diff --git a/crates/fj-core/src/algorithms/approx/edge.rs b/crates/fj-core/src/algorithms/approx/edge.rs
index de5f1e9d8..a66d02e78 100644
--- a/crates/fj-core/src/algorithms/approx/edge.rs
+++ b/crates/fj-core/src/algorithms/approx/edge.rs
@@ -10,7 +10,7 @@ use std::collections::BTreeMap;
 use fj_math::Point;
 
 use crate::{
-    geometry::{BoundaryOnCurve, GlobalPath, SurfacePath},
+    geometry::{CurveBoundary, GlobalPath, SurfacePath},
     objects::{GlobalEdge, HalfEdge, Surface, Vertex},
     storage::{Handle, HandleWrapper},
 };
@@ -141,7 +141,7 @@ impl HalfEdgeApprox {
 fn approx_edge(
     path: &SurfacePath,
     surface: &Surface,
-    boundary: BoundaryOnCurve,
+    boundary: CurveBoundary<Point<1>>,
     tolerance: impl Into<Tolerance>,
 ) -> GlobalEdgeApprox {
     // There are different cases of varying complexity. Circles are the hard
@@ -185,7 +185,7 @@ fn approx_edge(
         }
         (SurfacePath::Line(line), _) => {
             let range_u =
-                BoundaryOnCurve::from(boundary.inner.map(|point_curve| {
+                CurveBoundary::from(boundary.inner.map(|point_curve| {
                     [path.point_from_path_coords(point_curve).u]
                 }));
 
@@ -218,7 +218,7 @@ fn approx_edge(
 #[derive(Default)]
 pub struct EdgeCache {
     edge_approx: BTreeMap<
-        (HandleWrapper<GlobalEdge>, BoundaryOnCurve),
+        (HandleWrapper<GlobalEdge>, CurveBoundary<Point<1>>),
         GlobalEdgeApprox,
     >,
     vertex_approx: BTreeMap<HandleWrapper<Vertex>, Point<3>>,
@@ -234,7 +234,7 @@ impl EdgeCache {
     fn get_edge(
         &self,
         handle: Handle<GlobalEdge>,
-        boundary: BoundaryOnCurve,
+        boundary: CurveBoundary<Point<1>>,
     ) -> Option<GlobalEdgeApprox> {
         if let Some(approx) =
             self.edge_approx.get(&(handle.clone().into(), boundary))
@@ -256,7 +256,7 @@ impl EdgeCache {
     fn insert_edge(
         &mut self,
         handle: Handle<GlobalEdge>,
-        boundary: BoundaryOnCurve,
+        boundary: CurveBoundary<Point<1>>,
         approx: GlobalEdgeApprox,
     ) -> GlobalEdgeApprox {
         self.edge_approx
@@ -302,7 +302,7 @@ mod tests {
 
     use crate::{
         algorithms::approx::{Approx, ApproxPoint},
-        geometry::{BoundaryOnCurve, GlobalPath, SurfaceGeometry},
+        geometry::{CurveBoundary, GlobalPath, SurfaceGeometry},
         objects::{HalfEdge, Surface},
         operations::BuildHalfEdge,
         services::Services,
@@ -344,7 +344,7 @@ mod tests {
         let mut services = Services::new();
 
         let path = GlobalPath::circle_from_radius(1.);
-        let boundary = BoundaryOnCurve::from([[0.], [TAU]]);
+        let boundary = CurveBoundary::from([[0.], [TAU]]);
 
         let surface = Surface::new(SurfaceGeometry {
             u: path,
@@ -384,7 +384,7 @@ mod tests {
         let approx = (&half_edge, surface.deref()).approx(tolerance);
 
         let expected_approx =
-            (&half_edge.path(), BoundaryOnCurve::from([[0.], [TAU]]))
+            (&half_edge.path(), CurveBoundary::from([[0.], [TAU]]))
                 .approx(tolerance)
                 .into_iter()
                 .map(|(_, point_surface)| {
diff --git a/crates/fj-core/src/algorithms/approx/path.rs b/crates/fj-core/src/algorithms/approx/path.rs
index b06415248..ad55f9eeb 100644
--- a/crates/fj-core/src/algorithms/approx/path.rs
+++ b/crates/fj-core/src/algorithms/approx/path.rs
@@ -32,11 +32,11 @@ use std::iter;
 
 use fj_math::{Circle, Point, Scalar, Sign};
 
-use crate::geometry::{BoundaryOnCurve, GlobalPath, SurfacePath};
+use crate::geometry::{CurveBoundary, GlobalPath, SurfacePath};
 
 use super::{Approx, Tolerance};
 
-impl Approx for (&SurfacePath, BoundaryOnCurve) {
+impl Approx for (&SurfacePath, CurveBoundary<Point<1>>) {
     type Approximation = Vec<(Point<1>, Point<2>)>;
     type Cache = ();
 
@@ -56,7 +56,7 @@ impl Approx for (&SurfacePath, BoundaryOnCurve) {
     }
 }
 
-impl Approx for (GlobalPath, BoundaryOnCurve) {
+impl Approx for (GlobalPath, CurveBoundary<Point<1>>) {
     type Approximation = Vec<(Point<1>, Point<3>)>;
     type Cache = ();
 
@@ -82,7 +82,7 @@ impl Approx for (GlobalPath, BoundaryOnCurve) {
 /// from the circle.
 fn approx_circle<const D: usize>(
     circle: &Circle<D>,
-    boundary: impl Into<BoundaryOnCurve>,
+    boundary: impl Into<CurveBoundary<Point<1>>>,
     tolerance: Tolerance,
 ) -> Vec<(Point<1>, Point<D>)> {
     let boundary = boundary.into();
@@ -127,7 +127,7 @@ impl PathApproxParams {
 
     pub fn points(
         &self,
-        boundary: impl Into<BoundaryOnCurve>,
+        boundary: impl Into<CurveBoundary<Point<1>>>,
     ) -> impl Iterator<Item = Point<1>> + '_ {
         let boundary = boundary.into();
 
@@ -170,7 +170,7 @@ mod tests {
 
     use fj_math::{Circle, Point, Scalar};
 
-    use crate::algorithms::approx::{path::BoundaryOnCurve, Tolerance};
+    use crate::algorithms::approx::{path::CurveBoundary, Tolerance};
 
     use super::PathApproxParams;
 
@@ -220,7 +220,7 @@ mod tests {
         test_path([[TAU - 2.], [0.]], [2., 1.]);
 
         fn test_path(
-            boundary: impl Into<BoundaryOnCurve>,
+            boundary: impl Into<CurveBoundary<Point<1>>>,
             expected_coords: impl IntoIterator<Item = impl Into<Scalar>>,
         ) {
             // Choose radius and tolerance such, that we need 4 vertices to
diff --git a/crates/fj-core/src/geometry/boundary.rs b/crates/fj-core/src/geometry/boundary.rs
index c38a02563..bb5ed9935 100644
--- a/crates/fj-core/src/geometry/boundary.rs
+++ b/crates/fj-core/src/geometry/boundary.rs
@@ -1,26 +1,95 @@
+use std::{
+    cmp::Ordering,
+    hash::{Hash, Hasher},
+};
+
 use fj_math::Point;
 
+use crate::{objects::Vertex, storage::HandleWrapper};
+
 /// A boundary on a curve
-#[derive(Clone, Copy, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
-pub struct BoundaryOnCurve {
+///
+/// This struct is generic, because different situations require different
+/// representations of a boundary. In some cases, curve coordinates are enough,
+/// in other cases, vertices are required, and sometimes you need both.
+#[derive(Clone, Copy, Debug)]
+pub struct CurveBoundary<T: CurveBoundaryElement> {
     /// The raw representation of the boundary
-    pub inner: [Point<1>; 2],
+    pub inner: [T::Repr; 2],
 }
 
-impl BoundaryOnCurve {
+impl<T: CurveBoundaryElement> CurveBoundary<T> {
     /// Reverse the direction of the boundary
+    ///
+    /// Returns a new instance of this struct, which has its direction reversed.
+    #[must_use]
     pub fn reverse(self) -> Self {
         let [a, b] = self.inner;
         Self { inner: [b, a] }
     }
+
+    /// Normalize the boundary
+    ///
+    /// Returns a new instance of this struct, which has the bounding elements
+    /// in a defined order. This can be used to compare a boundary while
+    /// disregarding its direction.
+    #[must_use]
+    pub fn normalize(mut self) -> Self {
+        self.inner.sort();
+        self
+    }
+}
+
+impl<T: CurveBoundaryElement> Eq for CurveBoundary<T> {}
+
+impl<T: CurveBoundaryElement> PartialEq for CurveBoundary<T> {
+    fn eq(&self, other: &Self) -> bool {
+        self.inner == other.inner
+    }
 }
 
-impl<T> From<[T; 2]> for BoundaryOnCurve
+impl<S, T: CurveBoundaryElement> From<[S; 2]> for CurveBoundary<T>
 where
-    T: Into<Point<1>>,
+    S: Into<T::Repr>,
 {
-    fn from(boundary: [T; 2]) -> Self {
+    fn from(boundary: [S; 2]) -> Self {
         let inner = boundary.map(Into::into);
         Self { inner }
     }
 }
+
+impl<T: CurveBoundaryElement> Hash for CurveBoundary<T> {
+    fn hash<H: Hasher>(&self, state: &mut H) {
+        self.inner.hash(state);
+    }
+}
+
+impl<T: CurveBoundaryElement> Ord for CurveBoundary<T> {
+    fn cmp(&self, other: &Self) -> Ordering {
+        self.inner.cmp(&other.inner)
+    }
+}
+
+impl<T: CurveBoundaryElement> PartialOrd for CurveBoundary<T> {
+    fn partial_cmp(&self, other: &Self) -> Option<Ordering> {
+        self.inner.partial_cmp(&other.inner)
+    }
+}
+
+/// An element of a curve boundary
+///
+/// Used for the type parameter of [`CurveBoundary`].
+pub trait CurveBoundaryElement {
+    /// The representation the curve boundary element
+    ///
+    /// This is the actual data stored in [`CurveBoundary`].
+    type Repr: Eq + Hash + Ord;
+}
+
+impl CurveBoundaryElement for Point<1> {
+    type Repr = Self;
+}
+
+impl CurveBoundaryElement for Vertex {
+    type Repr = HandleWrapper<Vertex>;
+}
diff --git a/crates/fj-core/src/geometry/bounding_vertices.rs b/crates/fj-core/src/geometry/bounding_vertices.rs
deleted file mode 100644
index fd6afb6d4..000000000
--- a/crates/fj-core/src/geometry/bounding_vertices.rs
+++ /dev/null
@@ -1,31 +0,0 @@
-use crate::{
-    objects::Vertex,
-    storage::{Handle, HandleWrapper},
-};
-
-/// The bounding vertices of an edge
-#[derive(Debug, Eq, PartialEq, Ord, PartialOrd)]
-pub struct BoundingVertices {
-    /// The bounding vertices
-    pub inner: [HandleWrapper<Vertex>; 2],
-}
-
-impl BoundingVertices {
-    /// Normalize the bounding vertices
-    ///
-    /// Returns a new instance of this struct, which has the vertices in a
-    /// defined order. This can be used to compare bounding vertices while
-    /// disregarding their order.
-    pub fn normalize(mut self) -> Self {
-        self.inner.sort();
-        self
-    }
-}
-
-impl From<[Handle<Vertex>; 2]> for BoundingVertices {
-    fn from(vertices: [Handle<Vertex>; 2]) -> Self {
-        Self {
-            inner: vertices.map(Into::into),
-        }
-    }
-}
diff --git a/crates/fj-core/src/geometry/mod.rs b/crates/fj-core/src/geometry/mod.rs
index 732dbbb7f..0ed948703 100644
--- a/crates/fj-core/src/geometry/mod.rs
+++ b/crates/fj-core/src/geometry/mod.rs
@@ -1,13 +1,11 @@
 //! Types that are tied to objects, but aren't objects themselves
 
 mod boundary;
-mod bounding_vertices;
 mod path;
 mod surface;
 
 pub use self::{
-    boundary::BoundaryOnCurve,
-    bounding_vertices::BoundingVertices,
+    boundary::{CurveBoundary, CurveBoundaryElement},
     path::{GlobalPath, SurfacePath},
     surface::SurfaceGeometry,
 };
diff --git a/crates/fj-core/src/objects/kinds/edge.rs b/crates/fj-core/src/objects/kinds/edge.rs
index 153f82ac2..419e2b2b6 100644
--- a/crates/fj-core/src/objects/kinds/edge.rs
+++ b/crates/fj-core/src/objects/kinds/edge.rs
@@ -1,7 +1,7 @@
 use fj_math::Point;
 
 use crate::{
-    geometry::{BoundaryOnCurve, SurfacePath},
+    geometry::{CurveBoundary, SurfacePath},
     objects::{Curve, Vertex},
     storage::{Handle, HandleWrapper},
 };
@@ -41,7 +41,7 @@ use crate::{
 #[derive(Clone, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
 pub struct HalfEdge {
     path: SurfacePath,
-    boundary: BoundaryOnCurve,
+    boundary: CurveBoundary<Point<1>>,
     curve: HandleWrapper<Curve>,
     start_vertex: HandleWrapper<Vertex>,
     global_form: HandleWrapper<GlobalEdge>,
@@ -51,7 +51,7 @@ impl HalfEdge {
     /// Create an instance of `HalfEdge`
     pub fn new(
         path: SurfacePath,
-        boundary: impl Into<BoundaryOnCurve>,
+        boundary: impl Into<CurveBoundary<Point<1>>>,
         curve: Handle<Curve>,
         start_vertex: Handle<Vertex>,
         global_form: Handle<GlobalEdge>,
@@ -71,7 +71,7 @@ impl HalfEdge {
     }
 
     /// Access the boundary points of the half-edge on the curve
-    pub fn boundary(&self) -> BoundaryOnCurve {
+    pub fn boundary(&self) -> CurveBoundary<Point<1>> {
         self.boundary
     }
 
diff --git a/crates/fj-core/src/operations/build/edge.rs b/crates/fj-core/src/operations/build/edge.rs
index 3552ada11..0ffe35cff 100644
--- a/crates/fj-core/src/operations/build/edge.rs
+++ b/crates/fj-core/src/operations/build/edge.rs
@@ -2,7 +2,7 @@ use fj_interop::ext::ArrayExt;
 use fj_math::{Arc, Point, Scalar};
 
 use crate::{
-    geometry::{BoundaryOnCurve, SurfacePath},
+    geometry::{CurveBoundary, SurfacePath},
     objects::{Curve, GlobalEdge, HalfEdge, Vertex},
     operations::Insert,
     services::Services,
@@ -13,7 +13,7 @@ pub trait BuildHalfEdge {
     /// Create a half-edge that is not joined to another
     fn unjoined(
         path: SurfacePath,
-        boundary: impl Into<BoundaryOnCurve>,
+        boundary: impl Into<CurveBoundary<Point<1>>>,
         services: &mut Services,
     ) -> HalfEdge {
         let curve = Curve::new().insert(services);
diff --git a/crates/fj-core/src/operations/join/cycle.rs b/crates/fj-core/src/operations/join/cycle.rs
index 316880f52..44e5bbda6 100644
--- a/crates/fj-core/src/operations/join/cycle.rs
+++ b/crates/fj-core/src/operations/join/cycle.rs
@@ -1,9 +1,10 @@
 use std::ops::RangeInclusive;
 
+use fj_math::Point;
 use itertools::Itertools;
 
 use crate::{
-    geometry::{BoundaryOnCurve, SurfacePath},
+    geometry::{CurveBoundary, SurfacePath},
     objects::{Cycle, HalfEdge},
     operations::{BuildHalfEdge, Insert, UpdateCycle, UpdateHalfEdge},
     services::Services,
@@ -17,7 +18,7 @@ pub trait JoinCycle {
     fn add_joined_edges<Es>(&self, edges: Es, services: &mut Services) -> Self
     where
         Es: IntoIterator<
-            Item = (Handle<HalfEdge>, SurfacePath, BoundaryOnCurve),
+            Item = (Handle<HalfEdge>, SurfacePath, CurveBoundary<Point<1>>),
         >,
         Es::IntoIter: Clone + ExactSizeIterator;
 
@@ -64,7 +65,7 @@ impl JoinCycle for Cycle {
     fn add_joined_edges<Es>(&self, edges: Es, services: &mut Services) -> Self
     where
         Es: IntoIterator<
-            Item = (Handle<HalfEdge>, SurfacePath, BoundaryOnCurve),
+            Item = (Handle<HalfEdge>, SurfacePath, CurveBoundary<Point<1>>),
         >,
         Es::IntoIter: Clone + ExactSizeIterator,
     {
diff --git a/crates/fj-core/src/queries.rs b/crates/fj-core/src/queries.rs
index a0116aa36..e4d880b1c 100644
--- a/crates/fj-core/src/queries.rs
+++ b/crates/fj-core/src/queries.rs
@@ -10,8 +10,8 @@
 //! them for various objects that have the information to answer the query.
 
 use crate::{
-    geometry::BoundingVertices,
-    objects::{Cycle, Face, HalfEdge, Region, Shell},
+    geometry::CurveBoundary,
+    objects::{Cycle, Face, HalfEdge, Region, Shell, Vertex},
     storage::Handle,
 };
 
@@ -24,18 +24,18 @@ pub trait BoundingVerticesOfEdge {
     fn bounding_vertices_of_edge(
         &self,
         edge: &Handle<HalfEdge>,
-    ) -> Option<BoundingVertices>;
+    ) -> Option<CurveBoundary<Vertex>>;
 }
 
 impl BoundingVerticesOfEdge for Cycle {
     fn bounding_vertices_of_edge(
         &self,
         edge: &Handle<HalfEdge>,
-    ) -> Option<BoundingVertices> {
+    ) -> Option<CurveBoundary<Vertex>> {
         let start = edge.start_vertex().clone();
         let end = self.half_edge_after(edge)?.start_vertex().clone();
 
-        Some(BoundingVertices::from([start, end]))
+        Some(CurveBoundary::from([start, end]))
     }
 }
 
@@ -43,7 +43,7 @@ impl BoundingVerticesOfEdge for Region {
     fn bounding_vertices_of_edge(
         &self,
         edge: &Handle<HalfEdge>,
-    ) -> Option<BoundingVertices> {
+    ) -> Option<CurveBoundary<Vertex>> {
         for cycle in self.all_cycles() {
             if let Some(vertices) = cycle.bounding_vertices_of_edge(edge) {
                 return Some(vertices);
@@ -58,7 +58,7 @@ impl BoundingVerticesOfEdge for Face {
     fn bounding_vertices_of_edge(
         &self,
         edge: &Handle<HalfEdge>,
-    ) -> Option<BoundingVertices> {
+    ) -> Option<CurveBoundary<Vertex>> {
         self.region().bounding_vertices_of_edge(edge)
     }
 }
@@ -67,7 +67,7 @@ impl BoundingVerticesOfEdge for Shell {
     fn bounding_vertices_of_edge(
         &self,
         edge: &Handle<HalfEdge>,
-    ) -> Option<BoundingVertices> {
+    ) -> Option<CurveBoundary<Vertex>> {
         for face in self.faces() {
             if let Some(vertices) = face.bounding_vertices_of_edge(edge) {
                 return Some(vertices);