diff --git a/crates/fj-kernel/src/algorithms/intersect/curve_face.rs b/crates/fj-kernel/src/algorithms/intersect/curve_face.rs
index 06d049cc2..76d839a07 100644
--- a/crates/fj-kernel/src/algorithms/intersect/curve_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/curve_face.rs
@@ -188,8 +188,8 @@ mod tests {
         let face = {
             let mut face = PartialFace::default();
             face.exterior.write().surface = surface;
-            face.update_exterior_as_polygon(exterior);
-            face.add_interior_polygon(interior);
+            face.update_exterior_as_polygon_from_points(exterior);
+            face.add_interior_polygon_from_points(interior);
 
             face.build(&mut services.objects)
         };
diff --git a/crates/fj-kernel/src/algorithms/intersect/face_face.rs b/crates/fj-kernel/src/algorithms/intersect/face_face.rs
index adf9a5496..a8ba4668a 100644
--- a/crates/fj-kernel/src/algorithms/intersect/face_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/face_face.rs
@@ -95,7 +95,7 @@ mod tests {
         .map(|surface| {
             let mut face = PartialFace::default();
             face.exterior.write().surface = Partial::from(surface);
-            face.update_exterior_as_polygon(points);
+            face.update_exterior_as_polygon_from_points(points);
 
             face.build(&mut services.objects)
         });
@@ -124,7 +124,7 @@ mod tests {
         let [a, b] = surfaces.clone().map(|surface| {
             let mut face = PartialFace::default();
             face.exterior.write().surface = Partial::from(surface);
-            face.update_exterior_as_polygon(points);
+            face.update_exterior_as_polygon_from_points(points);
 
             face.build(&mut services.objects)
         });
diff --git a/crates/fj-kernel/src/algorithms/intersect/face_point.rs b/crates/fj-kernel/src/algorithms/intersect/face_point.rs
index 7e2c74c9f..c7192eb2e 100644
--- a/crates/fj-kernel/src/algorithms/intersect/face_point.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/face_point.rs
@@ -149,7 +149,11 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon([[0., 0.], [1., 1.], [0., 2.]]);
+        face.update_exterior_as_polygon_from_points([
+            [0., 0.],
+            [1., 1.],
+            [0., 2.],
+        ]);
         let face = face
             .build(&mut services.objects)
             .insert(&mut services.objects);
@@ -166,7 +170,11 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon([[0., 0.], [2., 1.], [0., 2.]]);
+        face.update_exterior_as_polygon_from_points([
+            [0., 0.],
+            [2., 1.],
+            [0., 2.],
+        ]);
         let face = face
             .build(&mut services.objects)
             .insert(&mut services.objects);
@@ -186,7 +194,11 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon([[4., 2.], [0., 4.], [0., 0.]]);
+        face.update_exterior_as_polygon_from_points([
+            [4., 2.],
+            [0., 4.],
+            [0., 0.],
+        ]);
         let face = face
             .build(&mut services.objects)
             .insert(&mut services.objects);
@@ -206,7 +218,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon([
+        face.update_exterior_as_polygon_from_points([
             [0., 0.],
             [2., 1.],
             [3., 0.],
@@ -231,7 +243,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon([
+        face.update_exterior_as_polygon_from_points([
             [0., 0.],
             [2., 1.],
             [3., 1.],
@@ -256,7 +268,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon([
+        face.update_exterior_as_polygon_from_points([
             [0., 0.],
             [2., 1.],
             [3., 1.],
@@ -282,7 +294,11 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon([[0., 0.], [2., 0.], [0., 1.]]);
+        face.update_exterior_as_polygon_from_points([
+            [0., 0.],
+            [2., 0.],
+            [0., 1.],
+        ]);
         let face = face
             .build(&mut services.objects)
             .insert(&mut services.objects);
@@ -312,7 +328,11 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon([[0., 0.], [1., 0.], [0., 1.]]);
+        face.update_exterior_as_polygon_from_points([
+            [0., 0.],
+            [1., 0.],
+            [0., 1.],
+        ]);
         let face = face
             .build(&mut services.objects)
             .insert(&mut services.objects);
diff --git a/crates/fj-kernel/src/algorithms/intersect/ray_face.rs b/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
index cb9572487..3391e7a18 100644
--- a/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/ray_face.rs
@@ -167,7 +167,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.yz_plane());
-        face.update_exterior_as_polygon([
+        face.update_exterior_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
@@ -190,7 +190,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.yz_plane());
-        face.update_exterior_as_polygon([
+        face.update_exterior_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
@@ -216,7 +216,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.yz_plane());
-        face.update_exterior_as_polygon([
+        face.update_exterior_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
@@ -239,7 +239,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.yz_plane());
-        face.update_exterior_as_polygon([
+        face.update_exterior_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
@@ -274,7 +274,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.yz_plane());
-        face.update_exterior_as_polygon([
+        face.update_exterior_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
@@ -308,7 +308,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon([
+        face.update_exterior_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
@@ -333,7 +333,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon([
+        face.update_exterior_as_polygon_from_points([
             [-1., -1.],
             [1., -1.],
             [1., 1.],
diff --git a/crates/fj-kernel/src/algorithms/sweep/face.rs b/crates/fj-kernel/src/algorithms/sweep/face.rs
index 799803854..65d0f298f 100644
--- a/crates/fj-kernel/src/algorithms/sweep/face.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/face.rs
@@ -116,7 +116,7 @@ mod tests {
 
         let mut bottom = PartialFace::default();
         bottom.exterior.write().surface = Partial::from(surface.clone());
-        bottom.update_exterior_as_polygon(TRIANGLE);
+        bottom.update_exterior_as_polygon_from_points(TRIANGLE);
         let bottom = bottom
             .build(&mut services.objects)
             .insert(&mut services.objects)
@@ -124,7 +124,7 @@ mod tests {
         let mut top = PartialFace::default();
         top.exterior.write().surface =
             Partial::from(surface.translate(UP, &mut services.objects));
-        top.update_exterior_as_polygon(TRIANGLE);
+        top.update_exterior_as_polygon_from_points(TRIANGLE);
         let top = top
             .build(&mut services.objects)
             .insert(&mut services.objects);
@@ -169,14 +169,14 @@ mod tests {
         bottom.exterior.write().surface = Partial::from(
             surface.clone().translate(DOWN, &mut services.objects),
         );
-        bottom.update_exterior_as_polygon(TRIANGLE);
+        bottom.update_exterior_as_polygon_from_points(TRIANGLE);
         let bottom = bottom
             .build(&mut services.objects)
             .insert(&mut services.objects)
             .reverse(&mut services.objects);
         let mut top = PartialFace::default();
         top.exterior.write().surface = Partial::from(surface);
-        top.update_exterior_as_polygon(TRIANGLE);
+        top.update_exterior_as_polygon_from_points(TRIANGLE);
         let top = top
             .build(&mut services.objects)
             .insert(&mut services.objects);
diff --git a/crates/fj-kernel/src/algorithms/triangulate/mod.rs b/crates/fj-kernel/src/algorithms/triangulate/mod.rs
index 695c7aad0..c252bb7fc 100644
--- a/crates/fj-kernel/src/algorithms/triangulate/mod.rs
+++ b/crates/fj-kernel/src/algorithms/triangulate/mod.rs
@@ -99,7 +99,7 @@ mod tests {
         let mut face = PartialFace::default();
         face.exterior.write().surface =
             Partial::from(services.objects.surfaces.xy_plane());
-        face.update_exterior_as_polygon([a, b, c, d]);
+        face.update_exterior_as_polygon_from_points([a, b, c, d]);
         let face = face
             .build(&mut services.objects)
             .insert(&mut services.objects);
@@ -136,8 +136,8 @@ mod tests {
         let surface = services.objects.surfaces.xy_plane();
         let mut face = PartialFace::default();
         face.exterior.write().surface = Partial::from(surface.clone());
-        face.update_exterior_as_polygon([a, b, c, d]);
-        face.add_interior_polygon([e, f, g, h]);
+        face.update_exterior_as_polygon_from_points([a, b, c, d]);
+        face.add_interior_polygon_from_points([e, f, g, h]);
         let face = face
             .build(&mut services.objects)
             .insert(&mut services.objects);
@@ -196,7 +196,7 @@ mod tests {
         let surface = services.objects.surfaces.xy_plane();
         let mut face = PartialFace::default();
         face.exterior.write().surface = Partial::from(surface.clone());
-        face.update_exterior_as_polygon([a, b, c, d, e]);
+        face.update_exterior_as_polygon_from_points([a, b, c, d, e]);
         let face = face
             .build(&mut services.objects)
             .insert(&mut services.objects);
diff --git a/crates/fj-kernel/src/builder/cycle.rs b/crates/fj-kernel/src/builder/cycle.rs
index da31da187..68749a828 100644
--- a/crates/fj-kernel/src/builder/cycle.rs
+++ b/crates/fj-kernel/src/builder/cycle.rs
@@ -1,6 +1,7 @@
 use fj_math::Point;
 
 use crate::{
+    builder::SurfaceBuilder,
     objects::HalfEdge,
     partial::{Partial, PartialCycle},
 };
@@ -9,17 +10,6 @@ use super::HalfEdgeBuilder;
 
 /// Builder API for [`PartialCycle`]
 pub trait CycleBuilder {
-    /// Create a cycle as a polygonal chain from the provided points
-    fn update_as_polygon_from_points(
-        &mut self,
-        points: impl IntoIterator<Item = impl Into<Point<2>>>,
-    ) -> Vec<Partial<HalfEdge>>;
-
-    /// Update cycle to be a polygon
-    ///
-    /// Will update each half-edge in the cycle to be a line segment.
-    fn update_as_polygon(&mut self);
-
     /// Add a new half-edge to the cycle
     ///
     /// Creates a half-edge and adds it to the cycle. The new half-edge is
@@ -43,31 +33,35 @@ pub trait CycleBuilder {
         &mut self,
         point: impl Into<Point<2>>,
     ) -> Partial<HalfEdge>;
-}
 
-impl CycleBuilder for PartialCycle {
+    /// Update cycle as a polygon from the provided points
     fn update_as_polygon_from_points(
         &mut self,
         points: impl IntoIterator<Item = impl Into<Point<2>>>,
-    ) -> Vec<Partial<HalfEdge>> {
-        let mut half_edges = Vec::new();
-
-        for point in points {
-            let half_edge = self.add_half_edge_from_point_to_start(point);
-            half_edges.push(half_edge);
-        }
-
-        self.update_as_polygon();
+    ) -> Vec<Partial<HalfEdge>>;
 
-        half_edges
-    }
+    /// Update cycle as a polygon
+    ///
+    /// Will update each half-edge in the cycle to be a line segment.
+    fn update_as_polygon(&mut self);
 
-    fn update_as_polygon(&mut self) {
-        for half_edge in &mut self.half_edges {
-            half_edge.write().update_as_line_segment();
-        }
-    }
+    /// Update cycle as a triangle, from global (3D) points
+    ///
+    /// Uses the three points to infer a plane that is used as the surface.
+    ///
+    /// # Implementation Note
+    ///
+    /// This method is probably just temporary, and will be generalized into a
+    /// "update as polygon from global points" method sooner or later. For now,
+    /// I didn't want to deal with the question of how to infer the surface, and
+    /// how to handle points that don't fit that surface.
+    fn update_as_triangle_from_global_points(
+        &mut self,
+        points: [impl Into<Point<3>>; 3],
+    ) -> [Partial<HalfEdge>; 3];
+}
 
+impl CycleBuilder for PartialCycle {
     fn add_half_edge(&mut self) -> Partial<HalfEdge> {
         let mut new_half_edge = Partial::<HalfEdge>::new();
 
@@ -129,4 +123,46 @@ impl CycleBuilder for PartialCycle {
 
         half_edge
     }
+
+    fn update_as_polygon_from_points(
+        &mut self,
+        points: impl IntoIterator<Item = impl Into<Point<2>>>,
+    ) -> Vec<Partial<HalfEdge>> {
+        let mut half_edges = Vec::new();
+
+        for point in points {
+            let half_edge = self.add_half_edge_from_point_to_start(point);
+            half_edges.push(half_edge);
+        }
+
+        self.update_as_polygon();
+
+        half_edges
+    }
+
+    fn update_as_polygon(&mut self) {
+        for half_edge in &mut self.half_edges {
+            half_edge.write().update_as_line_segment();
+        }
+    }
+
+    fn update_as_triangle_from_global_points(
+        &mut self,
+        points_global: [impl Into<Point<3>>; 3],
+    ) -> [Partial<HalfEdge>; 3] {
+        let points_surface = self
+            .surface
+            .write()
+            .update_as_plane_from_points(points_global);
+        let mut edges = self.update_as_polygon_from_points(points_surface);
+
+        // None of the following should panic, as we just created a polygon from
+        // three points, so we should have exactly three edges.
+        let c = edges.pop().unwrap();
+        let b = edges.pop().unwrap();
+        let a = edges.pop().unwrap();
+        assert!(edges.pop().is_none());
+
+        [a, b, c]
+    }
 }
diff --git a/crates/fj-kernel/src/builder/face.rs b/crates/fj-kernel/src/builder/face.rs
index bc1c49ee9..fb5e74f61 100644
--- a/crates/fj-kernel/src/builder/face.rs
+++ b/crates/fj-kernel/src/builder/face.rs
@@ -9,28 +9,59 @@ use super::CycleBuilder;
 
 /// Builder API for [`PartialFace`]
 pub trait FaceBuilder {
-    /// Update the [`PartialFace`] with an exterior polygon
-    fn update_exterior_as_polygon(
+    /// Update the face exterior as a polygon from the provided points
+    fn update_exterior_as_polygon_from_points(
         &mut self,
         points: impl IntoIterator<Item = impl Into<Point<2>>>,
     ) -> Vec<Partial<HalfEdge>>;
 
-    /// Update the [`PartialFace`] with an interior polygon
-    fn add_interior_polygon(
+    /// Update the face exterior as a polygon
+    fn update_exterior_as_polygon(&mut self);
+
+    /// Update the face exterior as a triangle, from 3D points
+    ///
+    /// Uses the three points to infer a plane that is used as the surface.
+    ///
+    /// # Implementation Note
+    ///
+    /// This method is probably just temporary, and will be generalized into a
+    /// "update as polygon from global points" method sooner or later. For now,
+    /// I didn't want to deal with the question of how to infer the surface, and
+    /// how to handle points that don't fit that surface.
+    fn update_exterior_as_triangle_from_global_points(
+        &mut self,
+        points: [impl Into<Point<3>>; 3],
+    ) -> [Partial<HalfEdge>; 3];
+
+    /// Add an interior polygon, from the provided points
+    fn add_interior_polygon_from_points(
         &mut self,
         points: impl IntoIterator<Item = impl Into<Point<2>>>,
     );
 }
 
 impl FaceBuilder for PartialFace {
-    fn update_exterior_as_polygon(
+    fn update_exterior_as_polygon_from_points(
         &mut self,
         points: impl IntoIterator<Item = impl Into<Point<2>>>,
     ) -> Vec<Partial<HalfEdge>> {
         self.exterior.write().update_as_polygon_from_points(points)
     }
 
-    fn add_interior_polygon(
+    fn update_exterior_as_polygon(&mut self) {
+        self.exterior.write().update_as_polygon()
+    }
+
+    fn update_exterior_as_triangle_from_global_points(
+        &mut self,
+        points_global: [impl Into<Point<3>>; 3],
+    ) -> [Partial<HalfEdge>; 3] {
+        self.exterior
+            .write()
+            .update_as_triangle_from_global_points(points_global)
+    }
+
+    fn add_interior_polygon_from_points(
         &mut self,
         points: impl IntoIterator<Item = impl Into<Point<2>>>,
     ) {
diff --git a/crates/fj-kernel/src/builder/shell.rs b/crates/fj-kernel/src/builder/shell.rs
index c673449b4..073d12fa1 100644
--- a/crates/fj-kernel/src/builder/shell.rs
+++ b/crates/fj-kernel/src/builder/shell.rs
@@ -55,7 +55,7 @@ impl ShellBuilder for PartialShell {
 
             let mut face = PartialFace::default();
             face.exterior.write().surface = Partial::from(surface);
-            face.update_exterior_as_polygon([
+            face.update_exterior_as_polygon_from_points([
                 [-h, -h],
                 [h, -h],
                 [h, h],
diff --git a/crates/fj-kernel/src/builder/sketch.rs b/crates/fj-kernel/src/builder/sketch.rs
index 1436ad2cd..57ca036fb 100644
--- a/crates/fj-kernel/src/builder/sketch.rs
+++ b/crates/fj-kernel/src/builder/sketch.rs
@@ -25,7 +25,7 @@ impl SketchBuilder for PartialSketch {
     ) {
         let mut face = PartialFace::default();
         face.exterior.write().surface = surface.into();
-        face.update_exterior_as_polygon(points);
+        face.update_exterior_as_polygon_from_points(points);
 
         self.faces.extend([Partial::from_partial(face)]);
     }
diff --git a/crates/fj-kernel/src/validate/face.rs b/crates/fj-kernel/src/validate/face.rs
index 893683beb..0effcee19 100644
--- a/crates/fj-kernel/src/validate/face.rs
+++ b/crates/fj-kernel/src/validate/face.rs
@@ -122,8 +122,16 @@ mod tests {
         let valid = {
             let mut face = PartialFace::default();
             face.exterior.write().surface = Partial::from(surface);
-            face.update_exterior_as_polygon([[0., 0.], [3., 0.], [0., 3.]]);
-            face.add_interior_polygon([[1., 1.], [1., 2.], [2., 1.]]);
+            face.update_exterior_as_polygon_from_points([
+                [0., 0.],
+                [3., 0.],
+                [0., 3.],
+            ]);
+            face.add_interior_polygon_from_points([
+                [1., 1.],
+                [1., 2.],
+                [2., 1.],
+            ]);
 
             face.build(&mut services.objects)
         };
@@ -154,8 +162,16 @@ mod tests {
         let valid = {
             let mut face = PartialFace::default();
             face.exterior.write().surface = Partial::from(surface);
-            face.update_exterior_as_polygon([[0., 0.], [3., 0.], [0., 3.]]);
-            face.add_interior_polygon([[1., 1.], [1., 2.], [2., 1.]]);
+            face.update_exterior_as_polygon_from_points([
+                [0., 0.],
+                [3., 0.],
+                [0., 3.],
+            ]);
+            face.add_interior_polygon_from_points([
+                [1., 1.],
+                [1., 2.],
+                [2., 1.],
+            ]);
             face.build(&mut services.objects)
         };
         let invalid = {
diff --git a/crates/fj-operations/src/sketch.rs b/crates/fj-operations/src/sketch.rs
index 61318cce6..4cd863d8e 100644
--- a/crates/fj-operations/src/sketch.rs
+++ b/crates/fj-operations/src/sketch.rs
@@ -64,7 +64,7 @@ impl Shape for fj::Sketch {
 
                 let mut face = PartialFace::default();
                 face.exterior.write().surface = Partial::from(surface);
-                face.update_exterior_as_polygon(points);
+                face.update_exterior_as_polygon_from_points(points);
                 face.color = Some(Color(self.color()));
 
                 face