diff --git a/crates/fj-operations/src/difference_2d.rs b/crates/fj-operations/src/difference_2d.rs
index a1df9dc6f..9c24fbe67 100644
--- a/crates/fj-operations/src/difference_2d.rs
+++ b/crates/fj-operations/src/difference_2d.rs
@@ -10,6 +10,8 @@ use fj_kernel::{
 };
 use fj_math::Aabb;
 
+use crate::planes::Planes;
+
 use super::Shape;
 
 impl Shape for fj::Difference2d {
@@ -19,6 +21,7 @@ impl Shape for fj::Difference2d {
         &self,
         config: &ValidationConfig,
         stores: &Stores,
+        planes: &Planes,
         debug_info: &mut DebugInfo,
     ) -> Result<Validated<Self::Brep>, ValidationError> {
         // This method assumes that `b` is fully contained within `a`:
@@ -33,8 +36,9 @@ impl Shape for fj::Difference2d {
         // - https://doc.rust-lang.org/std/primitive.array.html#method.each_ref
         // - https://doc.rust-lang.org/std/primitive.array.html#method.try_map
         let [a, b] = self.shapes();
-        let [a, b] =
-            [a, b].map(|shape| shape.compute_brep(config, stores, debug_info));
+        let [a, b] = [a, b].map(|shape| {
+            shape.compute_brep(config, stores, planes, debug_info)
+        });
         let [a, b] = [a?, b?];
 
         if let Some(face) = a.face_iter().next() {
diff --git a/crates/fj-operations/src/group.rs b/crates/fj-operations/src/group.rs
index 7c4e5672d..bf4670fd6 100644
--- a/crates/fj-operations/src/group.rs
+++ b/crates/fj-operations/src/group.rs
@@ -8,6 +8,8 @@ use fj_kernel::{
 };
 use fj_math::Aabb;
 
+use crate::planes::Planes;
+
 use super::Shape;
 
 impl Shape for fj::Group {
@@ -17,12 +19,13 @@ impl Shape for fj::Group {
         &self,
         config: &ValidationConfig,
         stores: &Stores,
+        planes: &Planes,
         debug_info: &mut DebugInfo,
     ) -> Result<Validated<Self::Brep>, ValidationError> {
         let mut faces = Faces::new();
 
-        let a = self.a.compute_brep(config, stores, debug_info)?;
-        let b = self.b.compute_brep(config, stores, debug_info)?;
+        let a = self.a.compute_brep(config, stores, planes, debug_info)?;
+        let b = self.b.compute_brep(config, stores, planes, debug_info)?;
 
         faces.extend(a.into_inner());
         faces.extend(b.into_inner());
diff --git a/crates/fj-operations/src/lib.rs b/crates/fj-operations/src/lib.rs
index 8b79b1e67..c87387787 100644
--- a/crates/fj-operations/src/lib.rs
+++ b/crates/fj-operations/src/lib.rs
@@ -20,10 +20,13 @@ pub mod shape_processor;
 
 mod difference_2d;
 mod group;
+mod planes;
 mod sketch;
 mod sweep;
 mod transform;
 
+pub use self::planes::Planes;
+
 use fj_interop::debug::DebugInfo;
 use fj_kernel::{
     algorithms::validate::{
@@ -44,6 +47,7 @@ pub trait Shape {
         &self,
         config: &ValidationConfig,
         stores: &Stores,
+        planes: &Planes,
         debug_info: &mut DebugInfo,
     ) -> Result<Validated<Self::Brep>, ValidationError>;
 
@@ -61,19 +65,20 @@ impl Shape for fj::Shape {
         &self,
         config: &ValidationConfig,
         stores: &Stores,
+        planes: &Planes,
         debug_info: &mut DebugInfo,
     ) -> Result<Validated<Self::Brep>, ValidationError> {
         match self {
             Self::Shape2d(shape) => shape
-                .compute_brep(config, stores, debug_info)?
+                .compute_brep(config, stores, planes, debug_info)?
                 .into_inner()
                 .into_faces()
                 .validate_with_config(config),
             Self::Group(shape) => {
-                shape.compute_brep(config, stores, debug_info)
+                shape.compute_brep(config, stores, planes, debug_info)
             }
             Self::Sweep(shape) => shape
-                .compute_brep(config, stores, debug_info)?
+                .compute_brep(config, stores, planes, debug_info)?
                 .into_inner()
                 .into_shells()
                 .map(|shell| shell.into_faces())
@@ -84,7 +89,7 @@ impl Shape for fj::Shape {
                 .unwrap_or_default()
                 .validate_with_config(config),
             Self::Transform(shape) => {
-                shape.compute_brep(config, stores, debug_info)
+                shape.compute_brep(config, stores, planes, debug_info)
             }
         }
     }
@@ -106,14 +111,15 @@ impl Shape for fj::Shape2d {
         &self,
         config: &ValidationConfig,
         stores: &Stores,
+        planes: &Planes,
         debug_info: &mut DebugInfo,
     ) -> Result<Validated<Self::Brep>, ValidationError> {
         match self {
             Self::Difference(shape) => {
-                shape.compute_brep(config, stores, debug_info)
+                shape.compute_brep(config, stores, planes, debug_info)
             }
             Self::Sketch(shape) => {
-                shape.compute_brep(config, stores, debug_info)
+                shape.compute_brep(config, stores, planes, debug_info)
             }
         }
     }
diff --git a/crates/fj-operations/src/planes.rs b/crates/fj-operations/src/planes.rs
new file mode 100644
index 000000000..584e2b1ec
--- /dev/null
+++ b/crates/fj-operations/src/planes.rs
@@ -0,0 +1,50 @@
+use fj_kernel::{
+    objects::Surface,
+    stores::{Handle, Stores},
+};
+
+/// The static planes
+///
+/// Keeps [`Handle`]s to the xy-, xz- and yz-planes. The purpose of this struct
+/// is to provide these handles to implementations of [`Shape`], so they don't
+/// have to create a duplicate `Surface` whenever they need one of those.
+///
+/// [`Shape`]: crate::Shape
+pub struct Planes {
+    xy: Handle<Surface>,
+    xz: Handle<Surface>,
+    yz: Handle<Surface>,
+}
+
+impl Planes {
+    /// Create a new instance of `Planes`
+    ///
+    /// Please note that the whole point of this struct is to not duplicate the
+    /// standard planes, and creating multiple instances of it defeats that
+    /// point.
+    ///
+    /// Create one instance of this struct, then share it everywhere it's
+    /// needed.
+    pub fn new(stores: &Stores) -> Self {
+        let xy = stores.surfaces.insert(Surface::xy_plane());
+        let xz = stores.surfaces.insert(Surface::xz_plane());
+        let yz = stores.surfaces.insert(Surface::yz_plane());
+
+        Self { xy, xz, yz }
+    }
+
+    /// Access the xy-plane
+    pub fn xy(&self) -> Handle<Surface> {
+        self.xy.clone()
+    }
+
+    /// Access the xz-plane
+    pub fn xz(&self) -> Handle<Surface> {
+        self.xz.clone()
+    }
+
+    /// Access the yz-plane
+    pub fn yz(&self) -> Handle<Surface> {
+        self.yz.clone()
+    }
+}
diff --git a/crates/fj-operations/src/shape_processor.rs b/crates/fj-operations/src/shape_processor.rs
index e66f89bd3..0800fcbfd 100644
--- a/crates/fj-operations/src/shape_processor.rs
+++ b/crates/fj-operations/src/shape_processor.rs
@@ -11,7 +11,7 @@ use fj_kernel::{
 };
 use fj_math::Scalar;
 
-use crate::Shape as _;
+use crate::{planes::Planes, Shape as _};
 
 /// Processes an [`fj::Shape`] into a [`ProcessedShape`]
 pub struct ShapeProcessor {
@@ -44,8 +44,10 @@ impl ShapeProcessor {
 
         let config = ValidationConfig::default();
         let stores = Stores::new();
+        let planes = Planes::new(&stores);
         let mut debug_info = DebugInfo::new();
-        let shape = shape.compute_brep(&config, &stores, &mut debug_info)?;
+        let shape =
+            shape.compute_brep(&config, &stores, &planes, &mut debug_info)?;
         let mesh = (&shape.into_inner(), tolerance).triangulate();
 
         Ok(ProcessedShape {
diff --git a/crates/fj-operations/src/sketch.rs b/crates/fj-operations/src/sketch.rs
index 7ab36767d..62ee23263 100644
--- a/crates/fj-operations/src/sketch.rs
+++ b/crates/fj-operations/src/sketch.rs
@@ -3,12 +3,14 @@ use fj_kernel::{
     algorithms::validate::{
         Validate, Validated, ValidationConfig, ValidationError,
     },
-    objects::{Cycle, Face, HalfEdge, Sketch, Surface},
+    objects::{Cycle, Face, HalfEdge, Sketch},
     partial::HasPartial,
     stores::Stores,
 };
 use fj_math::{Aabb, Point};
 
+use crate::planes::Planes;
+
 use super::Shape;
 
 impl Shape for fj::Sketch {
@@ -18,9 +20,10 @@ impl Shape for fj::Sketch {
         &self,
         config: &ValidationConfig,
         stores: &Stores,
+        planes: &Planes,
         _: &mut DebugInfo,
     ) -> Result<Validated<Self::Brep>, ValidationError> {
-        let surface = stores.surfaces.insert(Surface::xy_plane());
+        let surface = planes.xy();
 
         let face = match self.chain() {
             fj::Chain::Circle(circle) => {
diff --git a/crates/fj-operations/src/sweep.rs b/crates/fj-operations/src/sweep.rs
index 67a7f00a7..15438e4a3 100644
--- a/crates/fj-operations/src/sweep.rs
+++ b/crates/fj-operations/src/sweep.rs
@@ -9,6 +9,8 @@ use fj_kernel::{
 };
 use fj_math::{Aabb, Vector};
 
+use crate::planes::Planes;
+
 use super::Shape;
 
 impl Shape for fj::Sweep {
@@ -18,9 +20,12 @@ impl Shape for fj::Sweep {
         &self,
         config: &ValidationConfig,
         stores: &Stores,
+        planes: &Planes,
         debug_info: &mut DebugInfo,
     ) -> Result<Validated<Self::Brep>, ValidationError> {
-        let sketch = self.shape().compute_brep(config, stores, debug_info)?;
+        let sketch = self
+            .shape()
+            .compute_brep(config, stores, planes, debug_info)?;
         let path = Vector::from(self.path());
 
         let solid = sketch.into_inner().sweep(path, stores);
diff --git a/crates/fj-operations/src/transform.rs b/crates/fj-operations/src/transform.rs
index 5b79c1b53..959993883 100644
--- a/crates/fj-operations/src/transform.rs
+++ b/crates/fj-operations/src/transform.rs
@@ -9,6 +9,8 @@ use fj_kernel::{
 };
 use fj_math::{Aabb, Transform, Vector};
 
+use crate::planes::Planes;
+
 use super::Shape;
 
 impl Shape for fj::Transform {
@@ -18,11 +20,12 @@ impl Shape for fj::Transform {
         &self,
         config: &ValidationConfig,
         stores: &Stores,
+        planes: &Planes,
         debug_info: &mut DebugInfo,
     ) -> Result<Validated<Self::Brep>, ValidationError> {
         let faces = self
             .shape
-            .compute_brep(config, stores, debug_info)?
+            .compute_brep(config, stores, planes, debug_info)?
             .into_inner()
             .transform(&make_transform(self), stores);