diff --git a/crates/fj-kernel/src/algorithms/approx/edge.rs b/crates/fj-kernel/src/algorithms/approx/edge.rs
index 95bc0f724..f8899ed87 100644
--- a/crates/fj-kernel/src/algorithms/approx/edge.rs
+++ b/crates/fj-kernel/src/algorithms/approx/edge.rs
@@ -231,10 +231,12 @@ mod tests {
 
     use crate::{
         algorithms::approx::{path::RangeOnPath, Approx, ApproxPoint},
-        builder::{HalfEdgeBuilder, SurfaceBuilder},
+        builder::{CycleBuilder, HalfEdgeBuilder, SurfaceBuilder},
         geometry::curve::GlobalPath,
         insert::Insert,
-        partial::{PartialHalfEdge, PartialObject, PartialSurface},
+        partial::{
+            PartialCycle, PartialHalfEdge, PartialObject, PartialSurface,
+        },
         services::Services,
     };
 
@@ -244,17 +246,16 @@ mod tests {
 
         let surface = services.objects.surfaces.xz_plane();
         let half_edge = {
-            let mut half_edge = PartialHalfEdge::default();
+            let mut cycle = PartialCycle::default();
 
-            half_edge.update_as_line_segment_from_points(
-                [[1., 1.], [2., 1.]],
-                half_edge.end_vertex.clone(),
-            );
-            half_edge.infer_vertex_positions_if_necessary(
+            let [mut half_edge, next_half_edge, _] = cycle
+                .update_as_polygon_from_points([[1., 1.], [2., 1.], [1., 2.]]);
+            half_edge.write().infer_vertex_positions_if_necessary(
                 &surface.geometry(),
-                half_edge.end_vertex.clone(),
+                next_half_edge.read().start_vertex.clone(),
             );
 
+            let half_edge = half_edge.read().clone();
             half_edge
                 .build(&mut services.objects)
                 .insert(&mut services.objects)
@@ -277,17 +278,16 @@ mod tests {
         .build(&mut services.objects)
         .insert(&mut services.objects);
         let half_edge = {
-            let mut half_edge = PartialHalfEdge::default();
+            let mut cycle = PartialCycle::default();
 
-            half_edge.update_as_line_segment_from_points(
-                [[1., 1.], [2., 1.]],
-                half_edge.end_vertex.clone(),
-            );
-            half_edge.infer_vertex_positions_if_necessary(
+            let [mut half_edge, next_half_edge, _] = cycle
+                .update_as_polygon_from_points([[1., 1.], [2., 1.], [1., 2.]]);
+            half_edge.write().infer_vertex_positions_if_necessary(
                 &surface.geometry(),
-                half_edge.end_vertex.clone(),
+                next_half_edge.read().start_vertex.clone(),
             );
 
+            let half_edge = half_edge.read().clone();
             half_edge
                 .build(&mut services.objects)
                 .insert(&mut services.objects)
@@ -310,21 +310,20 @@ mod tests {
             .build(&mut services.objects)
             .insert(&mut services.objects);
         let half_edge = {
-            let mut half_edge = PartialHalfEdge::default();
+            let mut cycle = PartialCycle::default();
 
-            half_edge.update_as_line_segment_from_points(
-                [[0., 1.], [1., 1.]],
-                half_edge.end_vertex.clone(),
-            );
+            let [mut half_edge, next_half_edge, _] = cycle
+                .update_as_polygon_from_points([[0., 1.], [1., 1.], [1., 2.]]);
 
-            half_edge.boundary[0] = Some(range.boundary[0]);
-            half_edge.boundary[1] = Some(range.boundary[1]);
+            half_edge.write().boundary[0] = Some(range.boundary[0]);
+            half_edge.write().boundary[1] = Some(range.boundary[1]);
 
-            half_edge.infer_vertex_positions_if_necessary(
+            half_edge.write().infer_vertex_positions_if_necessary(
                 &surface.geometry(),
-                half_edge.end_vertex.clone(),
+                next_half_edge.read().start_vertex.clone(),
             );
 
+            let half_edge = half_edge.read().clone();
             half_edge
                 .build(&mut services.objects)
                 .insert(&mut services.objects)
diff --git a/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs b/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
index 05f5f363c..ae5b67fd4 100644
--- a/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
+++ b/crates/fj-kernel/src/algorithms/intersect/curve_edge.rs
@@ -72,9 +72,9 @@ mod tests {
     use fj_math::Point;
 
     use crate::{
-        builder::HalfEdgeBuilder,
+        builder::{CycleBuilder, HalfEdgeBuilder},
         geometry::curve::Curve,
-        partial::{PartialHalfEdge, PartialObject},
+        partial::PartialCycle,
         services::Services,
     };
 
@@ -87,14 +87,13 @@ mod tests {
         let surface = services.objects.surfaces.xy_plane();
         let curve = Curve::u_axis();
         let half_edge = {
-            let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                [[1., -1.], [1., 1.]],
-                half_edge.end_vertex.clone(),
-            );
-            half_edge.infer_vertex_positions_if_necessary(
+            let mut cycle = PartialCycle::default();
+
+            let [mut half_edge, next_half_edge, _] = cycle
+                .update_as_polygon_from_points([[1., -1.], [1., 1.], [0., 1.]]);
+            half_edge.write().infer_vertex_positions_if_necessary(
                 &surface.geometry(),
-                half_edge.end_vertex.clone(),
+                next_half_edge.read().start_vertex.clone(),
             );
 
             half_edge.build(&mut services.objects)
@@ -117,14 +116,17 @@ mod tests {
         let surface = services.objects.surfaces.xy_plane();
         let curve = Curve::u_axis();
         let half_edge = {
-            let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                [[-1., -1.], [-1., 1.]],
-                half_edge.end_vertex.clone(),
-            );
-            half_edge.infer_vertex_positions_if_necessary(
+            let mut cycle = PartialCycle::default();
+
+            let [mut half_edge, next_half_edge, _] = cycle
+                .update_as_polygon_from_points([
+                    [-1., -1.],
+                    [-1., 1.],
+                    [0., 1.],
+                ]);
+            half_edge.write().infer_vertex_positions_if_necessary(
                 &surface.geometry(),
-                half_edge.end_vertex.clone(),
+                next_half_edge.read().start_vertex.clone(),
             );
 
             half_edge.build(&mut services.objects)
@@ -147,14 +149,17 @@ mod tests {
         let surface = services.objects.surfaces.xy_plane();
         let curve = Curve::u_axis();
         let half_edge = {
-            let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                [[-1., -1.], [1., -1.]],
-                half_edge.end_vertex.clone(),
-            );
-            half_edge.infer_vertex_positions_if_necessary(
+            let mut cycle = PartialCycle::default();
+
+            let [mut half_edge, next_half_edge, _] = cycle
+                .update_as_polygon_from_points([
+                    [-1., -1.],
+                    [1., -1.],
+                    [1., 1.],
+                ]);
+            half_edge.write().infer_vertex_positions_if_necessary(
                 &surface.geometry(),
-                half_edge.end_vertex.clone(),
+                next_half_edge.read().start_vertex.clone(),
             );
 
             half_edge.build(&mut services.objects)
@@ -172,14 +177,13 @@ mod tests {
         let surface = services.objects.surfaces.xy_plane();
         let curve = Curve::u_axis();
         let half_edge = {
-            let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                [[-1., 0.], [1., 0.]],
-                half_edge.end_vertex.clone(),
-            );
-            half_edge.infer_vertex_positions_if_necessary(
+            let mut cycle = PartialCycle::default();
+
+            let [mut half_edge, next_half_edge, _] = cycle
+                .update_as_polygon_from_points([[-1., 0.], [1., 0.], [1., 1.]]);
+            half_edge.write().infer_vertex_positions_if_necessary(
                 &surface.geometry(),
-                half_edge.end_vertex.clone(),
+                next_half_edge.read().start_vertex.clone(),
             );
 
             half_edge.build(&mut services.objects)
diff --git a/crates/fj-kernel/src/algorithms/sweep/edge.rs b/crates/fj-kernel/src/algorithms/sweep/edge.rs
index d3de6176a..6cb9a21e9 100644
--- a/crates/fj-kernel/src/algorithms/sweep/edge.rs
+++ b/crates/fj-kernel/src/algorithms/sweep/edge.rs
@@ -128,8 +128,7 @@ impl Sweep for (Handle<HalfEdge>, &Handle<SurfaceVertex>, &Surface, Color) {
         .into_iter()
         .circular_tuple_windows()
         {
-            edge.write()
-                .update_as_line_segment(next.read().start_vertex.clone());
+            edge.write().update_as_line_segment(next.clone());
         }
 
         // Finally, we can make sure that all edges refer to the correct global
diff --git a/crates/fj-kernel/src/builder/cycle.rs b/crates/fj-kernel/src/builder/cycle.rs
index 4fb5e7659..f11b06f17 100644
--- a/crates/fj-kernel/src/builder/cycle.rs
+++ b/crates/fj-kernel/src/builder/cycle.rs
@@ -140,9 +140,7 @@ impl CycleBuilder for PartialCycle {
         for (mut half_edge, next) in
             self.half_edges.iter().cloned().circular_tuple_windows()
         {
-            half_edge
-                .write()
-                .update_as_line_segment(next.read().start_vertex.clone());
+            half_edge.write().update_as_line_segment(next.clone());
         }
     }
 
diff --git a/crates/fj-kernel/src/builder/edge.rs b/crates/fj-kernel/src/builder/edge.rs
index b061d3ecb..4f12910d1 100644
--- a/crates/fj-kernel/src/builder/edge.rs
+++ b/crates/fj-kernel/src/builder/edge.rs
@@ -27,20 +27,13 @@ pub trait HalfEdgeBuilder {
     fn update_as_arc(
         &mut self,
         angle_rad: impl Into<Scalar>,
-        next_vertex: Partial<SurfaceVertex>,
+        next_half_edge: Partial<HalfEdge>,
     );
 
-    /// Update partial half-edge to be a line segment, from the given points
-    fn update_as_line_segment_from_points(
-        &mut self,
-        points: [impl Into<Point<2>>; 2],
-        next_vertex: Partial<SurfaceVertex>,
-    ) -> Curve;
-
     /// Update partial half-edge to be a line segment
     fn update_as_line_segment(
         &mut self,
-        next_vertex: Partial<SurfaceVertex>,
+        next_half_edge: Partial<HalfEdge>,
     ) -> Curve;
 
     /// Infer the global form of the half-edge
@@ -104,17 +97,18 @@ impl HalfEdgeBuilder for PartialHalfEdge {
     fn update_as_arc(
         &mut self,
         angle_rad: impl Into<Scalar>,
-        mut next_vertex: Partial<SurfaceVertex>,
+        mut next_half_edge: Partial<HalfEdge>,
     ) {
         let angle_rad = angle_rad.into();
         if angle_rad <= -Scalar::TAU || angle_rad >= Scalar::TAU {
             panic!("arc angle must be in the range (-2pi, 2pi) radians");
         }
-        let [start, end] = [&self.start_vertex, &next_vertex].map(|vertex| {
-            vertex
-                .read()
-                .position
-                .expect("Can't infer arc without surface position")
+        let [start, end] = [
+            &self.start_position(),
+            &next_half_edge.read().start_position(),
+        ]
+        .map(|position| {
+            position.expect("Can't infer arc without surface position")
         });
 
         let arc = fj_math::Arc::from_endpoints_and_angle(start, end, angle_rad);
@@ -128,7 +122,10 @@ impl HalfEdgeBuilder for PartialHalfEdge {
         for ((point_boundary, surface_vertex), point_curve) in self
             .boundary
             .each_mut_ext()
-            .zip_ext([&mut self.start_vertex, &mut next_vertex])
+            .zip_ext([
+                &mut self.start_vertex,
+                &mut next_half_edge.write().start_vertex,
+            ])
             .zip_ext([a_curve, b_curve])
         {
             *point_boundary = Some(point_curve);
@@ -136,34 +133,20 @@ impl HalfEdgeBuilder for PartialHalfEdge {
                 Some(path.point_from_path_coords(point_curve));
         }
 
-        self.infer_global_form(next_vertex);
-    }
-
-    fn update_as_line_segment_from_points(
-        &mut self,
-        points: [impl Into<Point<2>>; 2],
-        mut next_vertex: Partial<SurfaceVertex>,
-    ) -> Curve {
-        for (vertex, point) in
-            [&mut self.start_vertex, &mut next_vertex].zip_ext(points)
-        {
-            let mut surface_form = vertex.write();
-            surface_form.position = Some(point.into());
-        }
-
-        self.update_as_line_segment(next_vertex)
+        self.infer_global_form(next_half_edge.read().start_vertex.clone());
     }
 
     fn update_as_line_segment(
         &mut self,
-        next_vertex: Partial<SurfaceVertex>,
+        next_half_edge: Partial<HalfEdge>,
     ) -> Curve {
         let boundary = self.boundary;
-        let points_surface = [&self.start_vertex, &next_vertex].map(|vertex| {
-            vertex
-                .read()
-                .position
-                .expect("Can't infer line segment without surface position")
+        let points_surface = [
+            &self.start_position(),
+            &next_half_edge.read().start_position(),
+        ]
+        .map(|position| {
+            position.expect("Can't infer line segment without surface position")
         });
 
         let path = if let [Some(start), Some(end)] = boundary {
@@ -186,7 +169,7 @@ impl HalfEdgeBuilder for PartialHalfEdge {
             path
         };
 
-        self.infer_global_form(next_vertex);
+        self.infer_global_form(next_half_edge.read().start_vertex.clone());
 
         path
     }
diff --git a/crates/fj-kernel/src/objects/full/edge.rs b/crates/fj-kernel/src/objects/full/edge.rs
index a40b1b193..23c476171 100644
--- a/crates/fj-kernel/src/objects/full/edge.rs
+++ b/crates/fj-kernel/src/objects/full/edge.rs
@@ -177,8 +177,8 @@ mod tests {
     use pretty_assertions::assert_eq;
 
     use crate::{
-        builder::HalfEdgeBuilder,
-        partial::{PartialHalfEdge, PartialObject},
+        builder::{CycleBuilder, HalfEdgeBuilder},
+        partial::PartialCycle,
         services::Services,
     };
 
@@ -190,29 +190,28 @@ mod tests {
 
         let a = [0., 0.];
         let b = [1., 0.];
+        let c = [0., 1.];
 
         let a_to_b = {
-            let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                [a, b],
-                half_edge.end_vertex.clone(),
-            );
-            half_edge.infer_vertex_positions_if_necessary(
+            let mut cycle = PartialCycle::default();
+
+            let [mut half_edge, next_half_edge, _] =
+                cycle.update_as_polygon_from_points([a, b, c]);
+            half_edge.write().infer_vertex_positions_if_necessary(
                 &surface.geometry(),
-                half_edge.end_vertex.clone(),
+                next_half_edge.read().start_vertex.clone(),
             );
 
             half_edge.build(&mut services.objects)
         };
         let b_to_a = {
-            let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                [b, a],
-                half_edge.end_vertex.clone(),
-            );
-            half_edge.infer_vertex_positions_if_necessary(
+            let mut cycle = PartialCycle::default();
+
+            let [mut half_edge, next_half_edge, _] =
+                cycle.update_as_polygon_from_points([b, a, c]);
+            half_edge.write().infer_vertex_positions_if_necessary(
                 &surface.geometry(),
-                half_edge.end_vertex.clone(),
+                next_half_edge.read().start_vertex.clone(),
             );
 
             half_edge.build(&mut services.objects)
diff --git a/crates/fj-kernel/src/partial/objects/edge.rs b/crates/fj-kernel/src/partial/objects/edge.rs
index 5c4ad3c8b..45808b722 100644
--- a/crates/fj-kernel/src/partial/objects/edge.rs
+++ b/crates/fj-kernel/src/partial/objects/edge.rs
@@ -25,6 +25,13 @@ pub struct PartialHalfEdge {
     pub global_form: Partial<GlobalEdge>,
 }
 
+impl PartialHalfEdge {
+    /// Compute the surface position where the half-edge starts
+    pub fn start_position(&self) -> Option<Point<2>> {
+        self.start_vertex.read().position
+    }
+}
+
 impl PartialObject for PartialHalfEdge {
     type Full = HalfEdge;
 
diff --git a/crates/fj-kernel/src/validate/edge.rs b/crates/fj-kernel/src/validate/edge.rs
index cd8709ad1..d473182e5 100644
--- a/crates/fj-kernel/src/validate/edge.rs
+++ b/crates/fj-kernel/src/validate/edge.rs
@@ -143,9 +143,9 @@ mod tests {
     use fj_math::Point;
 
     use crate::{
-        builder::HalfEdgeBuilder,
+        builder::{CycleBuilder, HalfEdgeBuilder},
         objects::HalfEdge,
-        partial::{Partial, PartialHalfEdge, PartialObject},
+        partial::{Partial, PartialCycle},
         services::Services,
         validate::Validate,
     };
@@ -157,14 +157,13 @@ mod tests {
         let valid = {
             let surface = services.objects.surfaces.xy_plane();
 
-            let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                [[0., 0.], [1., 0.]],
-                half_edge.end_vertex.clone(),
-            );
-            half_edge.infer_vertex_positions_if_necessary(
+            let mut cycle = PartialCycle::default();
+
+            let [mut half_edge, next_half_edge, _] = cycle
+                .update_as_polygon_from_points([[0., 0.], [1., 0.], [1., 1.]]);
+            half_edge.write().infer_vertex_positions_if_necessary(
                 &surface.geometry(),
-                half_edge.end_vertex.clone(),
+                next_half_edge.read().start_vertex.clone(),
             );
 
             half_edge.build(&mut services.objects)
@@ -208,14 +207,13 @@ mod tests {
         let valid = {
             let surface = services.objects.surfaces.xy_plane();
 
-            let mut half_edge = PartialHalfEdge::default();
-            half_edge.update_as_line_segment_from_points(
-                [[0., 0.], [1., 0.]],
-                half_edge.end_vertex.clone(),
-            );
-            half_edge.infer_vertex_positions_if_necessary(
+            let mut cycle = PartialCycle::default();
+
+            let [mut half_edge, next_half_edge, _] = cycle
+                .update_as_polygon_from_points([[0., 0.], [1., 0.], [1., 1.]]);
+            half_edge.write().infer_vertex_positions_if_necessary(
                 &surface.geometry(),
-                half_edge.end_vertex.clone(),
+                next_half_edge.read().start_vertex.clone(),
             );
 
             half_edge.build(&mut services.objects)
diff --git a/crates/fj-operations/src/sketch.rs b/crates/fj-operations/src/sketch.rs
index 6a1acbec2..f102b0032 100644
--- a/crates/fj-operations/src/sketch.rs
+++ b/crates/fj-operations/src/sketch.rs
@@ -73,19 +73,16 @@ impl Shape for fj::Sketch {
                     for ((mut half_edge, route), (next_half_edge, _)) in
                         half_edges.into_iter().circular_tuple_windows()
                     {
-                        let next_vertex =
-                            next_half_edge.read().start_vertex.clone();
-
                         match route {
                             fj::SketchSegmentRoute::Direct => {
                                 half_edge
                                     .write()
-                                    .update_as_line_segment(next_vertex);
+                                    .update_as_line_segment(next_half_edge);
                             }
                             fj::SketchSegmentRoute::Arc { angle } => {
                                 half_edge
                                     .write()
-                                    .update_as_arc(angle.rad(), next_vertex);
+                                    .update_as_arc(angle.rad(), next_half_edge);
                             }
                         }
                     }