(&mut services.objects)
         };
diff --git a/crates/fj-kernel/src/partial/objects/curve.rs b/crates/fj-kernel/src/partial/objects/curve.rs
index 3826d4cb2..4377536ae 100644
--- a/crates/fj-kernel/src/partial/objects/curve.rs
+++ b/crates/fj-kernel/src/partial/objects/curve.rs
@@ -46,7 +46,7 @@ impl PartialObject for PartialCurve {
 ///
 /// Can be a fully defined [`SurfacePath`], or just the type of path might be
 /// known.
-#[derive(Clone, Debug)]
+#[derive(Clone, Copy, Debug)]
 pub enum MaybeSurfacePath {
     /// The surface path is fully defined
     Defined(SurfacePath),
diff --git a/crates/fj-kernel/src/partial/objects/cycle.rs b/crates/fj-kernel/src/partial/objects/cycle.rs
index e1d109555..386fce9e1 100644
--- a/crates/fj-kernel/src/partial/objects/cycle.rs
+++ b/crates/fj-kernel/src/partial/objects/cycle.rs
@@ -1,4 +1,5 @@
 use crate::{
+    builder::HalfEdgeBuilder,
     objects::{Cycle, HalfEdge, Objects, Surface},
     partial::{FullToPartialCache, Partial, PartialObject},
     services::Service,
@@ -29,11 +30,15 @@ impl PartialObject for PartialCycle {
     }
 
     fn build(self, objects: &mut Service<Objects>) -> Self::Full {
-        let half_edges = self
-            .half_edges
-            .into_iter()
-            .map(|half_edge| half_edge.build(objects));
+        let surface = self.surface.build(objects);
+        let surface_geometry = surface.geometry();
+        let half_edges = self.half_edges.into_iter().map(|mut half_edge| {
+            half_edge
+                .write()
+                .infer_vertex_positions_if_necessary(&surface_geometry);
+            half_edge.build(objects)
+        });
 
-        Cycle::new(half_edges)
+        Cycle::new(surface, half_edges)
     }
 }
diff --git a/crates/fj-kernel/src/partial/objects/edge.rs b/crates/fj-kernel/src/partial/objects/edge.rs
index a0057c85a..db7ea206b 100644
--- a/crates/fj-kernel/src/partial/objects/edge.rs
+++ b/crates/fj-kernel/src/partial/objects/edge.rs
@@ -6,7 +6,7 @@ use fj_math::Point;
 use crate::{
     objects::{
         Curve, GlobalCurve, GlobalEdge, GlobalVertex, HalfEdge, Objects,
-        Surface, SurfaceVertex,
+        SurfaceVertex,
     },
     partial::{FullToPartialCache, Partial, PartialObject},
     services::Service,
@@ -15,9 +15,6 @@ use crate::{
 /// A partial [`HalfEdge`]
 #[derive(Clone, Debug)]
 pub struct PartialHalfEdge {
-    /// The surface that the half-edge is defined in
-    pub surface: Partial<Surface>,
-
     /// The curve that the half-edge is defined in
     pub curve: Partial<Curve>,
 
@@ -36,7 +33,6 @@ impl PartialObject for PartialHalfEdge {
         cache: &mut FullToPartialCache,
     ) -> Self {
         Self {
-            surface: Partial::from_full(half_edge.surface().clone(), cache),
             curve: Partial::from_full(half_edge.curve().clone(), cache),
             vertices: half_edge
                 .boundary()
@@ -55,53 +51,23 @@ impl PartialObject for PartialHalfEdge {
     }
 
     fn build(self, objects: &mut Service<Objects>) -> Self::Full {
-        let surface = self.surface.build(objects);
         let curve = self.curve.build(objects);
-        let vertices = self.vertices.map(|mut vertex| {
-            let position_surface = vertex.1.read().position;
-
-            // Infer surface position, if not available.
-            let position_surface = match position_surface {
-                Some(position_surface) => position_surface,
-                None => {
-                    let position_curve = vertex.0.expect(
-                        "Can't infer surface position without curve position",
-                    );
-                    let position_surface =
-                        curve.path().point_from_path_coords(position_curve);
-
-                    vertex.1.write().position = Some(position_surface);
-
-                    position_surface
-                }
-            };
-
-            // Infer global position, if not available.
-            let position_global = vertex.1.read().global_form.read().position;
-            if position_global.is_none() {
-                let position_global = surface
-                    .geometry()
-                    .point_from_surface_coords(position_surface);
-                vertex.1.write().global_form.write().position =
-                    Some(position_global);
-            }
-
-            let position =
-                vertex.0.expect("Can't build `Vertex` without position");
+        let vertices = self.vertices.map(|vertex| {
+            let position_curve = vertex
+                .0
+                .expect("Can't build `HalfEdge` without boundary positions");
             let surface_form = vertex.1.build(objects);
 
-            (position, surface_form)
+            (position_curve, surface_form)
         });
         let global_form = self.global_form.build(objects);
 
-        HalfEdge::new(surface, curve, vertices, global_form)
+        HalfEdge::new(curve, vertices, global_form)
     }
 }
 
 impl Default for PartialHalfEdge {
     fn default() -> Self {
-        let surface = Partial::new();
-
         let curve = Partial::<Curve>::new();
         let vertices = array::from_fn(|_| {
             let surface_form = Partial::default();
@@ -123,7 +89,6 @@ impl Default for PartialHalfEdge {
         });
 
         Self {
-            surface,
             curve,
             vertices,
             global_form,
diff --git a/crates/fj-kernel/src/validate/cycle.rs b/crates/fj-kernel/src/validate/cycle.rs
index c903e2ed0..264c4acf6 100644
--- a/crates/fj-kernel/src/validate/cycle.rs
+++ b/crates/fj-kernel/src/validate/cycle.rs
@@ -17,6 +17,7 @@ impl Validate for Cycle {
     ) {
         CycleValidationError::check_half_edge_connections(self, errors);
         CycleValidationError::check_half_edge_boundaries(self, config, errors);
+        CycleValidationError::check_vertex_positions(self, config, errors);
 
         // We don't need to check that all half-edges are defined in the same
         // surface. We already check that they are connected by identical
@@ -70,6 +71,33 @@ pub enum CycleValidationError {
         /// The half-edge
         half_edge: Handle<HalfEdge>,
     },
+
+    /// Mismatch between [`SurfaceVertex`] and `GlobalVertex` positions
+    #[error(
+        "`SurfaceVertex` position doesn't match position of its global form\n\
+        - Surface position: {surface_position:?}\n\
+        - Surface position converted to global position: \
+            {surface_position_as_global:?}\n\
+        - Global position: {global_position:?}\n\
+        - Distance between the positions: {distance}\n\
+        - `SurfaceVertex`: {surface_vertex:#?}"
+    )]
+    VertexSurfacePositionMismatch {
+        /// The position of the surface vertex
+        surface_position: Point<2>,
+
+        /// The surface position converted into a global position
+        surface_position_as_global: Point<3>,
+
+        /// The position of the global vertex
+        global_position: Point<3>,
+
+        /// The distance between the positions
+        distance: Scalar,
+
+        /// The surface vertex
+        surface_vertex: Handle<SurfaceVertex>,
+    },
 }
 
 impl CycleValidationError {
@@ -130,15 +158,49 @@ impl CycleValidationError {
             }
         }
     }
+
+    fn check_vertex_positions(
+        cycle: &Cycle,
+        config: &ValidationConfig,
+        errors: &mut Vec<ValidationError>,
+    ) {
+        for half_edge in cycle.half_edges() {
+            for surface_vertex in half_edge.surface_vertices() {
+                let surface_position_as_global = cycle
+                    .surface()
+                    .geometry()
+                    .point_from_surface_coords(surface_vertex.position());
+                let global_position = surface_vertex.global_form().position();
+
+                let distance =
+                    surface_position_as_global.distance_to(&global_position);
+
+                if distance > config.identical_max_distance {
+                    errors.push(
+                        Self::VertexSurfacePositionMismatch {
+                            surface_position: surface_vertex.position(),
+                            surface_position_as_global,
+                            global_position,
+                            distance,
+                            surface_vertex: surface_vertex.clone(),
+                        }
+                        .into(),
+                    );
+                }
+            }
+        }
+    }
 }
 
 #[cfg(test)]
 mod tests {
-    use fj_math::Point;
+    use fj_interop::ext::ArrayExt;
+    use fj_math::{Point, Scalar, Vector};
 
     use crate::{
-        builder::CycleBuilder,
-        objects::Cycle,
+        builder::{CycleBuilder, HalfEdgeBuilder},
+        insert::Insert,
+        objects::{Cycle, HalfEdge, SurfaceVertex},
         partial::{Partial, PartialCycle, PartialObject},
         services::Services,
         validate::Validate,
@@ -176,7 +238,7 @@ mod tests {
                 .into_iter()
                 .map(|half_edge| half_edge.build(&mut services.objects));
 
-            Cycle::new(half_edges)
+            Cycle::new(valid.surface().clone(), half_edges)
         };
 
         valid.validate_and_return_first_error()?;
@@ -212,7 +274,69 @@ mod tests {
                 .into_iter()
                 .map(|half_edge| half_edge.build(&mut services.objects));
 
-            Cycle::new(half_edges)
+            Cycle::new(valid.surface().clone(), half_edges)
+        };
+
+        valid.validate_and_return_first_error()?;
+        assert!(invalid.validate_and_return_first_error().is_err());
+
+        Ok(())
+    }
+
+    #[test]
+    fn surface_vertex_position_mismatch() -> anyhow::Result<()> {
+        let mut services = Services::new();
+
+        let valid = {
+            let surface = services.objects.surfaces.xy_plane();
+
+            let mut cycle = PartialCycle {
+                surface: Partial::from(surface.clone()),
+                ..Default::default()
+            };
+
+            let mut half_edge = cycle.add_half_edge();
+            half_edge.write().update_as_circle_from_radius(1.);
+            half_edge
+                .write()
+                .infer_vertex_positions_if_necessary(&surface.geometry());
+
+            cycle.build(&mut services.objects)
+        };
+        let invalid = {
+            let half_edge = {
+                let half_edge = valid.half_edges().next().unwrap();
+
+                let boundary = half_edge
+                    .boundary()
+                    .map(|point| point + Vector::from([Scalar::PI / 2.]));
+
+                let mut surface_vertices =
+                    half_edge.surface_vertices().map(Clone::clone);
+
+                let mut invalid = None;
+                for surface_vertex in surface_vertices.each_mut_ext() {
+                    let invalid = invalid.get_or_insert_with(|| {
+                        SurfaceVertex::new(
+                            [0., 1.],
+                            surface_vertex.global_form().clone(),
+                        )
+                        .insert(&mut services.objects)
+                    });
+                    *surface_vertex = invalid.clone();
+                }
+
+                let boundary = boundary.zip_ext(surface_vertices);
+
+                HalfEdge::new(
+                    half_edge.curve().clone(),
+                    boundary,
+                    half_edge.global_form().clone(),
+                )
+                .insert(&mut services.objects)
+            };
+
+            Cycle::new(valid.surface().clone(), [half_edge])
         };
 
         valid.validate_and_return_first_error()?;
diff --git a/crates/fj-kernel/src/validate/edge.rs b/crates/fj-kernel/src/validate/edge.rs
index d5658975f..e0f7e933c 100644
--- a/crates/fj-kernel/src/validate/edge.rs
+++ b/crates/fj-kernel/src/validate/edge.rs
@@ -1,9 +1,7 @@
 use fj_math::{Point, Scalar};
 
 use crate::{
-    objects::{
-        GlobalCurve, GlobalEdge, GlobalVertex, HalfEdge, Surface, SurfaceVertex,
-    },
+    objects::{GlobalCurve, GlobalEdge, GlobalVertex, HalfEdge, Surface},
     storage::Handle,
 };
 
@@ -18,7 +16,6 @@ impl Validate for HalfEdge {
         HalfEdgeValidationError::check_global_curve_identity(self, errors);
         HalfEdgeValidationError::check_global_vertex_identity(self, errors);
         HalfEdgeValidationError::check_vertex_coincidence(self, config, errors);
-        HalfEdgeValidationError::check_vertex_positions(self, config, errors);
     }
 }
 
@@ -107,33 +104,6 @@ pub enum HalfEdgeValidationError {
         /// The half-edge
         half_edge: HalfEdge,
     },
-
-    /// Mismatch between [`SurfaceVertex`] and [`GlobalVertex`] positions
-    #[error(
-        "`SurfaceVertex` position doesn't match position of its global form\n\
-        - Surface position: {surface_position:?}\n\
-        - Surface position converted to global position: \
-            {surface_position_as_global:?}\n\
-        - Global position: {global_position:?}\n\
-        - Distance between the positions: {distance}\n\
-        - `SurfaceVertex`: {surface_vertex:#?}"
-    )]
-    VertexSurfacePositionMismatch {
-        /// The position of the surface vertex
-        surface_position: Point<2>,
-
-        /// The surface position converted into a global position
-        surface_position_as_global: Point<3>,
-
-        /// The position of the global vertex
-        global_position: Point<3>,
-
-        /// The distance between the positions
-        distance: Scalar,
-
-        /// The surface vertex
-        surface_vertex: Handle<SurfaceVertex>,
-    },
 }
 
 impl HalfEdgeValidationError {
@@ -206,36 +176,6 @@ impl HalfEdgeValidationError {
             );
         }
     }
-
-    fn check_vertex_positions(
-        half_edge: &HalfEdge,
-        config: &ValidationConfig,
-        errors: &mut Vec<ValidationError>,
-    ) {
-        for surface_vertex in half_edge.surface_vertices() {
-            let surface_position_as_global = half_edge
-                .surface()
-                .geometry()
-                .point_from_surface_coords(surface_vertex.position());
-            let global_position = surface_vertex.global_form().position();
-
-            let distance =
-                surface_position_as_global.distance_to(&global_position);
-
-            if distance > config.identical_max_distance {
-                errors.push(
-                    Box::new(Self::VertexSurfacePositionMismatch {
-                        surface_position: surface_vertex.position(),
-                        surface_position_as_global,
-                        global_position,
-                        distance,
-                        surface_vertex: surface_vertex.clone(),
-                    })
-                    .into(),
-                );
-            }
-        }
-    }
 }
 
 #[cfg(test)]
@@ -246,7 +186,7 @@ mod tests {
     use crate::{
         builder::HalfEdgeBuilder,
         insert::Insert,
-        objects::{GlobalCurve, HalfEdge, SurfaceVertex},
+        objects::{GlobalCurve, HalfEdge},
         partial::{Partial, PartialHalfEdge, PartialObject},
         services::Services,
         validate::Validate,
@@ -257,11 +197,11 @@ mod tests {
         let mut services = Services::new();
 
         let valid = {
+            let surface = services.objects.surfaces.xy_plane();
+
             let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                services.objects.surfaces.xy_plane(),
-                [[0., 0.], [1., 0.]],
-            );
+            half_edge.update_as_line_segment_from_points([[0., 0.], [1., 0.]]);
+            half_edge.infer_vertex_positions_if_necessary(&surface.geometry());
 
             half_edge.build(&mut services.objects)
         };
@@ -277,12 +217,7 @@ mod tests {
                 .boundary()
                 .zip_ext(valid.surface_vertices().map(Clone::clone));
 
-            HalfEdge::new(
-                valid.surface().clone(),
-                valid.curve().clone(),
-                vertices,
-                global_form,
-            )
+            HalfEdge::new(valid.curve().clone(), vertices, global_form)
         };
 
         valid.validate_and_return_first_error()?;
@@ -296,11 +231,11 @@ mod tests {
         let mut services = Services::new();
 
         let valid = {
+            let surface = services.objects.surfaces.xy_plane();
+
             let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                services.objects.surfaces.xy_plane(),
-                [[0., 0.], [1., 0.]],
-            );
+            half_edge.update_as_line_segment_from_points([[0., 0.], [1., 0.]]);
+            half_edge.infer_vertex_positions_if_necessary(&surface.geometry());
 
             half_edge.build(&mut services.objects)
         };
@@ -324,12 +259,7 @@ mod tests {
                 .boundary()
                 .zip_ext(valid.surface_vertices().map(Clone::clone));
 
-            HalfEdge::new(
-                valid.surface().clone(),
-                valid.curve().clone(),
-                vertices,
-                global_form,
-            )
+            HalfEdge::new(valid.curve().clone(), vertices, global_form)
         };
 
         valid.validate_and_return_first_error()?;
@@ -343,11 +273,11 @@ mod tests {
         let mut services = Services::new();
 
         let valid = {
+            let surface = services.objects.surfaces.xy_plane();
+
             let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                services.objects.surfaces.xy_plane(),
-                [[0., 0.], [1., 0.]],
-            );
+            half_edge.update_as_line_segment_from_points([[0., 0.], [1., 0.]]);
+            half_edge.infer_vertex_positions_if_necessary(&surface.geometry());
 
             half_edge.build(&mut services.objects)
         };
@@ -357,7 +287,6 @@ mod tests {
             });
 
             HalfEdge::new(
-                valid.surface().clone(),
                 valid.curve().clone(),
                 vertices,
                 valid.global_form().clone(),
@@ -369,44 +298,4 @@ mod tests {
 
         Ok(())
     }
-
-    #[test]
-    fn surface_vertex_position_mismatch() -> anyhow::Result<()> {
-        let mut services = Services::new();
-
-        let valid = {
-            let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                services.objects.surfaces.xy_plane(),
-                [[0., 0.], [1., 0.]],
-            );
-
-            half_edge.build(&mut services.objects)
-        };
-        let invalid = {
-            let mut surface_vertices =
-                valid.surface_vertices().map(Clone::clone);
-
-            let [_, surface_vertex] = surface_vertices.each_mut_ext();
-            *surface_vertex = SurfaceVertex::new(
-                [2., 0.],
-                surface_vertex.global_form().clone(),
-            )
-            .insert(&mut services.objects);
-
-            let boundary = valid.boundary().zip_ext(surface_vertices);
-
-            HalfEdge::new(
-                valid.surface().clone(),
-                valid.curve().clone(),
-                boundary,
-                valid.global_form().clone(),
-            )
-        };
-
-        valid.validate_and_return_first_error()?;
-        assert!(invalid.validate_and_return_first_error().is_err());
-
-        Ok(())
-    }
 }
diff --git a/crates/fj-operations/src/sketch.rs b/crates/fj-operations/src/sketch.rs
index a790d391c..f9e402aeb 100644
--- a/crates/fj-operations/src/sketch.rs
+++ b/crates/fj-operations/src/sketch.rs
@@ -27,19 +27,19 @@ impl Shape for fj::Sketch {
 
         let face = match self.chain() {
             fj::Chain::Circle(circle) => {
-                let half_edge = {
-                    let surface = Partial::from(surface);
+                let surface = Partial::from(surface);
 
-                    let mut half_edge = PartialHalfEdge {
-                        surface,
-                        ..Default::default()
-                    };
+                let half_edge = {
+                    let mut half_edge = PartialHalfEdge::default();
                     half_edge.update_as_circle_from_radius(circle.radius());
 
                     Partial::from_partial(half_edge)
                 };
                 let exterior = {
-                    let mut cycle = PartialCycle::default();
+                    let mut cycle = PartialCycle {
+                        surface,
+                        ..Default::default()
+                    };
                     cycle.half_edges.push(half_edge);
                     Partial::from_partial(cycle)
                 };