diff --git a/crates/fj-core/src/geometry/curve.rs b/crates/fj-core/src/geometry/curve.rs
index 392ef64cc..67ccd34d5 100644
--- a/crates/fj-core/src/geometry/curve.rs
+++ b/crates/fj-core/src/geometry/curve.rs
@@ -1,3 +1,5 @@
+use std::collections::BTreeMap;
+
 use crate::{storage::Handle, topology::Surface};
 
 use super::SurfacePath;
@@ -22,7 +24,7 @@ pub struct CurveGeom {
     /// which is a prerequisite for making the required tooling practical:
     ///
     /// <https://github.com/hannobraun/fornjot/issues/2118>
-    pub definitions: Vec<LocalCurveGeom>,
+    pub definitions: BTreeMap<Handle<Surface>, LocalCurveGeom>,
 }
 
 impl CurveGeom {
@@ -31,9 +33,10 @@ impl CurveGeom {
         path: SurfacePath,
         surface: Handle<Surface>,
     ) -> Self {
-        Self {
-            definitions: vec![LocalCurveGeom { path, surface }],
-        }
+        let mut definitions = BTreeMap::new();
+        definitions.insert(surface, LocalCurveGeom { path });
+
+        Self { definitions }
     }
 }
 
@@ -42,7 +45,4 @@ impl CurveGeom {
 pub struct LocalCurveGeom {
     /// The path that defines the curve on its surface
     pub path: SurfacePath,
-
-    /// The surface that the curve is defined on
-    pub surface: Handle<Surface>,
 }