diff --git a/crates/fj-core/src/operations/build/shell.rs b/crates/fj-core/src/operations/build/shell.rs
index 0c846fda6..80027a368 100644
--- a/crates/fj-core/src/operations/build/shell.rs
+++ b/crates/fj-core/src/operations/build/shell.rs
@@ -166,123 +166,141 @@ pub trait BuildShell {
         let [a, b, c, d] = points.map(Into::into);
 
         let abc = Face::triangle([a, b, c], core);
-        let bad = Face::triangle([b, a, d], core).update_region(
-            |region, core| {
-                region.update_exterior(
-                    |cycle, core| {
-                        cycle
-                            .update_half_edge(
-                                cycle.half_edges().nth_circular(0),
-                                |edge, core| {
-                                    [edge
+        let bad = {
+            let bad = Face::triangle([b, a, d], core);
+            bad.update_region(
+                |region, core| {
+                    region.update_exterior(
+                        |cycle, core| {
+                            cycle
+                                .update_half_edge(
+                                    cycle.half_edges().nth_circular(0),
+                                    |edge, core| {
+                                        [edge.reverse_curve_coordinate_systems(
+                                            core,
+                                        )]
+                                    },
+                                    core,
+                                )
+                                .join_to(
+                                    abc.face.region().exterior(),
+                                    0..=0,
+                                    0..=0,
+                                    bad.face.surface().clone(),
+                                    core,
+                                )
+                        },
+                        core,
+                    )
+                },
+                core,
+            )
+        };
+        let dac =
+            {
+                let dac = Face::triangle([d, a, c], core);
+                dac.update_region(
+                    |region, core| {
+                        region.update_exterior(
+                            |cycle, core| {
+                                cycle
+                                    .update_half_edge(
+                                        cycle.half_edges().nth_circular(1),
+                                        |edge, core| {
+                                            [edge
                                         .reverse_curve_coordinate_systems(core)]
-                                },
-                                core,
-                            )
-                            .join_to(
-                                abc.face.region().exterior(),
-                                0..=0,
-                                0..=0,
-                                core,
-                            )
+                                        },
+                                        core,
+                                    )
+                                    .join_to(
+                                        abc.face.region().exterior(),
+                                        1..=1,
+                                        2..=2,
+                                        dac.face.surface().clone(),
+                                        core,
+                                    )
+                                    .update_half_edge(
+                                        cycle.half_edges().nth_circular(0),
+                                        |edge, core| {
+                                            [edge
+                                        .reverse_curve_coordinate_systems(core)]
+                                        },
+                                        core,
+                                    )
+                                    .join_to(
+                                        bad.face.region().exterior(),
+                                        0..=0,
+                                        1..=1,
+                                        dac.face.surface().clone(),
+                                        core,
+                                    )
+                            },
+                            core,
+                        )
                     },
                     core,
                 )
-            },
-            core,
-        );
-        let dac = Face::triangle([d, a, c], core).update_region(
-            |region, core| {
-                region.update_exterior(
-                    |cycle, core| {
-                        cycle
-                            .update_half_edge(
-                                cycle.half_edges().nth_circular(1),
-                                |edge, core| {
-                                    [edge
+            };
+        let cbd =
+            {
+                let cbd = Face::triangle([c, b, d], core);
+                cbd.update_region(
+                    |region, core| {
+                        region.update_exterior(
+                            |cycle, core| {
+                                cycle
+                                    .update_half_edge(
+                                        cycle.half_edges().nth_circular(0),
+                                        |edge, core| {
+                                            [edge
                                         .reverse_curve_coordinate_systems(core)]
-                                },
-                                core,
-                            )
-                            .join_to(
-                                abc.face.region().exterior(),
-                                1..=1,
-                                2..=2,
-                                core,
-                            )
-                            .update_half_edge(
-                                cycle.half_edges().nth_circular(0),
-                                |edge, core| {
-                                    [edge
+                                        },
+                                        core,
+                                    )
+                                    .update_half_edge(
+                                        cycle.half_edges().nth_circular(1),
+                                        |edge, core| {
+                                            [edge
                                         .reverse_curve_coordinate_systems(core)]
-                                },
-                                core,
-                            )
-                            .join_to(
-                                bad.face.region().exterior(),
-                                0..=0,
-                                1..=1,
-                                core,
-                            )
+                                        },
+                                        core,
+                                    )
+                                    .update_half_edge(
+                                        cycle.half_edges().nth_circular(2),
+                                        |edge, core| {
+                                            [edge
+                                        .reverse_curve_coordinate_systems(core)]
+                                        },
+                                        core,
+                                    )
+                                    .join_to(
+                                        abc.face.region().exterior(),
+                                        0..=0,
+                                        1..=1,
+                                        cbd.face.surface().clone(),
+                                        core,
+                                    )
+                                    .join_to(
+                                        bad.face.region().exterior(),
+                                        1..=1,
+                                        2..=2,
+                                        cbd.face.surface().clone(),
+                                        core,
+                                    )
+                                    .join_to(
+                                        dac.face.region().exterior(),
+                                        2..=2,
+                                        2..=2,
+                                        cbd.face.surface().clone(),
+                                        core,
+                                    )
+                            },
+                            core,
+                        )
                     },
                     core,
                 )
-            },
-            core,
-        );
-        let cbd = Face::triangle([c, b, d], core).update_region(
-            |region, core| {
-                region.update_exterior(
-                    |cycle, core| {
-                        cycle
-                            .update_half_edge(
-                                cycle.half_edges().nth_circular(0),
-                                |edge, core| {
-                                    [edge
-                                        .reverse_curve_coordinate_systems(core)]
-                                },
-                                core,
-                            )
-                            .update_half_edge(
-                                cycle.half_edges().nth_circular(1),
-                                |edge, core| {
-                                    [edge
-                                        .reverse_curve_coordinate_systems(core)]
-                                },
-                                core,
-                            )
-                            .update_half_edge(
-                                cycle.half_edges().nth_circular(2),
-                                |edge, core| {
-                                    [edge
-                                        .reverse_curve_coordinate_systems(core)]
-                                },
-                                core,
-                            )
-                            .join_to(
-                                abc.face.region().exterior(),
-                                0..=0,
-                                1..=1,
-                                core,
-                            )
-                            .join_to(
-                                bad.face.region().exterior(),
-                                1..=1,
-                                2..=2,
-                                core,
-                            )
-                            .join_to(
-                                dac.face.region().exterior(),
-                                2..=2,
-                                2..=2,
-                                core,
-                            )
-                    },
-                    core,
-                )
-            },
-            core,
-        );
+            };
 
         let triangles =
             [abc, bad, dac, cbd].map(|triangle| triangle.insert(core));
diff --git a/crates/fj-core/src/operations/join/cycle.rs b/crates/fj-core/src/operations/join/cycle.rs
index 45122513b..107b75a7f 100644
--- a/crates/fj-core/src/operations/join/cycle.rs
+++ b/crates/fj-core/src/operations/join/cycle.rs
@@ -84,6 +84,7 @@ pub trait JoinCycle {
         other: &Cycle,
         range: RangeInclusive<usize>,
         other_range: RangeInclusive<usize>,
+        surface_self: Handle<Surface>,
         core: &mut Core,
     ) -> Self;
 }
@@ -131,6 +132,7 @@ impl JoinCycle for Cycle {
         other: &Cycle,
         range: RangeInclusive<usize>,
         range_other: RangeInclusive<usize>,
+        surface_self: Handle<Surface>,
         core: &mut Core,
     ) -> Self {
         assert_eq!(
@@ -148,6 +150,30 @@ impl JoinCycle for Cycle {
                     .update_half_edge(
                         self.half_edges().nth_circular(index),
                         |half_edge, core| {
+                            // The curve of the other half-edge we're joining
+                            // this one to already has a curve geometry,
+                            // presumably. But it might not have a local
+                            // definition for the surface that *this* half-edge
+                            // is in.
+                            //
+                            // We need to make sure that any local definition
+                            // that our current curve already has, moves over to
+                            // the new one.
+                            let curve_geom = core
+                                .layers
+                                .geometry
+                                .of_curve(half_edge.curve())
+                                .and_then(|curve_geom| {
+                                    curve_geom.local_on(&surface_self)
+                                });
+                            if let Some(curve_geom) = curve_geom {
+                                core.layers.geometry.define_curve(
+                                    edge_other.curve().clone(),
+                                    surface_self.clone(),
+                                    curve_geom.clone(),
+                                );
+                            }
+
                             [half_edge
                                 .update_curve(
                                     |_, _| edge_other.curve().clone(),