diff --git a/crates/fj-kernel/src/algorithms/approx/curve.rs b/crates/fj-kernel/src/algorithms/approx/curve.rs
index 870925bcc..00e77f588 100644
--- a/crates/fj-kernel/src/algorithms/approx/curve.rs
+++ b/crates/fj-kernel/src/algorithms/approx/curve.rs
@@ -12,8 +12,8 @@
 use std::collections::BTreeMap;
 
 use crate::{
+    geometry::path::{GlobalPath, SurfacePath},
     objects::{Curve, GlobalCurve},
-    path::{GlobalPath, SurfacePath},
     storage::{Handle, ObjectId},
 };
 
@@ -62,7 +62,7 @@ fn approx_global_curve(
     // This will probably all be unified eventually, as `SurfacePath` and
     // `GlobalPath` grow APIs that are better suited to implementing this code
     // in a more abstract way.
-    let points = match (curve.path(), curve.surface().u()) {
+    let points = match (curve.path(), curve.surface().geometry().u) {
         (SurfacePath::Circle(_), GlobalPath::Circle(_)) => {
             todo!(
                 "Approximating a circle on a curved surface not supported yet."
@@ -90,6 +90,7 @@ fn approx_global_curve(
 
                     let point_global = curve
                         .surface()
+                        .geometry()
                         .point_from_surface_coords(point_surface);
                     (point_curve, point_global)
                 })
@@ -101,15 +102,17 @@ fn approx_global_curve(
                     [curve.path().point_from_path_coords(point_curve).u]
                 }));
 
-            let approx_u = (curve.surface().u(), range_u)
+            let approx_u = (curve.surface().geometry().u, range_u)
                 .approx_with_cache(tolerance, &mut ());
 
             let mut points = Vec::new();
             for (u, _) in approx_u {
                 let t = (u.t - line.origin().u) / line.direction().u;
                 let point_surface = curve.path().point_from_path_coords([t]);
-                let point_global =
-                    curve.surface().point_from_surface_coords(point_surface);
+                let point_global = curve
+                    .surface()
+                    .geometry()
+                    .point_from_surface_coords(point_surface);
                 points.push((u, point_global));
             }
 
@@ -197,11 +200,11 @@ mod tests {
 
     use crate::{
         algorithms::approx::{path::RangeOnPath, Approx, ApproxPoint},
-        builder::CurveBuilder,
+        builder::{CurveBuilder, SurfaceBuilder},
+        geometry::path::GlobalPath,
         insert::Insert,
-        objects::{Objects, Surface},
-        partial::PartialCurve,
-        path::GlobalPath,
+        objects::Objects,
+        partial::{PartialCurve, PartialSurface},
     };
 
     use super::CurveApprox;
@@ -210,9 +213,10 @@ mod tests {
     fn approx_line_on_flat_surface() -> anyhow::Result<()> {
         let objects = Objects::new();
 
-        let surface = objects
-            .surfaces
-            .insert(Surface::new(GlobalPath::x_axis(), [0., 0., 1.]))?;
+        let surface =
+            PartialSurface::from_axes(GlobalPath::x_axis(), [0., 0., 1.])
+                .build(&objects)?
+                .insert(&objects)?;
         let mut curve = PartialCurve {
             surface: Some(surface),
             ..Default::default()
@@ -232,10 +236,12 @@ mod tests {
     {
         let objects = Objects::new();
 
-        let surface = objects.surfaces.insert(Surface::new(
+        let surface = PartialSurface::from_axes(
             GlobalPath::circle_from_radius(1.),
             [0., 0., 1.],
-        ))?;
+        )
+        .build(&objects)?
+        .insert(&objects)?;
         let mut curve = PartialCurve {
             surface: Some(surface),
             ..Default::default()
@@ -255,8 +261,9 @@ mod tests {
         let objects = Objects::new();
 
         let path = GlobalPath::circle_from_radius(1.);
-        let surface =
-            objects.surfaces.insert(Surface::new(path, [0., 0., 1.]))?;
+        let surface = PartialSurface::from_axes(path, [0., 0., 1.])
+            .build(&objects)?
+            .insert(&objects)?;
         let mut curve = PartialCurve {
             surface: Some(surface.clone()),
             ..Default::default()
@@ -276,7 +283,7 @@ mod tests {
                 let point_surface =
                     curve.path().point_from_path_coords(point_local);
                 let point_global =
-                    surface.point_from_surface_coords(point_surface);
+                    surface.geometry().point_from_surface_coords(point_surface);
                 ApproxPoint::new(point_surface, point_global)
             })
             .collect::<Vec<_>>();
@@ -288,9 +295,10 @@ mod tests {
     fn approx_circle_on_flat_surface() -> anyhow::Result<()> {
         let objects = Objects::new();
 
-        let surface = objects
-            .surfaces
-            .insert(Surface::new(GlobalPath::x_axis(), [0., 0., 1.]))?;
+        let surface =
+            PartialSurface::from_axes(GlobalPath::x_axis(), [0., 0., 1.])
+                .build(&objects)?
+                .insert(&objects)?;
         let mut curve = PartialCurve {
             surface: Some(surface),
             ..Default::default()
@@ -306,8 +314,10 @@ mod tests {
             .approx(tolerance)
             .into_iter()
             .map(|(_, point_surface)| {
-                let point_global =
-                    curve.surface().point_from_surface_coords(point_surface);
+                let point_global = curve
+                    .surface()
+                    .geometry()
+                    .point_from_surface_coords(point_surface);
                 ApproxPoint::new(point_surface, point_global)
             })
             .collect::<Vec<_>>();
diff --git a/crates/fj-kernel/src/algorithms/approx/path.rs b/crates/fj-kernel/src/algorithms/approx/path.rs
index 1ce214b79..85335fa76 100644
--- a/crates/fj-kernel/src/algorithms/approx/path.rs
+++ b/crates/fj-kernel/src/algorithms/approx/path.rs
@@ -32,7 +32,7 @@ use std::iter;
 
 use fj_math::{Circle, Point, Scalar, Sign};
 
-use crate::path::{GlobalPath, SurfacePath};
+use crate::geometry::path::{GlobalPath, SurfacePath};
 
 use super::{Approx, Tolerance};
 
diff --git a/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs b/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
index ac53e7fff..0ebeafcaf 100644
--- a/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
@@ -1,8 +1,8 @@
 use fj_math::{Point, Segment};
 
 use crate::{
+    geometry::path::SurfacePath,
     objects::{Curve, HalfEdge},
-    path::SurfacePath,
 };
 
 use super::LineSegmentIntersection;
diff --git a/crates/fj-kernel/src/algorithms/intersect/ray_edge.rs b/crates/fj-kernel/src/algorithms/intersect/ray_edge.rs
index b3a79f44d..962fed93b 100644
--- a/crates/fj-kernel/src/algorithms/intersect/ray_edge.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/ray_edge.rs
@@ -4,8 +4,8 @@ use fj_math::Segment;
 
 use crate::{
     algorithms::intersect::{HorizontalRayToTheRight, Intersect},
+    geometry::path::SurfacePath,
     objects::HalfEdge,
-    path::SurfacePath,
     storage::Handle,
 };
 
diff --git a/crates/fj-kernel/src/algorithms/intersect/ray_face.rs b/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
index f5526972f..175abbaff 100644
--- a/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
@@ -4,8 +4,8 @@ use fj_math::{Plane, Point, Scalar};
 
 use crate::{
     algorithms::intersect::face_point::FacePointIntersection,
+    geometry::path::GlobalPath,
     objects::{Face, HalfEdge, Vertex},
-    path::GlobalPath,
     storage::Handle,
 };
 
@@ -17,14 +17,14 @@ impl Intersect for (&HorizontalRayToTheRight<3>, &Handle<Face>) {
     fn intersect(self) -> Option<Self::Intersection> {
         let (ray, face) = self;
 
-        let plane = match face.surface().u() {
+        let plane = match face.surface().geometry().u {
             GlobalPath::Circle(_) => todo!(
                 "Casting a ray against a swept circle is not supported yet"
             ),
             GlobalPath::Line(line) => Plane::from_parametric(
                 line.origin(),
                 line.direction(),
-                face.surface().v(),
+                face.surface().geometry().v,
             ),
         };
 
diff --git a/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs b/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs
index 43ed585d6..5cdf55261 100644
--- a/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/surface_surface.rs
@@ -2,8 +2,8 @@ use fj_interop::ext::ArrayExt;
 use fj_math::{Line, Plane, Point, Scalar};
 
 use crate::{
+    geometry::path::{GlobalPath, SurfacePath},
     objects::{Curve, GlobalCurve, Objects, Surface},
-    path::{GlobalPath, SurfacePath},
     storage::Handle,
     validate::ValidationError,
 };
@@ -74,12 +74,12 @@ impl SurfaceSurfaceIntersection {
 
 fn plane_from_surface(surface: &Surface) -> Plane {
     let (line, path) = {
-        let line = match surface.u() {
+        let line = match surface.geometry().u {
             GlobalPath::Line(line) => line,
             _ => todo!("Only plane-plane intersection is currently supported."),
         };
 
-        (line, surface.v())
+        (line, surface.geometry().v)
     };
 
     Plane::from_parametric(line.origin(), line.direction(), path)
diff --git a/crates/fj-kernel/src/algorithms/sweep/curve.rs b/crates/fj-kernel/src/algorithms/sweep/curve.rs
index 76ec79719..8ffc6e469 100644
--- a/crates/fj-kernel/src/algorithms/sweep/curve.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/curve.rs
@@ -1,8 +1,11 @@
 use fj_math::{Circle, Line, Vector};
 
 use crate::{
+    builder::SurfaceBuilder,
+    geometry::path::{GlobalPath, SurfacePath},
+    insert::Insert,
     objects::{Curve, Objects, Surface},
-    path::{GlobalPath, SurfacePath},
+    partial::PartialSurface,
     storage::Handle,
     validate::ValidationError,
 };
@@ -18,7 +21,7 @@ impl Sweep for Handle<Curve> {
         _: &mut SweepCache,
         objects: &Objects,
     ) -> Result<Self::Swept, ValidationError> {
-        match self.surface().u() {
+        match self.surface().geometry().u {
             GlobalPath::Circle(_) => {
                 // Sweeping a `Curve` creates a `Surface`. The u-axis of that
                 // `Surface` is a `GlobalPath`, which we are computing below.
@@ -44,20 +47,32 @@ impl Sweep for Handle<Curve> {
 
         let u = match self.path() {
             SurfacePath::Circle(circle) => {
-                let center =
-                    self.surface().point_from_surface_coords(circle.center());
-                let a = self.surface().vector_from_surface_coords(circle.a());
-                let b = self.surface().vector_from_surface_coords(circle.b());
+                let center = self
+                    .surface()
+                    .geometry()
+                    .point_from_surface_coords(circle.center());
+                let a = self
+                    .surface()
+                    .geometry()
+                    .vector_from_surface_coords(circle.a());
+                let b = self
+                    .surface()
+                    .geometry()
+                    .vector_from_surface_coords(circle.b());
 
                 let circle = Circle::new(center, a, b);
 
                 GlobalPath::Circle(circle)
             }
             SurfacePath::Line(line) => {
-                let origin =
-                    self.surface().point_from_surface_coords(line.origin());
-                let direction =
-                    self.surface().vector_from_surface_coords(line.direction());
+                let origin = self
+                    .surface()
+                    .geometry()
+                    .point_from_surface_coords(line.origin());
+                let direction = self
+                    .surface()
+                    .geometry()
+                    .vector_from_surface_coords(line.direction());
 
                 let line = Line::from_origin_and_direction(origin, direction);
 
@@ -65,6 +80,9 @@ impl Sweep for Handle<Curve> {
             }
         };
 
-        Ok(objects.surfaces.insert(Surface::new(u, path))?)
+        let surface = PartialSurface::from_axes(u, path)
+            .build(objects)?
+            .insert(objects)?;
+        Ok(surface)
     }
 }
diff --git a/crates/fj-kernel/src/algorithms/sweep/edge.rs b/crates/fj-kernel/src/algorithms/sweep/edge.rs
index 901533c47..3d1489d24 100644
--- a/crates/fj-kernel/src/algorithms/sweep/edge.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/edge.rs
@@ -4,13 +4,13 @@ use iter_fixed::IntoIteratorFixed;
 
 use crate::{
     algorithms::{reverse::Reverse, transform::TransformObject},
+    geometry::path::SurfacePath,
     insert::Insert,
     objects::{
         Curve, Cycle, Face, GlobalEdge, HalfEdge, Objects, SurfaceVertex,
         Vertex,
     },
     partial::HasPartial,
-    path::SurfacePath,
     storage::Handle,
     validate::ValidationError,
 };
diff --git a/crates/fj-kernel/src/algorithms/sweep/face.rs b/crates/fj-kernel/src/algorithms/sweep/face.rs
index cc99370b6..930e19ccf 100644
--- a/crates/fj-kernel/src/algorithms/sweep/face.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/face.rs
@@ -2,8 +2,8 @@ use fj_math::{Scalar, Vector};
 
 use crate::{
     algorithms::{reverse::Reverse, transform::TransformObject},
+    geometry::path::GlobalPath,
     objects::{Face, Objects, Shell},
-    path::GlobalPath,
     storage::Handle,
     validate::ValidationError,
 };
@@ -24,14 +24,14 @@ impl Sweep for Handle<Face> {
         let mut faces = Vec::new();
 
         let is_negative_sweep = {
-            let u = match self.surface().u() {
+            let u = match self.surface().geometry().u {
                 GlobalPath::Circle(_) => todo!(
                     "Sweeping from faces defined in round surfaces is not \
                     supported"
                 ),
                 GlobalPath::Line(line) => line.direction(),
             };
-            let v = self.surface().v();
+            let v = self.surface().geometry().v;
 
             let normal = u.cross(&v);
 
diff --git a/crates/fj-kernel/src/algorithms/sweep/vertex.rs b/crates/fj-kernel/src/algorithms/sweep/vertex.rs
index e07754a62..f6e622d1e 100644
--- a/crates/fj-kernel/src/algorithms/sweep/vertex.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/vertex.rs
@@ -3,11 +3,11 @@ use fj_math::{Line, Point, Scalar, Vector};
 use try_insert_ext::EntryInsertExt;
 
 use crate::{
+    geometry::path::SurfacePath,
     objects::{
         Curve, GlobalCurve, GlobalEdge, GlobalVertex, HalfEdge, Objects,
         Surface, SurfaceVertex, Vertex,
     },
-    path::SurfacePath,
     storage::Handle,
     validate::ValidationError,
 };
@@ -56,7 +56,7 @@ impl Sweep for (Handle<Vertex>, Handle<Surface>) {
         // not, we have no way of knowing the surface coordinates of the input
         // `Vertex` on the `Surface`, and we're going to need to do that further
         // down. There's no way to check for that, unfortunately.
-        assert_eq!(path, surface.v());
+        assert_eq!(path, surface.geometry().v);
 
         // With that out of the way, let's start by creating the `GlobalEdge`,
         // as that is the most straight-forward part of this operations, and
diff --git a/crates/fj-kernel/src/algorithms/transform/surface.rs b/crates/fj-kernel/src/algorithms/transform/surface.rs
index f05007520..10b6d1d6f 100644
--- a/crates/fj-kernel/src/algorithms/transform/surface.rs
+++ b/crates/fj-kernel/src/algorithms/transform/surface.rs
@@ -1,22 +1,25 @@
 use fj_math::Transform;
 
 use crate::{
-    objects::{Objects, Surface},
-    storage::Handle,
-    validate::ValidationError,
+    geometry::surface::SurfaceGeometry, objects::Objects,
+    partial::PartialSurface, validate::ValidationError,
 };
 
 use super::TransformObject;
 
-impl TransformObject for Handle<Surface> {
+impl TransformObject for PartialSurface {
     fn transform(
         self,
         transform: &Transform,
-        objects: &Objects,
+        _: &Objects,
     ) -> Result<Self, ValidationError> {
-        Ok(objects.surfaces.insert(Surface::new(
-            self.u().transform(transform),
-            transform.transform_vector(&self.v()),
-        ))?)
+        let geometry = self.geometry.map(|geometry| {
+            let u = geometry.u.transform(transform);
+            let v = transform.transform_vector(&geometry.v);
+
+            SurfaceGeometry { u, v }
+        });
+
+        Ok(Self { geometry })
     }
 }
diff --git a/crates/fj-kernel/src/algorithms/triangulate/mod.rs b/crates/fj-kernel/src/algorithms/triangulate/mod.rs
index a1a0ebe50..a6ade6f9b 100644
--- a/crates/fj-kernel/src/algorithms/triangulate/mod.rs
+++ b/crates/fj-kernel/src/algorithms/triangulate/mod.rs
@@ -148,12 +148,12 @@ mod tests {
 
         let triangles = triangulate(face)?;
 
-        let a = surface.point_from_surface_coords(a);
-        let b = surface.point_from_surface_coords(b);
-        let e = surface.point_from_surface_coords(e);
-        let f = surface.point_from_surface_coords(f);
-        let g = surface.point_from_surface_coords(g);
-        let h = surface.point_from_surface_coords(h);
+        let a = surface.geometry().point_from_surface_coords(a);
+        let b = surface.geometry().point_from_surface_coords(b);
+        let e = surface.geometry().point_from_surface_coords(e);
+        let f = surface.geometry().point_from_surface_coords(f);
+        let g = surface.geometry().point_from_surface_coords(g);
+        let h = surface.geometry().point_from_surface_coords(h);
 
         // Let's test that some correct triangles are present. We don't need to
         // test them all.
@@ -209,11 +209,11 @@ mod tests {
 
         let triangles = triangulate(face)?;
 
-        let a3 = surface.point_from_surface_coords(a);
-        let b3 = surface.point_from_surface_coords(b);
-        let c3 = surface.point_from_surface_coords(c);
-        let d3 = surface.point_from_surface_coords(d);
-        let e3 = surface.point_from_surface_coords(e);
+        let a3 = surface.geometry().point_from_surface_coords(a);
+        let b3 = surface.geometry().point_from_surface_coords(b);
+        let c3 = surface.geometry().point_from_surface_coords(c);
+        let d3 = surface.geometry().point_from_surface_coords(d);
+        let e3 = surface.geometry().point_from_surface_coords(e);
 
         assert!(triangles.contains_triangle([a3, b3, d3]));
         assert!(triangles.contains_triangle([b3, c3, d3]));
diff --git a/crates/fj-kernel/src/builder/curve.rs b/crates/fj-kernel/src/builder/curve.rs
index 1380bfff8..85d71a63c 100644
--- a/crates/fj-kernel/src/builder/curve.rs
+++ b/crates/fj-kernel/src/builder/curve.rs
@@ -1,6 +1,6 @@
 use fj_math::{Point, Scalar, Vector};
 
-use crate::{partial::PartialCurve, path::SurfacePath};
+use crate::{geometry::path::SurfacePath, partial::PartialCurve};
 
 /// Builder API for [`PartialCurve`]
 pub trait CurveBuilder {
diff --git a/crates/fj-kernel/src/builder/edge.rs b/crates/fj-kernel/src/builder/edge.rs
index 74794c131..3d276c505 100644
--- a/crates/fj-kernel/src/builder/edge.rs
+++ b/crates/fj-kernel/src/builder/edge.rs
@@ -5,8 +5,9 @@ use crate::{
     insert::Insert,
     objects::{Curve, Objects, Surface, Vertex, VerticesInNormalizedOrder},
     partial::{
-        MaybePartial, PartialCurve, PartialGlobalEdge, PartialGlobalVertex,
-        PartialHalfEdge, PartialSurfaceVertex, PartialVertex,
+        MaybePartial, MergeWith, PartialCurve, PartialGlobalEdge,
+        PartialGlobalVertex, PartialHalfEdge, PartialSurfaceVertex,
+        PartialVertex,
     },
     storage::Handle,
     validate::ValidationError,
@@ -135,8 +136,8 @@ impl HalfEdgeBuilder for PartialHalfEdge {
         let surface = self
             .curve()
             .surface()
-            .or_else(|| from_surface.surface())
-            .or_else(|| to_surface.surface())
+            .merge_with(from_surface.surface())
+            .merge_with(to_surface.surface())
             .expect("Can't infer line segment without a surface");
         let points = [&from_surface, &to_surface].map(|vertex| {
             vertex
diff --git a/crates/fj-kernel/src/builder/mod.rs b/crates/fj-kernel/src/builder/mod.rs
index 1d96e73d7..7a7b4e70a 100644
--- a/crates/fj-kernel/src/builder/mod.rs
+++ b/crates/fj-kernel/src/builder/mod.rs
@@ -12,6 +12,7 @@ mod curve;
 mod cycle;
 mod edge;
 mod face;
+mod surface;
 mod vertex;
 
 pub use self::{
@@ -22,5 +23,6 @@ pub use self::{
     shell::ShellBuilder,
     sketch::SketchBuilder,
     solid::SolidBuilder,
+    surface::SurfaceBuilder,
     vertex::{GlobalVertexBuilder, SurfaceVertexBuilder, VertexBuilder},
 };
diff --git a/crates/fj-kernel/src/builder/shell.rs b/crates/fj-kernel/src/builder/shell.rs
index 343b45857..838295820 100644
--- a/crates/fj-kernel/src/builder/shell.rs
+++ b/crates/fj-kernel/src/builder/shell.rs
@@ -6,10 +6,13 @@ use iter_fixed::IntoIteratorFixed;
 
 use crate::{
     algorithms::transform::TransformObject,
-    builder::{FaceBuilder, HalfEdgeBuilder},
+    builder::{FaceBuilder, HalfEdgeBuilder, SurfaceBuilder},
     insert::Insert,
-    objects::{Cycle, Face, FaceSet, HalfEdge, Objects, Shell, Surface},
-    partial::{HasPartial, PartialCurve, PartialSurfaceVertex, PartialVertex},
+    objects::{Cycle, Face, FaceSet, HalfEdge, Objects, Shell},
+    partial::{
+        HasPartial, PartialCurve, PartialSurface, PartialSurfaceVertex,
+        PartialVertex,
+    },
     storage::Handle,
 };
 
@@ -78,9 +81,10 @@ impl<'a> ShellBuilder<'a> {
                         .map(|vertex| vertex.global_form().position());
                     let c = a + [Z, Z, edge_length];
 
-                    self.objects
-                        .surfaces
-                        .insert(Surface::plane_from_points([a, b, c]))
+                    PartialSurface::plane_from_points([a, b, c])
+                        .build(self.objects)
+                        .unwrap()
+                        .insert(self.objects)
                         .unwrap()
                 })
                 .collect::<Vec<_>>();
diff --git a/crates/fj-kernel/src/builder/surface.rs b/crates/fj-kernel/src/builder/surface.rs
new file mode 100644
index 000000000..cc1d57728
--- /dev/null
+++ b/crates/fj-kernel/src/builder/surface.rs
@@ -0,0 +1,36 @@
+use fj_math::{Line, Point, Vector};
+
+use crate::{
+    geometry::{path::GlobalPath, surface::SurfaceGeometry},
+    partial::PartialSurface,
+};
+
+/// Builder API for [`PartialSurface`]
+pub trait SurfaceBuilder {
+    /// Build a surface from its two axes
+    fn from_axes(u: GlobalPath, v: impl Into<Vector<3>>) -> Self;
+
+    /// Construct a plane from 3 points
+    fn plane_from_points(points: [impl Into<Point<3>>; 3]) -> Self;
+}
+
+impl SurfaceBuilder for PartialSurface {
+    fn from_axes(u: GlobalPath, v: impl Into<Vector<3>>) -> Self {
+        let v = v.into();
+
+        Self {
+            geometry: Some(SurfaceGeometry { u, v }),
+        }
+    }
+
+    fn plane_from_points(points: [impl Into<Point<3>>; 3]) -> Self {
+        let [a, b, c] = points.map(Into::into);
+
+        let u = GlobalPath::Line(Line::from_points([a, b]));
+        let v = c - a;
+
+        Self {
+            geometry: Some(SurfaceGeometry { u, v }),
+        }
+    }
+}
diff --git a/crates/fj-kernel/src/builder/vertex.rs b/crates/fj-kernel/src/builder/vertex.rs
index 879d5533f..9c3266b00 100644
--- a/crates/fj-kernel/src/builder/vertex.rs
+++ b/crates/fj-kernel/src/builder/vertex.rs
@@ -1,7 +1,8 @@
 use fj_math::Point;
 
 use crate::{
-    objects::{Curve, GlobalVertex, Surface},
+    geometry::surface::SurfaceGeometry,
+    objects::{Curve, GlobalVertex},
     partial::{
         HasPartial, MaybePartial, PartialGlobalVertex, PartialSurfaceVertex,
         PartialVertex,
@@ -44,7 +45,7 @@ pub trait GlobalVertexBuilder {
 
     /// Update partial global vertex from the given surface and position on it
     fn from_surface_and_position(
-        surface: &Surface,
+        surface: &SurfaceGeometry,
         position: impl Into<Point<2>>,
     ) -> Self;
 }
@@ -59,9 +60,12 @@ impl GlobalVertexBuilder for PartialGlobalVertex {
         let path = curve.path.expect(
             "Need path to create `GlobalVertex` from curve and position",
         );
-        let surface = curve.surface.expect(
-            "Need surface to create `GlobalVertex` from curve and position",
-        );
+        let surface = curve
+            .surface
+            .expect(
+                "Need surface to create `GlobalVertex` from curve and position",
+            )
+            .geometry();
 
         let position_surface = path.point_from_path_coords(position);
 
@@ -69,7 +73,7 @@ impl GlobalVertexBuilder for PartialGlobalVertex {
     }
 
     fn from_surface_and_position(
-        surface: &Surface,
+        surface: &SurfaceGeometry,
         position: impl Into<Point<2>>,
     ) -> Self {
         PartialGlobalVertex {
diff --git a/crates/fj-kernel/src/geometry/mod.rs b/crates/fj-kernel/src/geometry/mod.rs
new file mode 100644
index 000000000..654a3c108
--- /dev/null
+++ b/crates/fj-kernel/src/geometry/mod.rs
@@ -0,0 +1,4 @@
+//! Types that are tied to objects, but aren't objects themselves
+
+pub mod path;
+pub mod surface;
diff --git a/crates/fj-kernel/src/path.rs b/crates/fj-kernel/src/geometry/path.rs
similarity index 100%
rename from crates/fj-kernel/src/path.rs
rename to crates/fj-kernel/src/geometry/path.rs
diff --git a/crates/fj-kernel/src/geometry/surface.rs b/crates/fj-kernel/src/geometry/surface.rs
new file mode 100644
index 000000000..a17abaaa5
--- /dev/null
+++ b/crates/fj-kernel/src/geometry/surface.rs
@@ -0,0 +1,81 @@
+//! The geometry that defines a surface
+
+use fj_math::{Line, Point, Vector};
+
+use super::path::GlobalPath;
+
+/// The geometry that defines a surface
+#[derive(Clone, Copy, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
+pub struct SurfaceGeometry {
+    /// The u-axis of the surface
+    pub u: GlobalPath,
+
+    /// The v-axis of the surface
+    pub v: Vector<3>,
+}
+
+impl SurfaceGeometry {
+    /// Convert a point in surface coordinates to model coordinates
+    pub fn point_from_surface_coords(
+        &self,
+        point: impl Into<Point<2>>,
+    ) -> Point<3> {
+        let point = point.into();
+        self.u.point_from_path_coords([point.u])
+            + self.path_to_line().vector_from_line_coords([point.v])
+    }
+
+    /// Convert a vector in surface coordinates to model coordinates
+    pub fn vector_from_surface_coords(
+        &self,
+        vector: impl Into<Vector<2>>,
+    ) -> Vector<3> {
+        let vector = vector.into();
+        self.u.vector_from_path_coords([vector.u])
+            + self.path_to_line().vector_from_line_coords([vector.v])
+    }
+
+    fn path_to_line(&self) -> Line<3> {
+        Line::from_origin_and_direction(self.u.origin(), self.v)
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use fj_math::{Line, Point, Vector};
+    use pretty_assertions::assert_eq;
+
+    use crate::geometry::{path::GlobalPath, surface::SurfaceGeometry};
+
+    #[test]
+    fn point_from_surface_coords() {
+        let surface = SurfaceGeometry {
+            u: GlobalPath::Line(Line::from_origin_and_direction(
+                Point::from([1., 1., 1.]),
+                Vector::from([0., 2., 0.]),
+            )),
+            v: Vector::from([0., 0., 2.]),
+        };
+
+        assert_eq!(
+            surface.point_from_surface_coords([2., 4.]),
+            Point::from([1., 5., 9.]),
+        );
+    }
+
+    #[test]
+    fn vector_from_surface_coords() {
+        let surface = SurfaceGeometry {
+            u: GlobalPath::Line(Line::from_origin_and_direction(
+                Point::from([1., 0., 0.]),
+                Vector::from([0., 2., 0.]),
+            )),
+            v: Vector::from([0., 0., 2.]),
+        };
+
+        assert_eq!(
+            surface.vector_from_surface_coords([2., 4.]),
+            Vector::from([0., 4., 8.]),
+        );
+    }
+}
diff --git a/crates/fj-kernel/src/lib.rs b/crates/fj-kernel/src/lib.rs
index 607c7b11c..87091d3a1 100644
--- a/crates/fj-kernel/src/lib.rs
+++ b/crates/fj-kernel/src/lib.rs
@@ -89,10 +89,10 @@
 
 pub mod algorithms;
 pub mod builder;
+pub mod geometry;
 pub mod insert;
 pub mod iter;
 pub mod objects;
 pub mod partial;
-pub mod path;
 pub mod storage;
 pub mod validate;
diff --git a/crates/fj-kernel/src/objects/curve.rs b/crates/fj-kernel/src/objects/curve.rs
index 54837aa5f..2d1b439ef 100644
--- a/crates/fj-kernel/src/objects/curve.rs
+++ b/crates/fj-kernel/src/objects/curve.rs
@@ -1,5 +1,5 @@
 use crate::{
-    path::SurfacePath,
+    geometry::path::SurfacePath,
     storage::{Handle, HandleWrapper},
 };
 
diff --git a/crates/fj-kernel/src/objects/cycle.rs b/crates/fj-kernel/src/objects/cycle.rs
index 412f07b50..91633ef83 100644
--- a/crates/fj-kernel/src/objects/cycle.rs
+++ b/crates/fj-kernel/src/objects/cycle.rs
@@ -3,7 +3,7 @@ use std::slice;
 use fj_interop::ext::SliceExt;
 use fj_math::{Scalar, Winding};
 
-use crate::{path::SurfacePath, storage::Handle};
+use crate::{geometry::path::SurfacePath, storage::Handle};
 
 use super::{HalfEdge, Surface};
 
diff --git a/crates/fj-kernel/src/objects/mod.rs b/crates/fj-kernel/src/objects/mod.rs
index d0406c0c9..221c245ef 100644
--- a/crates/fj-kernel/src/objects/mod.rs
+++ b/crates/fj-kernel/src/objects/mod.rs
@@ -100,7 +100,7 @@ use std::convert::Infallible;
 use fj_math::Vector;
 
 use crate::{
-    path::GlobalPath,
+    geometry::{path::GlobalPath, surface::SurfaceGeometry},
     storage::{Handle, Store},
     validate::{
         CycleValidationError, FaceValidationError, HalfEdgeValidationError,
@@ -380,12 +380,18 @@ impl Default for Surfaces {
     fn default() -> Self {
         let store = Store::new();
 
-        let xy_plane =
-            store.insert(Surface::new(GlobalPath::x_axis(), Vector::unit_y()));
-        let xz_plane =
-            store.insert(Surface::new(GlobalPath::x_axis(), Vector::unit_z()));
-        let yz_plane =
-            store.insert(Surface::new(GlobalPath::y_axis(), Vector::unit_z()));
+        let xy_plane = store.insert(Surface::new(SurfaceGeometry {
+            u: GlobalPath::x_axis(),
+            v: Vector::unit_y(),
+        }));
+        let xz_plane = store.insert(Surface::new(SurfaceGeometry {
+            u: GlobalPath::x_axis(),
+            v: Vector::unit_z(),
+        }));
+        let yz_plane = store.insert(Surface::new(SurfaceGeometry {
+            u: GlobalPath::y_axis(),
+            v: Vector::unit_z(),
+        }));
 
         Self {
             store,
diff --git a/crates/fj-kernel/src/objects/surface.rs b/crates/fj-kernel/src/objects/surface.rs
index 6da597626..0905a0bef 100644
--- a/crates/fj-kernel/src/objects/surface.rs
+++ b/crates/fj-kernel/src/objects/surface.rs
@@ -1,104 +1,19 @@
-use fj_math::{Line, Point, Vector};
-
-use crate::path::GlobalPath;
+use crate::geometry::surface::SurfaceGeometry;
 
 /// A two-dimensional shape
 #[derive(Clone, Copy, Debug, Eq, PartialEq, Hash, Ord, PartialOrd)]
 pub struct Surface {
-    u: GlobalPath,
-    v: Vector<3>,
+    geometry: SurfaceGeometry,
 }
 
 impl Surface {
     /// Construct a `Surface` from two paths that define its coordinate system
-    pub fn new(u: GlobalPath, v: impl Into<Vector<3>>) -> Self {
-        let v = v.into();
-        Self { u, v }
+    pub fn new(geometry: SurfaceGeometry) -> Self {
+        Self { geometry }
     }
 
-    /// Construct a plane from 3 points
-    pub fn plane_from_points(points: [impl Into<Point<3>>; 3]) -> Self {
-        let [a, b, c] = points.map(Into::into);
-
-        let u = GlobalPath::Line(Line::from_points([a, b]));
-        let v = c - a;
-
-        Self { u, v }
-    }
-
-    /// Access the path that defines the u-coordinate of this surface
-    pub fn u(&self) -> GlobalPath {
-        self.u
-    }
-
-    /// Access the path that defines the v-coordinate of this surface
-    pub fn v(&self) -> Vector<3> {
-        self.v
-    }
-
-    /// Convert a point in surface coordinates to model coordinates
-    pub fn point_from_surface_coords(
-        &self,
-        point: impl Into<Point<2>>,
-    ) -> Point<3> {
-        let point = point.into();
-        self.u.point_from_path_coords([point.u])
-            + self.path_to_line().vector_from_line_coords([point.v])
-    }
-
-    /// Convert a vector in surface coordinates to model coordinates
-    pub fn vector_from_surface_coords(
-        &self,
-        vector: impl Into<Vector<2>>,
-    ) -> Vector<3> {
-        let vector = vector.into();
-        self.u.vector_from_path_coords([vector.u])
-            + self.path_to_line().vector_from_line_coords([vector.v])
-    }
-
-    fn path_to_line(&self) -> Line<3> {
-        Line::from_origin_and_direction(self.u.origin(), self.v)
-    }
-}
-
-#[cfg(test)]
-mod tests {
-    use fj_math::{Line, Point, Vector};
-    use pretty_assertions::assert_eq;
-
-    use crate::path::GlobalPath;
-
-    use super::Surface;
-
-    #[test]
-    fn point_from_surface_coords() {
-        let swept = Surface {
-            u: GlobalPath::Line(Line::from_origin_and_direction(
-                Point::from([1., 1., 1.]),
-                Vector::from([0., 2., 0.]),
-            )),
-            v: Vector::from([0., 0., 2.]),
-        };
-
-        assert_eq!(
-            swept.point_from_surface_coords([2., 4.]),
-            Point::from([1., 5., 9.]),
-        );
-    }
-
-    #[test]
-    fn vector_from_surface_coords() {
-        let swept = Surface {
-            u: GlobalPath::Line(Line::from_origin_and_direction(
-                Point::from([1., 0., 0.]),
-                Vector::from([0., 2., 0.]),
-            )),
-            v: Vector::from([0., 0., 2.]),
-        };
-
-        assert_eq!(
-            swept.vector_from_surface_coords([2., 4.]),
-            Vector::from([0., 4., 8.]),
-        );
+    /// Access the surface's geometry
+    pub fn geometry(&self) -> SurfaceGeometry {
+        self.geometry
     }
 }
diff --git a/crates/fj-kernel/src/partial/maybe_partial.rs b/crates/fj-kernel/src/partial/maybe_partial.rs
index 0fef15070..cad49c181 100644
--- a/crates/fj-kernel/src/partial/maybe_partial.rs
+++ b/crates/fj-kernel/src/partial/maybe_partial.rs
@@ -1,12 +1,12 @@
 use fj_math::Point;
 
 use crate::{
+    geometry::{path::SurfacePath, surface::SurfaceGeometry},
     insert::Insert,
     objects::{
         Curve, GlobalCurve, GlobalEdge, GlobalVertex, HalfEdge, Objects,
         Surface, SurfaceVertex, Vertex,
     },
-    path::SurfacePath,
     storage::Handle,
     validate::{Validate, ValidationError},
 };
@@ -212,6 +212,16 @@ impl MaybePartial<HalfEdge> {
     }
 }
 
+impl MaybePartial<Surface> {
+    /// Access the geometry
+    pub fn geometry(&self) -> Option<SurfaceGeometry> {
+        match self {
+            Self::Full(full) => Some(full.geometry()),
+            Self::Partial(partial) => partial.geometry,
+        }
+    }
+}
+
 impl MaybePartial<SurfaceVertex> {
     /// Access the position
     pub fn position(&self) -> Option<Point<2>> {
diff --git a/crates/fj-kernel/src/partial/mod.rs b/crates/fj-kernel/src/partial/mod.rs
index 2f81c0d25..19289522b 100644
--- a/crates/fj-kernel/src/partial/mod.rs
+++ b/crates/fj-kernel/src/partial/mod.rs
@@ -47,6 +47,7 @@ pub use self::{
         cycle::PartialCycle,
         edge::{PartialGlobalEdge, PartialHalfEdge},
         face::PartialFace,
+        surface::PartialSurface,
         vertex::{PartialGlobalVertex, PartialSurfaceVertex, PartialVertex},
     },
     traits::{HasPartial, Partial},
diff --git a/crates/fj-kernel/src/partial/objects/curve.rs b/crates/fj-kernel/src/partial/objects/curve.rs
index f5670e27a..a82a7af2f 100644
--- a/crates/fj-kernel/src/partial/objects/curve.rs
+++ b/crates/fj-kernel/src/partial/objects/curve.rs
@@ -1,7 +1,7 @@
 use crate::{
+    geometry::path::SurfacePath,
     objects::{Curve, GlobalCurve, Objects, Surface},
     partial::{MaybePartial, MergeWith, Mergeable},
-    path::SurfacePath,
     storage::Handle,
     validate::ValidationError,
 };
diff --git a/crates/fj-kernel/src/partial/objects/mod.rs b/crates/fj-kernel/src/partial/objects/mod.rs
index 917f53a56..a4acf77a6 100644
--- a/crates/fj-kernel/src/partial/objects/mod.rs
+++ b/crates/fj-kernel/src/partial/objects/mod.rs
@@ -2,17 +2,18 @@ pub mod curve;
 pub mod cycle;
 pub mod edge;
 pub mod face;
+pub mod surface;
 pub mod vertex;
 
 use crate::objects::{
     Curve, Cycle, Face, GlobalCurve, GlobalEdge, GlobalVertex, HalfEdge,
-    Objects, SurfaceVertex, Vertex,
+    Objects, Surface, SurfaceVertex, Vertex,
 };
 
 use super::{
     HasPartial, MaybePartial, Partial, PartialCurve, PartialCycle, PartialFace,
     PartialGlobalCurve, PartialGlobalEdge, PartialGlobalVertex,
-    PartialHalfEdge, PartialSurfaceVertex, PartialVertex,
+    PartialHalfEdge, PartialSurface, PartialSurfaceVertex, PartialVertex,
 };
 
 macro_rules! impl_traits {
@@ -52,6 +53,7 @@ impl_traits!(
     GlobalEdge, PartialGlobalEdge;
     GlobalVertex, PartialGlobalVertex;
     HalfEdge, PartialHalfEdge;
+    Surface, PartialSurface;
     SurfaceVertex, PartialSurfaceVertex;
     Vertex, PartialVertex;
 );
diff --git a/crates/fj-kernel/src/partial/objects/surface.rs b/crates/fj-kernel/src/partial/objects/surface.rs
new file mode 100644
index 000000000..29c4a0ba8
--- /dev/null
+++ b/crates/fj-kernel/src/partial/objects/surface.rs
@@ -0,0 +1,44 @@
+use crate::{
+    geometry::surface::SurfaceGeometry,
+    objects::{Objects, Surface},
+    partial::MergeWith,
+    validate::ValidationError,
+};
+
+/// A partial [`Surface`]
+///
+/// See [`crate::partial`] for more information
+#[derive(Clone, Debug, Default)]
+pub struct PartialSurface {
+    /// The geometry that defines the [`Surface`]
+    pub geometry: Option<SurfaceGeometry>,
+}
+
+impl PartialSurface {
+    /// Build a full [`Surface`] from the partial surface
+    pub fn build(self, _: &Objects) -> Result<Surface, ValidationError> {
+        let geometry = self
+            .geometry
+            .expect("Can't build `Surface` without geometry");
+
+        Ok(Surface::new(geometry))
+    }
+}
+
+impl MergeWith for PartialSurface {
+    fn merge_with(self, other: impl Into<Self>) -> Self {
+        let other = other.into();
+
+        Self {
+            geometry: self.geometry.merge_with(other.geometry),
+        }
+    }
+}
+
+impl From<&Surface> for PartialSurface {
+    fn from(surface: &Surface) -> Self {
+        Self {
+            geometry: Some(surface.geometry()),
+        }
+    }
+}
diff --git a/crates/fj-kernel/src/partial/objects/vertex.rs b/crates/fj-kernel/src/partial/objects/vertex.rs
index 111ddf7fc..75b6c7539 100644
--- a/crates/fj-kernel/src/partial/objects/vertex.rs
+++ b/crates/fj-kernel/src/partial/objects/vertex.rs
@@ -80,7 +80,7 @@ impl From<&Vertex> for PartialVertex {
 /// A partial [`SurfaceVertex`]
 ///
 /// See [`crate::partial`] for more information.
-#[derive(Clone, Debug, Default, Eq, PartialEq, Hash, Ord, PartialOrd)]
+#[derive(Clone, Debug, Default)]
 pub struct PartialSurfaceVertex {
     /// The position of the [`SurfaceVertex`]
     pub position: Option<Point<2>>,
@@ -108,7 +108,8 @@ impl PartialSurfaceVertex {
         let global_form = self
             .global_form
             .merge_with(PartialGlobalVertex::from_surface_and_position(
-                &surface, position,
+                &surface.geometry(),
+                position,
             ))
             .into_full(objects)?;
 
diff --git a/crates/fj-kernel/src/validate/vertex.rs b/crates/fj-kernel/src/validate/vertex.rs
index 186fab353..0ed7318e6 100644
--- a/crates/fj-kernel/src/validate/vertex.rs
+++ b/crates/fj-kernel/src/validate/vertex.rs
@@ -158,6 +158,7 @@ impl SurfaceVertexValidationError {
     ) -> Result<(), Self> {
         let surface_position_as_global = surface_vertex
             .surface()
+            .geometry()
             .point_from_surface_coords(surface_vertex.position());
         let global_position = surface_vertex.global_form().position();