diff --git a/crates/fj-core/src/operations/holes.rs b/crates/fj-core/src/operations/holes.rs
index 519305263..f46980348 100644
--- a/crates/fj-core/src/operations/holes.rs
+++ b/crates/fj-core/src/operations/holes.rs
@@ -75,6 +75,7 @@ impl AddHole for Shell {
                                     entry.clone(),
                                     *core.layers.geometry.of_half_edge(&entry),
                                 )],
+                                location.face.surface().clone(),
                                 core,
                             )],
                             core,
@@ -150,6 +151,7 @@ impl AddHole for Shell {
                                     entry.clone(),
                                     *core.layers.geometry.of_half_edge(&entry),
                                 )],
+                                entry_location.face.surface().clone(),
                                 core,
                             )],
                             core,
@@ -171,6 +173,7 @@ impl AddHole for Shell {
                                     exit.clone(),
                                     *core.layers.geometry.of_half_edge(exit),
                                 )],
+                                exit_location.face.surface().clone(),
                                 core,
                             )],
                             core,
diff --git a/crates/fj-core/src/operations/join/cycle.rs b/crates/fj-core/src/operations/join/cycle.rs
index 037b823c2..45122513b 100644
--- a/crates/fj-core/src/operations/join/cycle.rs
+++ b/crates/fj-core/src/operations/join/cycle.rs
@@ -3,7 +3,7 @@ use std::ops::RangeInclusive;
 use itertools::Itertools;
 
 use crate::{
-    geometry::HalfEdgeGeom,
+    geometry::{HalfEdgeGeom, LocalCurveGeom},
     operations::{
         build::BuildHalfEdge,
         geometry::UpdateHalfEdgeGeometry,
@@ -11,15 +11,28 @@ use crate::{
         update::{UpdateCycle, UpdateHalfEdge},
     },
     storage::Handle,
-    topology::{Cycle, HalfEdge},
+    topology::{Cycle, HalfEdge, Surface},
     Core,
 };
 
 /// Join a [`Cycle`] to another
 pub trait JoinCycle {
-    /// Add half-edges to the cycle that are joined to the provided ones
+    /// Add new half-edges to the cycle that are joined to the provided ones
+    ///
+    /// This method creates a new half-edge for each half-edge that is provided,
+    /// joins the new half-edge to the provided one, and adds the new half-edge
+    /// to the cycle.
+    ///
+    /// The geometry for each new half-edge needs to be provided as well.
+    ///
+    /// Also requires the surface that the cycle is defined in.
     #[must_use]
-    fn add_joined_edges<Es>(&self, edges: Es, core: &mut Core) -> Self
+    fn add_joined_edges<Es>(
+        &self,
+        edges: Es,
+        surface: Handle<Surface>,
+        core: &mut Core,
+    ) -> Self
     where
         Es: IntoIterator<Item = (Handle<HalfEdge>, HalfEdgeGeom)>,
         Es::IntoIter: Clone + ExactSizeIterator;
@@ -76,7 +89,12 @@ pub trait JoinCycle {
 }
 
 impl JoinCycle for Cycle {
-    fn add_joined_edges<Es>(&self, edges: Es, core: &mut Core) -> Self
+    fn add_joined_edges<Es>(
+        &self,
+        edges: Es,
+        surface: Handle<Surface>,
+        core: &mut Core,
+    ) -> Self
     where
         Es: IntoIterator<Item = (Handle<HalfEdge>, HalfEdgeGeom)>,
         Es::IntoIter: Clone + ExactSizeIterator,
@@ -85,14 +103,24 @@ impl JoinCycle for Cycle {
             .into_iter()
             .circular_tuple_windows()
             .map(|((prev_half_edge, _), (half_edge, geometry))| {
-                HalfEdge::unjoined(core)
+                let half_edge = HalfEdge::unjoined(core)
                     .update_curve(|_, _| half_edge.curve().clone(), core)
                     .update_start_vertex(
                         |_, _| prev_half_edge.start_vertex().clone(),
                         core,
                     )
                     .insert(core)
-                    .set_geometry(geometry, &mut core.layers.geometry)
+                    .set_geometry(geometry, &mut core.layers.geometry);
+
+                core.layers.geometry.define_curve(
+                    half_edge.curve().clone(),
+                    surface.clone(),
+                    LocalCurveGeom {
+                        path: geometry.path,
+                    },
+                );
+
+                half_edge
             })
             .collect::<Vec<_>>();
         self.add_half_edges(half_edges, core)
diff --git a/crates/fj-core/src/operations/sweep/cycle.rs b/crates/fj-core/src/operations/sweep/cycle.rs
index 0b7b1ffef..4d00d294a 100644
--- a/crates/fj-core/src/operations/sweep/cycle.rs
+++ b/crates/fj-core/src/operations/sweep/cycle.rs
@@ -39,6 +39,7 @@ pub trait SweepCycle {
     fn sweep_cycle(
         &self,
         surface: Handle<Surface>,
+        top_surface: Handle<Surface>,
         color: Option<Color>,
         path: impl Into<Vector<3>>,
         cache: &mut SweepCache,
@@ -50,6 +51,7 @@ impl SweepCycle for Cycle {
     fn sweep_cycle(
         &self,
         surface: Handle<Surface>,
+        top_surface: Handle<Surface>,
         color: Option<Color>,
         path: impl Into<Vector<3>>,
         cache: &mut SweepCache,
@@ -64,7 +66,7 @@ impl SweepCycle for Cycle {
             let (bottom_half_edge, bottom_half_edge_next) =
                 bottom_half_edge_pair;
 
-            let (side_face, top_edge) = bottom_half_edge.sweep_half_edge(
+            let (side_face, top_half_edge) = bottom_half_edge.sweep_half_edge(
                 bottom_half_edge_next.start_vertex().clone(),
                 surface.clone(),
                 color,
@@ -76,12 +78,13 @@ impl SweepCycle for Cycle {
             faces.push(side_face);
 
             top_edges.push((
-                top_edge,
+                top_half_edge,
                 *core.layers.geometry.of_half_edge(bottom_half_edge),
             ));
         }
 
-        let top_cycle = Cycle::empty().add_joined_edges(top_edges, core);
+        let top_cycle =
+            Cycle::empty().add_joined_edges(top_edges, top_surface, core);
 
         SweptCycle { faces, top_cycle }
     }
diff --git a/crates/fj-core/src/operations/sweep/region.rs b/crates/fj-core/src/operations/sweep/region.rs
index 05af614b7..b3be68d8b 100644
--- a/crates/fj-core/src/operations/sweep/region.rs
+++ b/crates/fj-core/src/operations/sweep/region.rs
@@ -52,9 +52,12 @@ impl SweepRegion for Region {
 
         let mut faces = Vec::new();
 
+        let top_surface = surface.translate(path, core).insert(core);
+
         let top_exterior = sweep_cycle(
             self.exterior(),
             surface.clone(),
+            top_surface.clone(),
             color,
             &mut faces,
             path,
@@ -69,6 +72,7 @@ impl SweepRegion for Region {
                 sweep_cycle(
                     bottom_cycle,
                     surface.clone(),
+                    top_surface.clone(),
                     color,
                     &mut faces,
                     path,
@@ -79,7 +83,6 @@ impl SweepRegion for Region {
             .collect::<Vec<_>>();
 
         let top_face = {
-            let top_surface = surface.translate(path, core).insert(core);
             let top_region =
                 Region::new(top_exterior, top_interiors).insert(core);
 
@@ -93,9 +96,11 @@ impl SweepRegion for Region {
     }
 }
 
+#[allow(clippy::too_many_arguments)]
 fn sweep_cycle(
     bottom_cycle: &Cycle,
     bottom_surface: Handle<Surface>,
+    top_surface: Handle<Surface>,
     color: Option<Color>,
     faces: &mut Vec<Face>,
     path: Vector<3>,
@@ -104,6 +109,7 @@ fn sweep_cycle(
 ) -> Handle<Cycle> {
     let swept_cycle = bottom_cycle.reverse(core).sweep_cycle(
         bottom_surface,
+        top_surface,
         color,
         path,
         cache,