diff --git a/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs b/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
index 065e7f44c..4b6376820 100644
--- a/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
@@ -92,8 +92,7 @@ mod tests {
             .update_as_u_axis()
             .build(&objects)?;
         let half_edge = HalfEdge::partial()
-            .with_surface(Some(surface))
-            .update_as_line_segment_from_points([[1., -1.], [1., 1.]])
+            .update_as_line_segment_from_points(surface, [[1., -1.], [1., 1.]])
             .build(&objects)?;
 
         let intersection = CurveEdgeIntersection::compute(&curve, &half_edge);
@@ -117,8 +116,10 @@ mod tests {
             .update_as_u_axis()
             .build(&objects)?;
         let half_edge = HalfEdge::partial()
-            .with_surface(Some(surface))
-            .update_as_line_segment_from_points([[-1., -1.], [-1., 1.]])
+            .update_as_line_segment_from_points(
+                surface,
+                [[-1., -1.], [-1., 1.]],
+            )
             .build(&objects)?;
 
         let intersection = CurveEdgeIntersection::compute(&curve, &half_edge);
@@ -142,8 +143,10 @@ mod tests {
             .update_as_u_axis()
             .build(&objects)?;
         let half_edge = HalfEdge::partial()
-            .with_surface(Some(surface))
-            .update_as_line_segment_from_points([[-1., -1.], [1., -1.]])
+            .update_as_line_segment_from_points(
+                surface,
+                [[-1., -1.], [1., -1.]],
+            )
             .build(&objects)?;
 
         let intersection = CurveEdgeIntersection::compute(&curve, &half_edge);
@@ -162,8 +165,7 @@ mod tests {
             .update_as_u_axis()
             .build(&objects)?;
         let half_edge = HalfEdge::partial()
-            .with_surface(Some(surface))
-            .update_as_line_segment_from_points([[-1., 0.], [1., 0.]])
+            .update_as_line_segment_from_points(surface, [[-1., 0.], [1., 0.]])
             .build(&objects)?;
 
         let intersection = CurveEdgeIntersection::compute(&curve, &half_edge);
diff --git a/crates/fj-kernel/src/algorithms/sweep/edge.rs b/crates/fj-kernel/src/algorithms/sweep/edge.rs
index 40b44d62d..2f3cce225 100644
--- a/crates/fj-kernel/src/algorithms/sweep/edge.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/edge.rs
@@ -199,8 +199,10 @@ mod tests {
         let objects = Objects::new();
 
         let half_edge = HalfEdge::partial()
-            .with_surface(Some(objects.surfaces.xy_plane()))
-            .update_as_line_segment_from_points([[0., 0.], [1., 0.]])
+            .update_as_line_segment_from_points(
+                objects.surfaces.xy_plane(),
+                [[0., 0.], [1., 0.]],
+            )
             .build(&objects)?;
 
         let face =
@@ -210,8 +212,10 @@ mod tests {
             let surface = objects.surfaces.xz_plane();
 
             let bottom = HalfEdge::partial()
-                .with_surface(Some(surface.clone()))
-                .update_as_line_segment_from_points([[0., 0.], [1., 0.]])
+                .update_as_line_segment_from_points(
+                    surface.clone(),
+                    [[0., 0.], [1., 0.]],
+                )
                 .build(&objects)?;
             let side_up = HalfEdge::partial()
                 .with_surface(Some(surface.clone()))
diff --git a/crates/fj-kernel/src/algorithms/sweep/face.rs b/crates/fj-kernel/src/algorithms/sweep/face.rs
index 2a95a7438..eb44fc87c 100644
--- a/crates/fj-kernel/src/algorithms/sweep/face.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/face.rs
@@ -125,8 +125,10 @@ mod tests {
             .array_windows_ext()
             .map(|&[a, b]| {
                 let half_edge = HalfEdge::partial()
-                    .with_surface(Some(objects.surfaces.xy_plane()))
-                    .update_as_line_segment_from_points([a, b])
+                    .update_as_line_segment_from_points(
+                        objects.surfaces.xy_plane(),
+                        [a, b],
+                    )
                     .build(&objects)?;
                 (half_edge, Color::default()).sweep(UP, &objects)
             })
@@ -167,8 +169,10 @@ mod tests {
             .array_windows_ext()
             .map(|&[a, b]| {
                 let half_edge = HalfEdge::partial()
-                    .with_surface(Some(objects.surfaces.xy_plane()))
-                    .update_as_line_segment_from_points([a, b])
+                    .update_as_line_segment_from_points(
+                        objects.surfaces.xy_plane(),
+                        [a, b],
+                    )
                     .build(&objects)?
                     .reverse(&objects)?;
                 (half_edge, Color::default()).sweep(DOWN, &objects)
diff --git a/crates/fj-kernel/src/algorithms/sweep/vertex.rs b/crates/fj-kernel/src/algorithms/sweep/vertex.rs
index 9946dc987..f348b561d 100644
--- a/crates/fj-kernel/src/algorithms/sweep/vertex.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/vertex.rs
@@ -191,8 +191,7 @@ mod tests {
             (vertex, surface.clone()).sweep([0., 0., 1.], &objects)?;
 
         let expected_half_edge = HalfEdge::partial()
-            .with_surface(Some(surface))
-            .update_as_line_segment_from_points([[0., 0.], [0., 1.]])
+            .update_as_line_segment_from_points(surface, [[0., 0.], [0., 1.]])
             .build(&objects)?;
         assert_eq!(half_edge, expected_half_edge);
         Ok(())
diff --git a/crates/fj-kernel/src/algorithms/transform/cycle.rs b/crates/fj-kernel/src/algorithms/transform/cycle.rs
index 1cbf50694..5f222af69 100644
--- a/crates/fj-kernel/src/algorithms/transform/cycle.rs
+++ b/crates/fj-kernel/src/algorithms/transform/cycle.rs
@@ -12,22 +12,11 @@ impl TransformObject for PartialCycle {
         transform: &Transform,
         objects: &Objects,
     ) -> Result<Self, ValidationError> {
-        let surface = self
-            .surface()
-            .map(|surface| surface.transform(transform, objects))
-            .transpose()?;
         let half_edges = self
             .half_edges()
-            .map(|edge| {
-                Ok(edge
-                    .into_partial()
-                    .transform(transform, objects)?
-                    .with_surface(surface.clone()))
-            })
+            .map(|edge| edge.into_partial().transform(transform, objects))
             .collect::<Result<Vec<_>, ValidationError>>()?;
 
-        Ok(Self::default()
-            .with_surface(surface)
-            .with_half_edges(half_edges))
+        Ok(Self::default().with_half_edges(half_edges))
     }
 }
diff --git a/crates/fj-kernel/src/algorithms/transform/edge.rs b/crates/fj-kernel/src/algorithms/transform/edge.rs
index 3e7dac5ae..71c72a8d1 100644
--- a/crates/fj-kernel/src/algorithms/transform/edge.rs
+++ b/crates/fj-kernel/src/algorithms/transform/edge.rs
@@ -15,15 +15,10 @@ impl TransformObject for PartialHalfEdge {
         transform: &Transform,
         objects: &Objects,
     ) -> Result<Self, ValidationError> {
-        let surface = self
-            .surface()
-            .map(|surface| surface.transform(transform, objects))
-            .transpose()?;
         let curve: MaybePartial<_> = self
             .curve()
             .into_partial()
             .transform(transform, objects)?
-            .with_surface(surface.clone())
             .into();
         let vertices = self.vertices().try_map_ext(
             |vertex| -> Result<_, ValidationError> {
@@ -41,7 +36,6 @@ impl TransformObject for PartialHalfEdge {
             .into();
 
         Ok(Self::default()
-            .with_surface(surface)
             .with_curve(Some(curve))
             .with_vertices(Some(vertices))
             .with_global_form(global_form))
diff --git a/crates/fj-kernel/src/builder/cycle.rs b/crates/fj-kernel/src/builder/cycle.rs
index 04251a2fc..86103d627 100644
--- a/crates/fj-kernel/src/builder/cycle.rs
+++ b/crates/fj-kernel/src/builder/cycle.rs
@@ -1,8 +1,9 @@
 use fj_math::Point;
 
 use crate::{
-    objects::{Curve, HalfEdge, SurfaceVertex, Vertex},
+    objects::{Curve, HalfEdge, Surface, SurfaceVertex, Vertex},
     partial::{HasPartial, MaybePartial, PartialCycle},
+    storage::Handle,
 };
 
 use super::{CurveBuilder, HalfEdgeBuilder};
@@ -18,6 +19,7 @@ pub trait CycleBuilder {
     /// Update the partial cycle with a polygonal chain from the provided points
     fn with_poly_chain_from_points(
         self,
+        surface: Handle<Surface>,
         points: impl IntoIterator<Item = impl Into<Point<2>>>,
     ) -> Self;
 
@@ -49,9 +51,8 @@ impl CycleBuilder for PartialCycle {
         let mut half_edges = Vec::new();
         for vertex_next in iter {
             if let Some(vertex_prev) = previous {
-                let surface = self
+                let surface = vertex_prev
                     .surface()
-                    .clone()
                     .expect("Need surface to extend cycle with poly-chain");
 
                 let position_prev = vertex_prev
@@ -61,12 +62,8 @@ impl CycleBuilder for PartialCycle {
                     .position()
                     .expect("Need surface position to extend cycle");
 
-                let from = vertex_prev.update_partial(|partial| {
-                    partial.with_surface(Some(surface.clone()))
-                });
-                let to = vertex_next.update_partial(|partial| {
-                    partial.with_surface(Some(surface.clone()))
-                });
+                let from = vertex_prev;
+                let to = vertex_next;
 
                 previous = Some(to.clone());
 
@@ -99,10 +96,13 @@ impl CycleBuilder for PartialCycle {
 
     fn with_poly_chain_from_points(
         self,
+        surface: Handle<Surface>,
         points: impl IntoIterator<Item = impl Into<Point<2>>>,
     ) -> Self {
         self.with_poly_chain(points.into_iter().map(|position| {
-            SurfaceVertex::partial().with_position(Some(position))
+            SurfaceVertex::partial()
+                .with_surface(Some(surface.clone()))
+                .with_position(Some(position))
         }))
     }
 
@@ -127,8 +127,7 @@ impl CycleBuilder for PartialCycle {
 
         self.with_half_edges(Some(
             HalfEdge::partial()
-                .with_surface(Some(surface))
-                .update_as_line_segment_from_points(vertices),
+                .update_as_line_segment_from_points(surface, vertices),
         ))
     }
 }
diff --git a/crates/fj-kernel/src/builder/edge.rs b/crates/fj-kernel/src/builder/edge.rs
index e23263f9c..65c20f4a0 100644
--- a/crates/fj-kernel/src/builder/edge.rs
+++ b/crates/fj-kernel/src/builder/edge.rs
@@ -3,7 +3,7 @@ use fj_math::{Point, Scalar};
 
 use crate::{
     objects::{
-        Curve, GlobalVertex, Objects, SurfaceVertex, Vertex,
+        Curve, GlobalVertex, Objects, Surface, SurfaceVertex, Vertex,
         VerticesInNormalizedOrder,
     },
     partial::{HasPartial, PartialGlobalEdge, PartialHalfEdge},
@@ -31,6 +31,7 @@ pub trait HalfEdgeBuilder: Sized {
     /// Update partial half-edge as a line segment, from the given points
     fn update_as_line_segment_from_points(
         self,
+        surface: Handle<Surface>,
         points: [impl Into<Point<2>>; 2],
     ) -> Self;
 
@@ -44,9 +45,10 @@ impl HalfEdgeBuilder for PartialHalfEdge {
         radius: impl Into<Scalar>,
         objects: &Objects,
     ) -> Result<Self, ValidationError> {
-        let curve = Curve::partial()
+        let curve = self
+            .curve()
+            .into_partial()
             .with_global_form(Some(self.extract_global_curve()))
-            .with_surface(self.surface())
             .update_as_circle_from_radius(radius);
 
         let path = curve.path().expect("Expected path that was just created");
@@ -55,7 +57,8 @@ impl HalfEdgeBuilder for PartialHalfEdge {
             [Scalar::ZERO, Scalar::TAU].map(|coord| Point::from([coord]));
 
         let global_vertex = self
-            .extract_global_vertices()
+            .global_form()
+            .vertices()
             .map(|[global_form, _]| global_form)
             .unwrap_or_else(|| {
                 GlobalVertex::partial()
@@ -65,7 +68,7 @@ impl HalfEdgeBuilder for PartialHalfEdge {
 
         let surface_vertex = SurfaceVertex::partial()
             .with_position(Some(path.point_from_path_coords(a_curve)))
-            .with_surface(self.surface())
+            .with_surface(curve.surface())
             .with_global_form(Some(global_vertex))
             .build(objects)?;
 
@@ -83,18 +86,20 @@ impl HalfEdgeBuilder for PartialHalfEdge {
 
     fn update_as_line_segment_from_points(
         self,
+        surface: Handle<Surface>,
         points: [impl Into<Point<2>>; 2],
     ) -> Self {
-        let surface = self.surface();
         let vertices = points.map(|point| {
             let surface_form = SurfaceVertex::partial()
-                .with_surface(surface.clone())
+                .with_surface(Some(surface.clone()))
                 .with_position(Some(point));
 
             Vertex::partial().with_surface_form(Some(surface_form))
         });
 
-        self.with_vertices(Some(vertices)).update_as_line_segment()
+        self.with_surface(Some(surface))
+            .with_vertices(Some(vertices))
+            .update_as_line_segment()
     }
 
     fn update_as_line_segment(self) -> Self {
@@ -103,6 +108,7 @@ impl HalfEdgeBuilder for PartialHalfEdge {
             [&from, &to].map(|vertex| vertex.surface_form());
 
         let surface = self
+            .curve()
             .surface()
             .or_else(|| from_surface.surface())
             .or_else(|| to_surface.surface())
@@ -147,7 +153,8 @@ impl HalfEdgeBuilder for PartialHalfEdge {
                     must_switch_order
                 };
 
-                self.extract_global_vertices()
+                self.global_form()
+                    .vertices()
                     .map(
                         |[a, b]| {
                             if must_switch_order {
diff --git a/crates/fj-kernel/src/builder/face.rs b/crates/fj-kernel/src/builder/face.rs
index bb34c3768..0f217d409 100644
--- a/crates/fj-kernel/src/builder/face.rs
+++ b/crates/fj-kernel/src/builder/face.rs
@@ -50,10 +50,14 @@ impl<'a> FaceBuilder<'a> {
         mut self,
         points: impl IntoIterator<Item = impl Into<Point<2>>>,
     ) -> Self {
+        let surface = self
+            .surface
+            .as_ref()
+            .expect("Need surface to create polygon");
+
         self.exterior = Some(
             Cycle::partial()
-                .with_surface(self.surface.clone())
-                .with_poly_chain_from_points(points)
+                .with_poly_chain_from_points(surface.clone(), points)
                 .close_with_line_segment()
                 .build(self.objects)
                 .unwrap(),
@@ -75,10 +79,14 @@ impl<'a> FaceBuilder<'a> {
         mut self,
         points: impl IntoIterator<Item = impl Into<Point<2>>>,
     ) -> Self {
+        let surface = self
+            .surface
+            .as_ref()
+            .expect("Need surface to build polygon.");
+
         self.interiors.push(
             Cycle::partial()
-                .with_surface(self.surface.clone())
-                .with_poly_chain_from_points(points)
+                .with_poly_chain_from_points(surface.clone(), points)
                 .close_with_line_segment()
                 .build(self.objects)
                 .unwrap(),
diff --git a/crates/fj-kernel/src/builder/shell.rs b/crates/fj-kernel/src/builder/shell.rs
index 83c73cf29..1785af97b 100644
--- a/crates/fj-kernel/src/builder/shell.rs
+++ b/crates/fj-kernel/src/builder/shell.rs
@@ -89,12 +89,11 @@ impl<'a> ShellBuilder<'a> {
                 .zip(&surfaces)
                 .map(|(half_edge, surface)| {
                     HalfEdge::partial()
-                        .with_surface(Some(surface.clone()))
                         .with_global_form(Some(half_edge.global_form().clone()))
-                        .update_as_line_segment_from_points([
-                            [Z, Z],
-                            [edge_length, Z],
-                        ])
+                        .update_as_line_segment_from_points(
+                            surface.clone(),
+                            [[Z, Z], [edge_length, Z]],
+                        )
                         .build(self.objects)
                         .unwrap()
                 })
@@ -188,10 +187,8 @@ impl<'a> ShellBuilder<'a> {
                 .zip(sides_up)
                 .zip(tops.clone())
                 .zip(sides_down)
-                .zip(surfaces)
-                .map(|((((bottom, side_up), top), side_down), surface)| {
+                .map(|(((bottom, side_up), top), side_down)| {
                     let cycle = Cycle::partial()
-                        .with_surface(Some(surface))
                         .with_half_edges([bottom, side_up, top, side_down])
                         .build(self.objects)
                         .unwrap();
diff --git a/crates/fj-kernel/src/iter.rs b/crates/fj-kernel/src/iter.rs
index 718f7c5f0..c27d3fbaf 100644
--- a/crates/fj-kernel/src/iter.rs
+++ b/crates/fj-kernel/src/iter.rs
@@ -399,8 +399,10 @@ mod tests {
 
         let surface = objects.surfaces.xy_plane();
         let object = Cycle::partial()
-            .with_surface(Some(surface))
-            .with_poly_chain_from_points([[0., 0.], [1., 0.], [0., 1.]])
+            .with_poly_chain_from_points(
+                surface,
+                [[0., 0.], [1., 0.], [0., 1.]],
+            )
             .close_with_line_segment()
             .build(&objects);
 
@@ -485,8 +487,10 @@ mod tests {
         let objects = Objects::new();
 
         let object = HalfEdge::partial()
-            .with_surface(Some(objects.surfaces.xy_plane()))
-            .update_as_line_segment_from_points([[0., 0.], [1., 0.]])
+            .update_as_line_segment_from_points(
+                objects.surfaces.xy_plane(),
+                [[0., 0.], [1., 0.]],
+            )
             .build(&objects);
 
         assert_eq!(1, object.curve_iter().count());
diff --git a/crates/fj-kernel/src/objects/edge.rs b/crates/fj-kernel/src/objects/edge.rs
index 5e5dc45fb..ea20d4e7d 100644
--- a/crates/fj-kernel/src/objects/edge.rs
+++ b/crates/fj-kernel/src/objects/edge.rs
@@ -165,12 +165,10 @@ mod tests {
         let b = [1., 0.];
 
         let a_to_b = HalfEdge::partial()
-            .with_surface(Some(surface.clone()))
-            .update_as_line_segment_from_points([a, b])
+            .update_as_line_segment_from_points(surface.clone(), [a, b])
             .build(&objects)?;
         let b_to_a = HalfEdge::partial()
-            .with_surface(Some(surface))
-            .update_as_line_segment_from_points([b, a])
+            .update_as_line_segment_from_points(surface, [b, a])
             .build(&objects)?;
 
         assert_eq!(a_to_b.global_form(), b_to_a.global_form());
diff --git a/crates/fj-kernel/src/partial/maybe_partial.rs b/crates/fj-kernel/src/partial/maybe_partial.rs
index 7431df451..5e7e7c1da 100644
--- a/crates/fj-kernel/src/partial/maybe_partial.rs
+++ b/crates/fj-kernel/src/partial/maybe_partial.rs
@@ -64,6 +64,20 @@ impl<T: HasPartial> MaybePartial<T> {
         }
     }
 
+    /// Merge this `MaybePartial` with another of the same type
+    pub fn merge_with(self, other: Self) -> Self {
+        match (self, other) {
+            (Self::Full(_), Self::Full(_)) => {
+                panic!("Can't merge two full objects")
+            }
+            (Self::Full(full), Self::Partial(_))
+            | (Self::Partial(_), Self::Full(full)) => Self::Full(full),
+            (Self::Partial(a), Self::Partial(b)) => {
+                Self::Partial(a.merge_with(b))
+            }
+        }
+    }
+
     /// Return or build a full object
     ///
     /// If this already is a full object, it is returned. If this is a partial
@@ -160,6 +174,14 @@ impl MaybePartial<GlobalEdge> {
 }
 
 impl MaybePartial<HalfEdge> {
+    /// Access the curve
+    pub fn curve(&self) -> MaybePartial<Curve> {
+        match self {
+            Self::Full(full) => full.curve().clone().into(),
+            Self::Partial(partial) => partial.curve(),
+        }
+    }
+
     /// Access the front vertex
     pub fn front(&self) -> MaybePartial<Vertex> {
         match self {
diff --git a/crates/fj-kernel/src/partial/mod.rs b/crates/fj-kernel/src/partial/mod.rs
index 87ac53f2d..172c05cef 100644
--- a/crates/fj-kernel/src/partial/mod.rs
+++ b/crates/fj-kernel/src/partial/mod.rs
@@ -37,6 +37,7 @@
 mod maybe_partial;
 mod objects;
 mod traits;
+mod util;
 
 pub use self::{
     maybe_partial::MaybePartial,
diff --git a/crates/fj-kernel/src/partial/objects/curve.rs b/crates/fj-kernel/src/partial/objects/curve.rs
index f55b97924..f8840e1a0 100644
--- a/crates/fj-kernel/src/partial/objects/curve.rs
+++ b/crates/fj-kernel/src/partial/objects/curve.rs
@@ -1,6 +1,6 @@
 use crate::{
     objects::{Curve, GlobalCurve, Objects, Surface},
-    partial::MaybePartial,
+    partial::{util::merge_options, MaybePartial},
     path::SurfacePath,
     storage::Handle,
     validate::ValidationError,
@@ -59,6 +59,26 @@ impl PartialCurve {
         self
     }
 
+    /// Merge this partial object with another
+    pub fn merge_with(self, other: Self) -> Self {
+        // This is harder than it should be, as `global_form` uses the redundant
+        // `Option<MaybePartial<_>>` representation. There's some code relying
+        // on that though, so we have to live with it for now.
+        let global_form = match (self.global_form, other.global_form) {
+            (Some(a), Some(b)) => Some(a.merge_with(b)),
+            (Some(global_form), None) | (None, Some(global_form)) => {
+                Some(global_form)
+            }
+            (None, None) => None,
+        };
+
+        Self {
+            path: merge_options(self.path, other.path),
+            surface: merge_options(self.surface, other.surface),
+            global_form,
+        }
+    }
+
     /// Build a full [`Curve`] from the partial curve
     pub fn build(
         self,
@@ -102,6 +122,11 @@ impl From<&Curve> for PartialCurve {
 pub struct PartialGlobalCurve;
 
 impl PartialGlobalCurve {
+    /// Merge this partial object with another
+    pub fn merge_with(self, _: Self) -> Self {
+        Self
+    }
+
     /// Build a full [`GlobalCurve`] from the partial global curve
     pub fn build(
         self,
diff --git a/crates/fj-kernel/src/partial/objects/cycle.rs b/crates/fj-kernel/src/partial/objects/cycle.rs
index fe2163112..7d2c146a3 100644
--- a/crates/fj-kernel/src/partial/objects/cycle.rs
+++ b/crates/fj-kernel/src/partial/objects/cycle.rs
@@ -1,6 +1,6 @@
 use crate::{
     objects::{Cycle, HalfEdge, Objects, Surface},
-    partial::MaybePartial,
+    partial::{util::merge_options, MaybePartial},
     storage::Handle,
     validate::ValidationError,
 };
@@ -10,46 +10,80 @@ use crate::{
 /// See [`crate::partial`] for more information.
 #[derive(Clone, Debug, Default)]
 pub struct PartialCycle {
-    surface: Option<Handle<Surface>>,
     half_edges: Vec<MaybePartial<HalfEdge>>,
 }
 
 impl PartialCycle {
-    /// Access the surface that the [`Cycle`] is defined in
-    pub fn surface(&self) -> Option<Handle<Surface>> {
-        self.surface.clone()
-    }
-
     /// Access the half-edges that make up the [`Cycle`]
     pub fn half_edges(&self) -> impl Iterator<Item = MaybePartial<HalfEdge>> {
         self.half_edges.clone().into_iter()
     }
 
-    /// Update the partial cycle with the given surface
-    pub fn with_surface(mut self, surface: Option<Handle<Surface>>) -> Self {
-        if let Some(surface) = surface {
-            self.surface = Some(surface);
-        }
-        self
+    /// Access the surface that the [`Cycle`]'s [`HalfEdge`]s are defined in
+    pub fn surface(&self) -> Option<Handle<Surface>> {
+        self.half_edges
+            .first()
+            .and_then(|half_edge| half_edge.curve().surface())
     }
 
-    /// Update the partial cycle with the given half-edges
+    /// Add the provided half-edges to the partial cycle
+    ///
+    /// This will merge all the surfaces of the added half-edges. All added
+    /// half-edges will end up with the same merged surface.
+    ///
+    /// # Panics
+    ///
+    /// Panics, if the surfaces can't be merged.
     pub fn with_half_edges(
         mut self,
         half_edges: impl IntoIterator<Item = impl Into<MaybePartial<HalfEdge>>>,
     ) -> Self {
-        self.half_edges
-            .extend(half_edges.into_iter().map(Into::into));
+        let half_edges = half_edges.into_iter().map(Into::into);
+
+        let mut surface = self.surface();
+        for half_edge in half_edges {
+            surface = merge_options(surface, half_edge.curve().surface());
+            self.half_edges.push(half_edge);
+        }
+
+        self.with_surface(surface)
+    }
+
+    /// Update the partial cycle with the provided surface
+    ///
+    /// All [`HalfEdge`]s will be updated with this surface.
+    pub fn with_surface(mut self, surface: Option<Handle<Surface>>) -> Self {
+        if let Some(surface) = surface {
+            for half_edge in &mut self.half_edges {
+                *half_edge = half_edge.clone().update_partial(|half_edge| {
+                    half_edge.with_surface(Some(surface.clone()))
+                });
+            }
+        }
         self
     }
 
+    /// Merge this partial object with another
+    pub fn merge_with(self, other: Self) -> Self {
+        let a_is_empty = self.half_edges.is_empty();
+        let b_is_empty = other.half_edges.is_empty();
+        let half_edges = match (a_is_empty, b_is_empty) {
+            (true, true) => {
+                panic!("Can't merge `PartialHalfEdge`, if both have half-edges")
+            }
+            (true, false) => self.half_edges,
+            (false, true) => other.half_edges,
+            (false, false) => self.half_edges, // doesn't matter which we use
+        };
+
+        Self { half_edges }
+    }
+
     /// Build a full [`Cycle`] from the partial cycle
     pub fn build(
         mut self,
         objects: &Objects,
     ) -> Result<Handle<Cycle>, ValidationError> {
-        let surface = self.surface.expect("Need surface to build `Cycle`");
-        let surface_for_edges = surface.clone();
         let half_edges = {
             let last_vertex = self
                 .half_edges
@@ -65,11 +99,7 @@ impl PartialCycle {
                 .map(|(half_edge, vertex, surface_vertex)|
                     -> Result<_, ValidationError>
                 {
-                    let surface_vertex = surface_vertex
-                        .update_partial(|surface_vertex| {
-                            surface_vertex.with_surface(Some(surface.clone()))
-                        })
-                        .into_full(objects)?;
+                    let surface_vertex = surface_vertex.into_full(objects)?;
 
                     *half_edge =
                         half_edge.clone().update_partial(|half_edge| {
@@ -98,9 +128,7 @@ impl PartialCycle {
                                 partial.with_surface_form(previous_vertex)
                             });
 
-                            half_edge
-                                .with_surface(Some(surface_for_edges.clone()))
-                                .with_back_vertex(Some(back))
+                            half_edge.with_back_vertex(Some(back))
                         })
                         .into_full(objects)?;
 
@@ -121,7 +149,6 @@ impl PartialCycle {
 impl From<&Cycle> for PartialCycle {
     fn from(cycle: &Cycle) -> Self {
         Self {
-            surface: Some(cycle.surface().clone()),
             half_edges: cycle.half_edges().cloned().map(Into::into).collect(),
         }
     }
diff --git a/crates/fj-kernel/src/partial/objects/edge.rs b/crates/fj-kernel/src/partial/objects/edge.rs
index 4265d158c..2aadf3bb4 100644
--- a/crates/fj-kernel/src/partial/objects/edge.rs
+++ b/crates/fj-kernel/src/partial/objects/edge.rs
@@ -6,7 +6,7 @@ use crate::{
         Curve, GlobalCurve, GlobalEdge, GlobalVertex, HalfEdge, Objects,
         Surface, Vertex,
     },
-    partial::MaybePartial,
+    partial::{util::merge_arrays, MaybePartial},
     storage::Handle,
     validate::ValidationError,
 };
@@ -16,18 +16,12 @@ use crate::{
 /// See [`crate::partial`] for more information.
 #[derive(Clone, Debug, Default)]
 pub struct PartialHalfEdge {
-    surface: Option<Handle<Surface>>,
     curve: MaybePartial<Curve>,
     vertices: [MaybePartial<Vertex>; 2],
     global_form: MaybePartial<GlobalEdge>,
 }
 
 impl PartialHalfEdge {
-    /// Access the surface that the [`HalfEdge`]'s [`Curve`] is defined in
-    pub fn surface(&self) -> Option<Handle<Surface>> {
-        self.surface.clone()
-    }
-
     /// Access the curve that the [`HalfEdge`] is defined in
     pub fn curve(&self) -> MaybePartial<Curve> {
         self.curve.clone()
@@ -52,17 +46,24 @@ impl PartialHalfEdge {
             .unwrap_or_else(|| self.global_form.curve())
     }
 
-    /// Access the vertices of the global form, if available
-    pub fn extract_global_vertices(
-        &self,
-    ) -> Option<[MaybePartial<GlobalVertex>; 2]> {
-        self.global_form.vertices()
-    }
-
     /// Update the partial half-edge with the given surface
     pub fn with_surface(mut self, surface: Option<Handle<Surface>>) -> Self {
         if let Some(surface) = surface {
-            self.surface = Some(surface);
+            self.curve = self.curve.update_partial(|curve| {
+                curve.with_surface(Some(surface.clone()))
+            });
+
+            self.vertices = self.vertices.map(|vertex| {
+                vertex.update_partial(|vertex| {
+                    let surface_form = vertex.surface_form().update_partial(
+                        |surface_vertex| {
+                            surface_vertex.with_surface(Some(surface.clone()))
+                        },
+                    );
+
+                    vertex.with_surface_form(Some(surface_form))
+                })
+            });
         }
         self
     }
@@ -78,7 +79,7 @@ impl PartialHalfEdge {
         self
     }
 
-    /// Update the partial half-edge with the given from vertex
+    /// Update the partial half-edge with the given back vertex
     pub fn with_back_vertex(
         mut self,
         vertex: Option<impl Into<MaybePartial<Vertex>>>,
@@ -90,7 +91,7 @@ impl PartialHalfEdge {
         self
     }
 
-    /// Update the partial half-edge with the given from vertex
+    /// Update the partial half-edge with the given front vertex
     pub fn with_front_vertex(
         mut self,
         vertex: Option<impl Into<MaybePartial<Vertex>>>,
@@ -125,16 +126,21 @@ impl PartialHalfEdge {
         self
     }
 
+    /// Merge this partial object with another
+    pub fn merge_with(self, other: Self) -> Self {
+        Self {
+            curve: self.curve.merge_with(other.curve),
+            vertices: merge_arrays(self.vertices, other.vertices),
+            global_form: self.global_form.merge_with(other.global_form),
+        }
+    }
+
     /// Build a full [`HalfEdge`] from the partial half-edge
     pub fn build(
         self,
         objects: &Objects,
     ) -> Result<Handle<HalfEdge>, ValidationError> {
-        let surface = self.surface;
-        let curve = self
-            .curve
-            .update_partial(|curve| curve.with_surface(surface))
-            .into_full(objects)?;
+        let curve = self.curve.into_full(objects)?;
         let vertices = self.vertices.try_map_ext(|vertex| {
             vertex
                 .update_partial(|vertex| vertex.with_curve(Some(curve.clone())))
@@ -160,7 +166,6 @@ impl From<&HalfEdge> for PartialHalfEdge {
             half_edge.vertices().clone().map(Into::into);
 
         Self {
-            surface: Some(half_edge.curve().surface().clone()),
             curve: half_edge.curve().clone().into(),
             vertices: [back_vertex, front_vertex],
             global_form: half_edge.global_form().clone().into(),
@@ -210,6 +215,23 @@ impl PartialGlobalEdge {
         self
     }
 
+    /// Merge this partial object with another
+    pub fn merge_with(self, other: Self) -> Self {
+        // This is harder than it needs to be, because `vertices` uses the
+        // redundant combination of `Option` and `MaybePartial`. There's some
+        // code relying on that, however, so we have to live with it for now.
+        let vertices = match (self.vertices, other.vertices) {
+            (Some(a), Some(b)) => Some(merge_arrays(a, b)),
+            (Some(vertices), None) | (None, Some(vertices)) => Some(vertices),
+            (None, None) => None,
+        };
+
+        Self {
+            curve: self.curve.merge_with(other.curve),
+            vertices,
+        }
+    }
+
     /// Build a full [`GlobalEdge`] from the partial global edge
     pub fn build(
         self,
diff --git a/crates/fj-kernel/src/partial/objects/mod.rs b/crates/fj-kernel/src/partial/objects/mod.rs
index 4032519a7..751063c23 100644
--- a/crates/fj-kernel/src/partial/objects/mod.rs
+++ b/crates/fj-kernel/src/partial/objects/mod.rs
@@ -27,6 +27,10 @@ macro_rules! impl_traits {
             impl Partial for $partial {
                 type Full = $full;
 
+                fn merge_with(self, other: Self) -> Self {
+                    self.merge_with(other)
+                }
+
                 fn build(self, objects: &Objects)
                     -> Result<
                         Handle<Self::Full>,
diff --git a/crates/fj-kernel/src/partial/objects/vertex.rs b/crates/fj-kernel/src/partial/objects/vertex.rs
index 778d9b068..b04a67b02 100644
--- a/crates/fj-kernel/src/partial/objects/vertex.rs
+++ b/crates/fj-kernel/src/partial/objects/vertex.rs
@@ -3,7 +3,7 @@ use fj_math::Point;
 use crate::{
     builder::GlobalVertexBuilder,
     objects::{Curve, GlobalVertex, Objects, Surface, SurfaceVertex, Vertex},
-    partial::MaybePartial,
+    partial::{util::merge_options, MaybePartial},
     storage::Handle,
     validate::ValidationError,
 };
@@ -67,6 +67,15 @@ impl PartialVertex {
         self
     }
 
+    /// Merge this partial object with another
+    pub fn merge_with(self, other: Self) -> Self {
+        Self {
+            position: merge_options(self.position, other.position),
+            curve: self.curve.merge_with(other.curve),
+            surface_form: self.surface_form.merge_with(other.surface_form),
+        }
+    }
+
     /// Build a full [`Vertex`] from the partial vertex
     ///
     /// # Panics
@@ -170,6 +179,15 @@ impl PartialSurfaceVertex {
         self
     }
 
+    /// Merge this partial object with another
+    pub fn merge_with(self, other: Self) -> Self {
+        Self {
+            position: merge_options(self.position, other.position),
+            surface: merge_options(self.surface, other.surface),
+            global_form: self.global_form.merge_with(other.global_form),
+        }
+    }
+
     /// Build a full [`SurfaceVertex`] from the partial surface vertex
     pub fn build(
         self,
@@ -232,6 +250,13 @@ impl PartialGlobalVertex {
         self
     }
 
+    /// Merge this partial object with another
+    pub fn merge_with(self, other: Self) -> Self {
+        Self {
+            position: merge_options(self.position, other.position),
+        }
+    }
+
     /// Build a full [`GlobalVertex`] from the partial global vertex
     pub fn build(
         self,
diff --git a/crates/fj-kernel/src/partial/traits.rs b/crates/fj-kernel/src/partial/traits.rs
index 6a15f8b48..e18bf1c4a 100644
--- a/crates/fj-kernel/src/partial/traits.rs
+++ b/crates/fj-kernel/src/partial/traits.rs
@@ -68,6 +68,9 @@ pub trait Partial: Default + for<'a> From<&'a Self::Full> {
     /// The type representing the full variant of this object
     type Full;
 
+    /// Merge another partial object of the same type into this one
+    fn merge_with(self, other: Self) -> Self;
+
     /// Build a full object from this partial one
     ///
     /// Implementations of this method will typically try to infer any missing
diff --git a/crates/fj-kernel/src/partial/util.rs b/crates/fj-kernel/src/partial/util.rs
new file mode 100644
index 000000000..e0605ee1e
--- /dev/null
+++ b/crates/fj-kernel/src/partial/util.rs
@@ -0,0 +1,26 @@
+use fj_interop::ext::ArrayExt;
+
+use super::{HasPartial, MaybePartial};
+
+pub fn merge_options<T>(a: Option<T>, b: Option<T>) -> Option<T>
+where
+    T: Eq,
+{
+    if a == b {
+        return a;
+    }
+
+    // We know that `a != b`, or we wouldn't have made it here.
+    if a.is_some() && b.is_some() {
+        panic!("Can't merge `Option`s if both are defined");
+    }
+
+    a.xor(b)
+}
+
+pub fn merge_arrays<T: HasPartial>(
+    a: [MaybePartial<T>; 2],
+    b: [MaybePartial<T>; 2],
+) -> [MaybePartial<T>; 2] {
+    a.zip_ext(b).map(|(a, b)| a.merge_with(b))
+}
diff --git a/crates/fj-kernel/src/validate/edge.rs b/crates/fj-kernel/src/validate/edge.rs
index 3f2f2e3ad..ffc50c29f 100644
--- a/crates/fj-kernel/src/validate/edge.rs
+++ b/crates/fj-kernel/src/validate/edge.rs
@@ -207,8 +207,10 @@ mod tests {
         let objects = Objects::new();
 
         let valid = HalfEdge::partial()
-            .with_surface(Some(objects.surfaces.xy_plane()))
-            .update_as_line_segment_from_points([[0., 0.], [1., 0.]])
+            .update_as_line_segment_from_points(
+                objects.surfaces.xy_plane(),
+                [[0., 0.], [1., 0.]],
+            )
             .build(&objects)?;
         let invalid = {
             let mut vertices = valid.vertices().clone();
@@ -232,8 +234,10 @@ mod tests {
         let objects = Objects::new();
 
         let valid = HalfEdge::partial()
-            .with_surface(Some(objects.surfaces.xy_plane()))
-            .update_as_line_segment_from_points([[0., 0.], [1., 0.]])
+            .update_as_line_segment_from_points(
+                objects.surfaces.xy_plane(),
+                [[0., 0.], [1., 0.]],
+            )
             .build(&objects)?;
         let invalid = HalfEdge::new(
             valid.vertices().clone(),
@@ -255,8 +259,10 @@ mod tests {
         let objects = Objects::new();
 
         let valid = HalfEdge::partial()
-            .with_surface(Some(objects.surfaces.xy_plane()))
-            .update_as_line_segment_from_points([[0., 0.], [1., 0.]])
+            .update_as_line_segment_from_points(
+                objects.surfaces.xy_plane(),
+                [[0., 0.], [1., 0.]],
+            )
             .build(&objects)?;
         let invalid = HalfEdge::new(
             valid.vertices().clone(),
@@ -285,8 +291,10 @@ mod tests {
         let objects = Objects::new();
 
         let valid = HalfEdge::partial()
-            .with_surface(Some(objects.surfaces.xy_plane()))
-            .update_as_line_segment_from_points([[0., 0.], [1., 0.]])
+            .update_as_line_segment_from_points(
+                objects.surfaces.xy_plane(),
+                [[0., 0.], [1., 0.]],
+            )
             .build(&objects)?;
         let invalid = HalfEdge::new(
             valid.vertices().clone().try_map_ext(|vertex| {