diff --git a/crates/fj-kernel/src/algorithms/mod.rs b/crates/fj-kernel/src/algorithms/mod.rs
index 559f22709..01928fca2 100644
--- a/crates/fj-kernel/src/algorithms/mod.rs
+++ b/crates/fj-kernel/src/algorithms/mod.rs
@@ -13,6 +13,6 @@ pub mod intersection;
 pub use self::{
     approx::{CycleApprox, FaceApprox, InvalidTolerance, Tolerance},
     sweep::sweep,
-    transform::transform_shape,
+    transform::transform,
     triangulation::triangulate,
 };
diff --git a/crates/fj-kernel/src/algorithms/sweep.rs b/crates/fj-kernel/src/algorithms/sweep.rs
index 93e343f74..0ee491af3 100644
--- a/crates/fj-kernel/src/algorithms/sweep.rs
+++ b/crates/fj-kernel/src/algorithms/sweep.rs
@@ -8,7 +8,7 @@ use crate::{
     shape::{LocalForm, Shape},
 };
 
-use super::{CycleApprox, Tolerance};
+use super::{transform::transform_cycles, CycleApprox, Tolerance};
 
 /// Create a solid by sweeping a sketch
 pub fn sweep(
@@ -101,9 +101,8 @@ fn create_top_face(
 
     surface = surface.transform(&translation);
 
-    let mut tmp = Shape::new();
-    exteriors = transform_cycle(&exteriors, &translation, &mut tmp);
-    interiors = transform_cycle(&interiors, &translation, &mut tmp);
+    exteriors = transform_cycles(&exteriors, &translation);
+    interiors = transform_cycles(&interiors, &translation);
 
     if is_sweep_along_negative_direction {
         surface = surface.reverse();
@@ -112,6 +111,7 @@ fn create_top_face(
         interiors = reverse_local_coordinates_in_cycle(&interiors);
     };
 
+    let mut tmp = Shape::new();
     let surface = tmp.insert(surface);
 
     let face = Face::new(
@@ -149,84 +149,6 @@ fn reverse_local_coordinates_in_cycle(cycles: &CyclesInFace) -> CyclesInFace {
     CyclesInFace::new(cycles)
 }
 
-fn transform_cycle(
-    cycles: &CyclesInFace,
-    transform: &Transform,
-    target: &mut Shape,
-) -> CyclesInFace {
-    let cycles = cycles.as_local_form().map(|cycle| {
-        let edges_local = cycle
-            .local()
-            .edges
-            .iter()
-            .map(|edge| {
-                let curve_local = *edge.local().curve.local();
-                let curve_canonical = target
-                    .merge(edge.canonical().get().curve().transform(transform));
-
-                let vertices = edge.canonical().get().vertices.map(|vertex| {
-                    let point = vertex.canonical().get().point;
-                    let point = transform.transform_point(&point);
-
-                    let local = *vertex.local();
-                    let canonical = target.merge(Vertex { point });
-
-                    LocalForm::new(local, canonical)
-                });
-
-                let edge_local = Edge {
-                    curve: LocalForm::new(curve_local, curve_canonical.clone()),
-                    vertices: vertices.clone(),
-                };
-                let edge_canonical = target.merge(Edge {
-                    curve: LocalForm::canonical_only(curve_canonical),
-                    vertices,
-                });
-
-                LocalForm::new(edge_local, edge_canonical)
-            })
-            .collect();
-        let edges_canonical = cycle
-            .canonical()
-            .get()
-            .edges
-            .iter()
-            .map(|edge| {
-                let edge = edge.canonical().get();
-
-                let curve = {
-                    let curve = edge.curve().transform(transform);
-
-                    let curve = target.merge(curve);
-                    LocalForm::canonical_only(curve)
-                };
-                let vertices = edge.vertices.map(|vertex| {
-                    let point = vertex.canonical().get().point;
-                    let point = transform.transform_point(&point);
-
-                    let local = *vertex.local();
-                    let canonical = target.merge(Vertex { point });
-
-                    LocalForm::new(local, canonical)
-                });
-
-                let edge = target.merge(Edge { curve, vertices });
-                LocalForm::canonical_only(edge)
-            })
-            .collect();
-
-        let cycle_local = Cycle { edges: edges_local };
-
-        let cycle_canonical = target.merge(Cycle {
-            edges: edges_canonical,
-        });
-
-        LocalForm::new(cycle_local, cycle_canonical)
-    });
-
-    CyclesInFace::new(cycles)
-}
-
 fn create_non_continuous_side_face(
     path: Vector<3>,
     is_sweep_along_negative_direction: bool,
diff --git a/crates/fj-kernel/src/algorithms/transform.rs b/crates/fj-kernel/src/algorithms/transform.rs
index 3da41920b..bd148a9ae 100644
--- a/crates/fj-kernel/src/algorithms/transform.rs
+++ b/crates/fj-kernel/src/algorithms/transform.rs
@@ -1,30 +1,125 @@
 use fj_math::Transform;
 
 use crate::{
-    objects::{Curve, Face, Surface, Vertex},
-    shape::Shape,
+    objects::{Cycle, CyclesInFace, Edge, Face, FaceBRep, Vertex},
+    shape::{LocalForm, Shape},
 };
 
-/// Transform the geometry of the shape
-///
-/// Since the topological types refer to geometry, and don't contain any
-/// geometry themselves, this transforms the whole shape.
-pub fn transform_shape(shape: &mut Shape, transform: &Transform) {
-    shape
-        .update()
-        .update_all(|vertex: &mut Vertex| {
-            vertex.point = transform.transform_point(&vertex.point)
-        })
-        .update_all(|curve: &mut Curve<3>| *curve = curve.transform(transform))
-        .update_all(|surface: &mut Surface| {
-            *surface = surface.transform(transform)
-        })
-        .update_all(|mut face: &mut Face| {
-            use std::ops::DerefMut as _;
-            if let Face::Triangles(triangles) = face.deref_mut() {
-                for (triangle, _) in triangles {
-                    *triangle = transform.transform_triangle(triangle);
-                }
+/// Transform a shape
+pub fn transform(faces: &[Face], transform: &Transform) -> Vec<Face> {
+    let mut target = Vec::new();
+
+    for face in faces {
+        let face = match face {
+            Face::Face(face) => {
+                let mut tmp = Shape::new();
+                let surface = face.surface.get().transform(transform);
+                let surface = tmp.insert(surface);
+
+                let exteriors = transform_cycles(&face.exteriors, transform);
+                let interiors = transform_cycles(&face.interiors, transform);
+
+                let color = face.color;
+
+                Face::Face(FaceBRep {
+                    surface,
+                    exteriors,
+                    interiors,
+                    color,
+                })
             }
-        });
+            Face::Triangles(triangles) => {
+                let mut target = Vec::new();
+
+                for &(triangle, color) in triangles {
+                    let triangle = transform.transform_triangle(&triangle);
+                    target.push((triangle, color));
+                }
+
+                Face::Triangles(target)
+            }
+        };
+        target.push(face);
+    }
+
+    target
+}
+
+pub fn transform_cycles(
+    cycles: &CyclesInFace,
+    transform: &Transform,
+) -> CyclesInFace {
+    let mut tmp = Shape::new();
+
+    let cycles = cycles.as_local_form().map(|cycle| {
+        let edges_local = cycle
+            .local()
+            .edges
+            .iter()
+            .map(|edge| {
+                let curve_local = *edge.local().curve.local();
+                let curve_canonical = tmp
+                    .merge(edge.canonical().get().curve().transform(transform));
+
+                let vertices = edge.canonical().get().vertices.map(|vertex| {
+                    let point = vertex.canonical().get().point;
+                    let point = transform.transform_point(&point);
+
+                    let local = *vertex.local();
+                    let canonical = tmp.merge(Vertex { point });
+
+                    LocalForm::new(local, canonical)
+                });
+
+                let edge_local = Edge {
+                    curve: LocalForm::new(curve_local, curve_canonical.clone()),
+                    vertices: vertices.clone(),
+                };
+                let edge_canonical = tmp.merge(Edge {
+                    curve: LocalForm::canonical_only(curve_canonical),
+                    vertices,
+                });
+
+                LocalForm::new(edge_local, edge_canonical)
+            })
+            .collect();
+        let edges_canonical = cycle
+            .canonical()
+            .get()
+            .edges
+            .iter()
+            .map(|edge| {
+                let edge = edge.canonical().get();
+
+                let curve = {
+                    let curve = edge.curve().transform(transform);
+
+                    let curve = tmp.merge(curve);
+                    LocalForm::canonical_only(curve)
+                };
+                let vertices = edge.vertices.map(|vertex| {
+                    let point = vertex.canonical().get().point;
+                    let point = transform.transform_point(&point);
+
+                    let local = *vertex.local();
+                    let canonical = tmp.merge(Vertex { point });
+
+                    LocalForm::new(local, canonical)
+                });
+
+                let edge = tmp.merge(Edge { curve, vertices });
+                LocalForm::canonical_only(edge)
+            })
+            .collect();
+
+        let cycle_local = Cycle { edges: edges_local };
+
+        let cycle_canonical = tmp.merge(Cycle {
+            edges: edges_canonical,
+        });
+
+        LocalForm::new(cycle_local, cycle_canonical)
+    });
+
+    CyclesInFace::new(cycles)
 }
diff --git a/crates/fj-operations/src/transform.rs b/crates/fj-operations/src/transform.rs
index bfd1a207c..39eb98380 100644
--- a/crates/fj-operations/src/transform.rs
+++ b/crates/fj-operations/src/transform.rs
@@ -1,6 +1,7 @@
 use fj_interop::debug::DebugInfo;
 use fj_kernel::{
-    algorithms::{transform_shape, Tolerance},
+    algorithms::{transform, Tolerance},
+    iter::ObjectIters,
     shape::Shape,
     validation::{validate, Validated, ValidationConfig, ValidationError},
 };
@@ -16,22 +17,27 @@ impl ToShape for fj::Transform {
         debug_info: &mut DebugInfo,
     ) -> Result<Validated<Shape>, ValidationError> {
         let shape = self.shape.to_shape(config, tolerance, debug_info)?;
-        let mut shape = shape.into_inner();
+        let shape = shape.into_inner();
 
-        let transform = transform(self);
+        let shape = shape.face_iter().collect::<Vec<_>>();
+        let faces = transform(&shape, &make_transform(self));
 
-        transform_shape(&mut shape, &transform);
-        let shape = validate(shape, config)?;
+        let mut target = Shape::new();
+        for face in faces {
+            target.merge(face);
+        }
+
+        let shape = validate(target, config)?;
 
         Ok(shape)
     }
 
     fn bounding_volume(&self) -> Aabb<3> {
-        transform(self).transform_aabb(&self.shape.bounding_volume())
+        make_transform(self).transform_aabb(&self.shape.bounding_volume())
     }
 }
 
-fn transform(transform: &fj::Transform) -> Transform {
+fn make_transform(transform: &fj::Transform) -> Transform {
     let axis = Vector::from(transform.axis).normalize();
     Transform::translation(transform.offset)
         * Transform::rotation(axis * transform.angle.rad())