diff --git a/src/transformation/mesh_intersection/mesh_intersection.rs b/src/transformation/mesh_intersection/mesh_intersection.rs
index 90033855..0a21905d 100644
--- a/src/transformation/mesh_intersection/mesh_intersection.rs
+++ b/src/transformation/mesh_intersection/mesh_intersection.rs
@@ -186,7 +186,10 @@ pub fn intersect_meshes_with_tolerances(
                 insert_point(pos1 * mesh1.vertices()[face[1] as usize]),
                 insert_point(pos1 * mesh1.vertices()[face[2] as usize]),
             ];
-            let _ = topology_indices.insert(idx.into(), idx);
+
+            if !is_topologically_degenerate(idx) {
+                insert_topology_indices(&mut topology_indices, idx);
+            }
         }
 
         // Add the inside vertices and triangles from mesh2
@@ -199,7 +202,10 @@ pub fn intersect_meshes_with_tolerances(
                 insert_point(pos2 * mesh2.vertices()[face[1] as usize]),
                 insert_point(pos2 * mesh2.vertices()[face[2] as usize]),
             ];
-            let _ = topology_indices.insert(idx.into(), idx);
+
+            if !is_topologically_degenerate(idx) {
+                insert_topology_indices(&mut topology_indices, idx);
+            }
         }
     }
 
@@ -677,43 +683,12 @@ fn merge_triangle_sets(
                 // This should *never* trigger. If it does
                 // it means the code has created a triangle with duplicate vertices,
                 // which means we encountered an unaccounted for edge case.
-                if new_tri_idx[0] == new_tri_idx[1]
-                    || new_tri_idx[0] == new_tri_idx[2]
-                    || new_tri_idx[1] == new_tri_idx[2]
+                if is_topologically_degenerate(new_tri_idx)
                 {
                     return Err(MeshIntersectionError::DuplicateVertices);
                 }
 
-                // Insert in the hashmap with sorted indices to avoid adding duplicates.
-                // We also check if we don’t keep pairs of triangles that have the same
-                // set of indices but opposite orientations.
-                match topology_indices.entry(new_tri_idx.into()) {
-                    Entry::Vacant(e) => {
-                        let _ = e.insert(new_tri_idx);
-                    }
-                    Entry::Occupied(e) => {
-                        fn same_orientation(a: &[u32; 3], b: &[u32; 3]) -> bool {
-                            let ib = if a[0] == b[0] {
-                                0
-                            } else if a[0] == b[1] {
-                                1
-                            } else {
-                                2
-                            };
-                            a[1] == b[(ib + 1) % 3]
-                        }
-
-                        if !same_orientation(e.get(), &new_tri_idx) {
-                            // If we are inserting two identical triangles but with mismatching
-                            // orientations, we can just ignore both because they cover a degenerate
-                            // 2D plane.
-                            #[cfg(feature = "enhanced-determinism")]
-                            let _ = e.swap_remove();
-                            #[cfg(not(feature = "enhanced-determinism"))]
-                            let _ = e.remove();
-                        }
-                    }
-                }
+                insert_topology_indices(topology_indices, new_tri_idx);
             }
         }
     }
@@ -721,6 +696,48 @@ fn merge_triangle_sets(
     Ok(())
 }
 
+// Insert in the hashmap with sorted indices to avoid adding duplicates.
+//
+// We also check if we don’t keep pairs of triangles that have the same
+// set of indices but opposite orientations. If this happens, both the new triangle, and the one it
+// matched with are removed (because they describe a degenerate piece of volume).
+fn insert_topology_indices(topology_indices: &mut HashMap<HashableTriangleIndices, [u32; 3]>,
+    new_tri_idx: [u32; 3]) {
+    match topology_indices.entry(new_tri_idx.into()) {
+        Entry::Vacant(e) => {
+            let _ = e.insert(new_tri_idx);
+        }
+        Entry::Occupied(e) => {
+            fn same_orientation(a: &[u32; 3], b: &[u32; 3]) -> bool {
+                let ib = if a[0] == b[0] {
+                    0
+                } else if a[0] == b[1] {
+                    1
+                } else {
+                    2
+                };
+                a[1] == b[(ib + 1) % 3]
+            }
+
+            if !same_orientation(e.get(), &new_tri_idx) {
+                // If we are inserting two identical triangles but with mismatching
+                // orientations, we can just ignore both because they cover a degenerate
+                // 2D plane.
+                #[cfg(feature = "enhanced-determinism")]
+                let _ = e.swap_remove();
+                #[cfg(not(feature = "enhanced-determinism"))]
+                let _ = e.remove();
+            }
+        }
+    }
+}
+
+fn is_topologically_degenerate(tri_idx: [u32; 3]) -> bool {
+    tri_idx[0] == tri_idx[1]
+        || tri_idx[0] == tri_idx[2]
+        || tri_idx[1] == tri_idx[2]
+}
+
 #[cfg(feature = "wavefront")]
 #[cfg(test)]
 mod tests {
diff --git a/src/utils/point_in_poly2d.rs b/src/utils/point_in_poly2d.rs
index c57f77e4..f3587e13 100644
--- a/src/utils/point_in_poly2d.rs
+++ b/src/utils/point_in_poly2d.rs
@@ -50,18 +50,31 @@ pub fn point_in_poly2d(pt: &Point2<Real>, poly: &[Point2<Real>]) -> bool {
         let perp = dpt.perp(&seg_dir);
         winding += match (dpt.y >= 0.0, b.y > pt.y) {
             (true, true) if perp < 0.0 => 1,
-            (false, false) if perp > 0.0 => -1,
+            (false, false) if perp > 0.0 => 1,
             _ => 0,
         };
     }
 
-    winding != 0
+    winding % 2 == 1
 }
 
 #[cfg(test)]
 mod tests {
     use super::*;
 
+    #[test]
+    fn point_in_poly2d_self_intersecting() {
+        let poly = [
+            [-1.0, -1.0],
+            [0.0, 1.0],
+            [1.0, -1.0],
+            [-1.0, 0.2],
+        ]
+            .map(Point2::from);
+        let pt = Point2::origin();
+        assert!(point_in_poly2d(&pt, &poly));
+    }
+
     #[test]
     fn point_in_poly2d_concave() {
         let poly = [