diff --git a/crates/fj-core/src/algorithms/approx/cycle.rs b/crates/fj-core/src/algorithms/approx/cycle.rs
index dfedcdf38..981fcb8f0 100644
--- a/crates/fj-core/src/algorithms/approx/cycle.rs
+++ b/crates/fj-core/src/algorithms/approx/cycle.rs
@@ -5,7 +5,7 @@
 use fj_math::Segment;
 
 use crate::{
-    geometry::Geometry,
+    geometry::{CurveBoundary, Geometry},
     storage::Handle,
     topology::{Cycle, Surface},
 };
@@ -28,9 +28,24 @@ pub fn approx_cycle(
 
     let half_edges = cycle
         .half_edges()
-        .iter()
-        .map(|half_edge| {
-            let boundary = geometry.of_half_edge(half_edge).boundary;
+        .pairs()
+        .map(|(half_edge, next_half_edge)| {
+            let boundary = CurveBoundary {
+                inner: [
+                    geometry
+                        .of_vertex(half_edge.start_vertex())
+                        .unwrap()
+                        .local_on(half_edge.curve())
+                        .unwrap()
+                        .position,
+                    geometry
+                        .of_vertex(next_half_edge.start_vertex())
+                        .unwrap()
+                        .local_on(half_edge.curve())
+                        .unwrap()
+                        .position,
+                ],
+            };
             let [start_position_curve, _] = boundary.inner;
 
             let start = approx_vertex(
diff --git a/crates/fj-core/src/algorithms/bounding_volume/cycle.rs b/crates/fj-core/src/algorithms/bounding_volume/cycle.rs
index a934a38a1..1c19db6ae 100644
--- a/crates/fj-core/src/algorithms/bounding_volume/cycle.rs
+++ b/crates/fj-core/src/algorithms/bounding_volume/cycle.rs
@@ -12,8 +12,8 @@ impl super::BoundingVolume<2> for (&Cycle, &Handle<Surface>) {
 
         let mut aabb: Option<Aabb<2>> = None;
 
-        for half_edge in cycle.half_edges() {
-            let new_aabb = (half_edge, surface)
+        for (half_edge, half_edge_next) in cycle.half_edges().pairs() {
+            let new_aabb = (half_edge, half_edge_next.start_vertex(), surface)
                 .aabb(geometry)
                 .expect("`HalfEdge` can always compute AABB");
             aabb = Some(aabb.map_or(new_aabb, |aabb| aabb.merged(&new_aabb)));
diff --git a/crates/fj-core/src/algorithms/bounding_volume/half_edge.rs b/crates/fj-core/src/algorithms/bounding_volume/half_edge.rs
index 5f5322f62..ba8ee4408 100644
--- a/crates/fj-core/src/algorithms/bounding_volume/half_edge.rs
+++ b/crates/fj-core/src/algorithms/bounding_volume/half_edge.rs
@@ -3,14 +3,15 @@ use fj_math::{Aabb, Vector};
 use crate::{
     geometry::{Geometry, SurfacePath},
     storage::Handle,
-    topology::{HalfEdge, Surface},
+    topology::{HalfEdge, Surface, Vertex},
 };
 
-impl super::BoundingVolume<2> for (&Handle<HalfEdge>, &Handle<Surface>) {
+impl super::BoundingVolume<2>
+    for (&Handle<HalfEdge>, &Handle<Vertex>, &Handle<Surface>)
+{
     fn aabb(self, geometry: &Geometry) -> Option<Aabb<2>> {
-        let (half_edge, surface) = self;
+        let (half_edge, end_vertex, surface) = self;
 
-        let half_edge_geom = geometry.of_half_edge(half_edge);
         let path = geometry
             .of_curve(half_edge.curve())
             .unwrap()
@@ -32,9 +33,17 @@ impl super::BoundingVolume<2> for (&Handle<HalfEdge>, &Handle<Surface>) {
                 })
             }
             SurfacePath::Line(_) => {
-                let points = half_edge_geom.boundary.inner.map(|point_curve| {
-                    path.point_from_path_coords(point_curve)
-                });
+                let points =
+                    [half_edge.start_vertex(), end_vertex].map(|vertex| {
+                        let point_curve = geometry
+                            .of_vertex(vertex)
+                            .unwrap()
+                            .local_on(half_edge.curve())
+                            .unwrap()
+                            .position;
+
+                        path.point_from_path_coords(point_curve)
+                    });
 
                 Some(Aabb::<2>::from_points(points))
             }
diff --git a/crates/fj-core/src/operations/split/face.rs b/crates/fj-core/src/operations/split/face.rs
index 8b8b53e67..c742ac478 100644
--- a/crates/fj-core/src/operations/split/face.rs
+++ b/crates/fj-core/src/operations/split/face.rs
@@ -104,29 +104,43 @@ impl SplitFace for Shell {
 
         // Build the edge that's going to divide the new faces.
         let dividing_half_edge_a_to_d = {
+            let start = core
+                .layers
+                .geometry
+                .of_curve(b.curve())
+                .unwrap()
+                .local_on(face.surface())
+                .unwrap()
+                .path
+                .point_from_path_coords(
+                    core.layers
+                        .geometry
+                        .of_vertex(b.start_vertex())
+                        .unwrap()
+                        .local_on(b.curve())
+                        .unwrap()
+                        .position,
+                );
+            let end = core
+                .layers
+                .geometry
+                .of_curve(d.curve())
+                .unwrap()
+                .local_on(face.surface())
+                .unwrap()
+                .path
+                .point_from_path_coords(
+                    core.layers
+                        .geometry
+                        .of_vertex(d.start_vertex())
+                        .unwrap()
+                        .local_on(d.curve())
+                        .unwrap()
+                        .position,
+                );
+
             let (half_edge, boundary) = HalfEdge::line_segment(
-                [
-                    core.layers.geometry.of_half_edge(&b).start_position(
-                        &core
-                            .layers
-                            .geometry
-                            .of_curve(b.curve())
-                            .unwrap()
-                            .local_on(face.surface())
-                            .unwrap()
-                            .path,
-                    ),
-                    core.layers.geometry.of_half_edge(&d).start_position(
-                        &core
-                            .layers
-                            .geometry
-                            .of_curve(d.curve())
-                            .unwrap()
-                            .local_on(face.surface())
-                            .unwrap()
-                            .path,
-                    ),
-                ],
+                [start, end],
                 face.surface().clone(),
                 core,
             );
diff --git a/crates/fj-core/src/operations/split/half_edge.rs b/crates/fj-core/src/operations/split/half_edge.rs
index 3e304e933..ba9ee5d64 100644
--- a/crates/fj-core/src/operations/split/half_edge.rs
+++ b/crates/fj-core/src/operations/split/half_edge.rs
@@ -46,7 +46,27 @@ impl SplitHalfEdge for Cycle {
         let point = point.into();
 
         let geometry = *core.layers.geometry.of_half_edge(half_edge);
-        let [start, end] = geometry.boundary.inner;
+        let [start, end] = [
+            core.layers
+                .geometry
+                .of_vertex(half_edge.start_vertex())
+                .unwrap()
+                .local_on(half_edge.curve())
+                .unwrap()
+                .position,
+            core.layers
+                .geometry
+                .of_vertex(
+                    self.half_edges()
+                        .after(half_edge)
+                        .expect("Expected half-edge to be in cycle")
+                        .start_vertex(),
+                )
+                .unwrap()
+                .local_on(half_edge.curve())
+                .unwrap()
+                .position,
+        ];
 
         let a = HalfEdge::new(
             half_edge.curve().clone(),
diff --git a/crates/fj-core/src/operations/sweep/half_edge.rs b/crates/fj-core/src/operations/sweep/half_edge.rs
index d04dc5bdf..522a33e23 100644
--- a/crates/fj-core/src/operations/sweep/half_edge.rs
+++ b/crates/fj-core/src/operations/sweep/half_edge.rs
@@ -59,7 +59,22 @@ impl SweepHalfEdge for Handle<HalfEdge> {
     ) -> SweptHalfEdge {
         let path = path.into();
 
-        let half_edge_geom = *core.layers.geometry.of_half_edge(self);
+        let boundary = [
+            core.layers
+                .geometry
+                .of_vertex(self.start_vertex())
+                .unwrap()
+                .local_on(self.curve())
+                .unwrap()
+                .position,
+            core.layers
+                .geometry
+                .of_vertex(&end_vertex)
+                .unwrap()
+                .local_on(self.curve())
+                .unwrap()
+                .position,
+        ];
         let curve_geom = core
             .layers
             .geometry
@@ -94,7 +109,7 @@ impl SweepHalfEdge for Handle<HalfEdge> {
 
         // Let's figure out the surface coordinates of the edge vertices.
         let surface_points = {
-            let [a, b] = half_edge_geom.boundary.inner;
+            let [a, b] = boundary;
 
             [
                 [a.t, Scalar::ZERO],
@@ -112,7 +127,7 @@ impl SweepHalfEdge for Handle<HalfEdge> {
 
         // Now, the boundaries of each edge.
         let boundaries = {
-            let [a, b] = half_edge_geom.boundary.inner;
+            let [a, b] = boundary;
             let [c, d] = [0., 1.].map(|coord| Point::from([coord]));
 
             [[a, b], [c, d], [b, a], [d, c]]
diff --git a/crates/fj-core/src/topology/objects/cycle.rs b/crates/fj-core/src/topology/objects/cycle.rs
index 2ad742a51..509072bee 100644
--- a/crates/fj-core/src/topology/objects/cycle.rs
+++ b/crates/fj-core/src/topology/objects/cycle.rs
@@ -46,7 +46,6 @@ impl Cycle {
                 .next()
                 .expect("Invalid cycle: expected at least one edge");
 
-            let half_edge_geom = geometry.of_half_edge(first);
             let curve_geom = geometry
                 .of_curve(first.curve())
                 .unwrap()
@@ -54,7 +53,29 @@ impl Cycle {
                 .unwrap()
                 .clone();
 
-            let [a, b] = half_edge_geom.boundary.inner;
+            let [a, b] = [
+                curve_geom.path.point_from_path_coords(
+                    geometry
+                        .of_vertex(first.start_vertex())
+                        .unwrap()
+                        .local_on(first.curve())
+                        .unwrap()
+                        .position,
+                ),
+                curve_geom.path.point_from_path_coords(
+                    geometry
+                        .of_vertex(
+                            self.half_edges()
+                                .after(first)
+                                .expect("Just got half-edge from this cycle")
+                                .start_vertex(),
+                        )
+                        .unwrap()
+                        .local_on(first.curve())
+                        .unwrap()
+                        .position,
+                ),
+            ];
             let edge_direction_positive = a < b;
 
             let circle = match curve_geom.path {
@@ -80,14 +101,20 @@ impl Cycle {
 
         for (a, b) in self.half_edges().pairs() {
             let [a, b] = [a, b].map(|half_edge| {
-                geometry.of_half_edge(half_edge).start_position(
-                    &geometry
-                        .of_curve(half_edge.curve())
-                        .unwrap()
-                        .local_on(surface)
-                        .unwrap()
-                        .path,
-                )
+                geometry
+                    .of_curve(half_edge.curve())
+                    .unwrap()
+                    .local_on(surface)
+                    .unwrap()
+                    .path
+                    .point_from_path_coords(
+                        geometry
+                            .of_vertex(half_edge.start_vertex())
+                            .unwrap()
+                            .local_on(half_edge.curve())
+                            .unwrap()
+                            .position,
+                    )
             });
 
             sum += (b.u - a.u) * (b.v + a.v);
diff --git a/crates/fj-core/src/validate/solid.rs b/crates/fj-core/src/validate/solid.rs
index f51191aa7..64d0ebae9 100644
--- a/crates/fj-core/src/validate/solid.rs
+++ b/crates/fj-core/src/validate/solid.rs
@@ -117,9 +117,14 @@ impl SolidValidationError {
 
                 Some((
                     geometry.of_surface(s).point_from_surface_coords(
-                        geometry
-                            .of_half_edge(&h)
-                            .start_position(&local_curve_geometry.path),
+                        local_curve_geometry.path.point_from_path_coords(
+                            geometry
+                                .of_vertex(h.start_vertex())
+                                .unwrap()
+                                .local_on(h.curve())
+                                .unwrap()
+                                .position,
+                        ),
                     ),
                     h.start_vertex().clone(),
                 ))
diff --git a/crates/fj-core/src/validation/checks/coincident_half_edges_are_not_siblings.rs b/crates/fj-core/src/validation/checks/coincident_half_edges_are_not_siblings.rs
index cb1529dfe..ca13ffd2c 100644
--- a/crates/fj-core/src/validation/checks/coincident_half_edges_are_not_siblings.rs
+++ b/crates/fj-core/src/validation/checks/coincident_half_edges_are_not_siblings.rs
@@ -5,7 +5,8 @@ use fj_math::{Point, Scalar};
 use crate::{
     geometry::{CurveBoundary, Geometry},
     queries::{
-        AllHalfEdgesWithSurface, BoundingVerticesOfHalfEdge, SiblingOfHalfEdge,
+        AllHalfEdgesWithSurface, BoundingVerticesOfHalfEdge, CycleOfHalfEdge,
+        SiblingOfHalfEdge,
     },
     storage::Handle,
     topology::{Curve, HalfEdge, Shell, Surface, Vertex},
@@ -109,8 +110,22 @@ impl ValidationCheck<Shell> for CoincidentHalfEdgesAreNotSiblings {
 
                 let Some(mut distances) = distances(
                     half_edge_a.clone(),
+                    object
+                        .find_cycle_of_half_edge(half_edge_a)
+                        .unwrap()
+                        .half_edges()
+                        .after(half_edge_a)
+                        .unwrap()
+                        .start_vertex(),
                     surface_a,
                     half_edge_b.clone(),
+                    object
+                        .find_cycle_of_half_edge(half_edge_b)
+                        .unwrap()
+                        .half_edges()
+                        .after(half_edge_b)
+                        .unwrap()
+                        .start_vertex(),
                     surface_b,
                     geometry,
                 ) else {
@@ -152,25 +167,39 @@ impl ValidationCheck<Shell> for CoincidentHalfEdgesAreNotSiblings {
 /// Returns an [`Iterator`] of the distance at each sample.
 fn distances(
     half_edge_a: Handle<HalfEdge>,
+    end_vertex_a: &Handle<Vertex>,
     surface_a: &Handle<Surface>,
     half_edge_b: Handle<HalfEdge>,
+    end_vertex_b: &Handle<Vertex>,
     surface_b: &Handle<Surface>,
     geometry: &Geometry,
 ) -> Option<impl Iterator<Item = Scalar>> {
     fn sample(
         percent: f64,
         half_edge: &Handle<HalfEdge>,
+        end_vertex: &Handle<Vertex>,
         surface: &Handle<Surface>,
         geometry: &Geometry,
     ) -> Option<Point<3>> {
-        let [start, end] = geometry.of_half_edge(half_edge).boundary.inner;
+        let [start, end] = [
+            geometry
+                .of_vertex(half_edge.start_vertex())
+                .unwrap()
+                .local_on(half_edge.curve())
+                .unwrap()
+                .position,
+            geometry
+                .of_vertex(end_vertex)
+                .unwrap()
+                .local_on(half_edge.curve())
+                .unwrap()
+                .position,
+        ];
         let path_coords = start + (end - start) * percent;
-        // let path = geometry.of_half_edge(half_edge).path;
         let path = geometry
             .of_curve(half_edge.curve())?
             .local_on(surface)?
             .path;
-        // assert_eq!(path, path_from_curve);
         let surface_coords = path.point_from_path_coords(path_coords);
         Some(
             geometry
@@ -188,8 +217,15 @@ fn distances(
     let mut distances = Vec::new();
     for i in 0..sample_count {
         let percent = i as f64 * step;
-        let sample1 = sample(percent, &half_edge_a, surface_a, geometry)?;
-        let sample2 = sample(1.0 - percent, &half_edge_b, surface_b, geometry)?;
+        let sample1 =
+            sample(percent, &half_edge_a, end_vertex_a, surface_a, geometry)?;
+        let sample2 = sample(
+            1.0 - percent,
+            &half_edge_b,
+            end_vertex_b,
+            surface_b,
+            geometry,
+        )?;
         distances.push(sample1.distance_to(&sample2))
     }
     Some(distances.into_iter())
diff --git a/crates/fj-core/src/validation/checks/curve_geometry_mismatch.rs b/crates/fj-core/src/validation/checks/curve_geometry_mismatch.rs
index bd8fb440b..adf1f8188 100644
--- a/crates/fj-core/src/validation/checks/curve_geometry_mismatch.rs
+++ b/crates/fj-core/src/validation/checks/curve_geometry_mismatch.rs
@@ -3,7 +3,7 @@ use itertools::Itertools;
 
 use crate::{
     geometry::Geometry,
-    queries::AllHalfEdgesWithSurface,
+    queries::{AllHalfEdgesWithSurface, CycleOfHalfEdge},
     storage::Handle,
     topology::{HalfEdge, Shell},
     validation::{ValidationCheck, ValidationConfig},
@@ -130,8 +130,28 @@ impl ValidationCheck<Shell> for CurveGeometryMismatch {
                     // we have right now are circles, 3 would be enough to check
                     // for coincidence. But the first and last might be
                     // identical, so let's add an extra one.
-                    let [a, d] =
-                        geometry.of_half_edge(&half_edge_a).boundary.inner;
+                    let [a, d] = [
+                        geometry
+                            .of_vertex(half_edge_a.start_vertex())
+                            .unwrap()
+                            .local_on(half_edge_a.curve())
+                            .unwrap()
+                            .position,
+                        geometry
+                            .of_vertex(
+                                object
+                                    .find_cycle_of_half_edge(&half_edge_a)
+                                    .unwrap()
+                                    .half_edges()
+                                    .after(&half_edge_a)
+                                    .unwrap()
+                                    .start_vertex(),
+                            )
+                            .unwrap()
+                            .local_on(half_edge_a.curve())
+                            .unwrap()
+                            .position,
+                    ];
                     let b = a + (d - a) * 1. / 3.;
                     let c = a + (d - a) * 2. / 3.;
 
diff --git a/crates/fj-core/src/validation/checks/half_edge_connection.rs b/crates/fj-core/src/validation/checks/half_edge_connection.rs
index 376cdf1e0..4a181b56f 100644
--- a/crates/fj-core/src/validation/checks/half_edge_connection.rs
+++ b/crates/fj-core/src/validation/checks/half_edge_connection.rs
@@ -99,7 +99,12 @@ fn check_cycle<'r>(
 ) -> impl Iterator<Item = AdjacentHalfEdgesNotConnected> + 'r {
     cycle.half_edges().pairs().filter_map(|(first, second)| {
         let end_pos_of_first_half_edge = {
-            let [_, end] = geometry.of_half_edge(first).boundary.inner;
+            let end = geometry
+                .of_vertex(second.start_vertex())
+                .unwrap()
+                .local_on(first.curve())
+                .unwrap()
+                .position;
             geometry
                 .of_curve(first.curve())
                 .unwrap()
@@ -117,9 +122,18 @@ fn check_cycle<'r>(
             return None;
         };
 
-        let start_pos_of_second_half_edge = geometry
-            .of_half_edge(second)
-            .start_position(&local_curve_geometry.path);
+        let start_pos_of_second_half_edge = {
+            let point_curve = geometry
+                .of_vertex(second.start_vertex())
+                .unwrap()
+                .local_on(second.curve())
+                .unwrap()
+                .position;
+
+            local_curve_geometry
+                .path
+                .point_from_path_coords(point_curve)
+        };
 
         let distance_between_positions = (end_pos_of_first_half_edge
             - start_pos_of_second_half_edge)