diff --git a/crates/fj-kernel/src/algorithms/contains/face_point.rs b/crates/fj-kernel/src/algorithms/contains/face_point.rs
new file mode 100644
index 000000000..25b5a875b
--- /dev/null
+++ b/crates/fj-kernel/src/algorithms/contains/face_point.rs
@@ -0,0 +1,156 @@
+use fj_math::Point;
+
+use crate::{
+    algorithms::cast_ray::{CastRay, HorizontalRayToTheRight, RaySegmentHit},
+    objects::Face,
+};
+
+use super::Contains;
+
+impl Contains<Point<2>> for Face {
+    fn contains(&self, point: &Point<2>) -> bool {
+        let ray = HorizontalRayToTheRight { origin: *point };
+
+        let mut num_hits = 0;
+
+        for cycle in self.all_cycles() {
+            // We need to properly detect the ray passing the boundary at the
+            // "seam" of the polygon, i.e. the vertex between the last and the
+            // first segment. The logic in the loop properly takes care of that,
+            // as long as we initialize the `previous_hit` variable with the
+            // result of the last segment.
+            let mut previous_hit = cycle
+                .edges()
+                .last()
+                .copied()
+                .and_then(|edge| edge.cast_ray(ray));
+
+            for edge in cycle.edges() {
+                let hit = edge.cast_ray(ray);
+
+                let count_hit = match (hit, previous_hit) {
+                    (Some(RaySegmentHit::Segment), _) => {
+                        // We're hitting a segment right-on. Clear case.
+                        true
+                    }
+                    (
+                        Some(RaySegmentHit::UpperVertex),
+                        Some(RaySegmentHit::LowerVertex),
+                    )
+                    | (
+                        Some(RaySegmentHit::LowerVertex),
+                        Some(RaySegmentHit::UpperVertex),
+                    ) => {
+                        // If we're hitting a vertex, only count it if we've hit
+                        // the other kind of vertex right before.
+                        //
+                        // That means, we're passing through the polygon
+                        // boundary at where two edges touch. Depending on the
+                        // order in which edges are checked, we're seeing this
+                        // as a hit to one edge's lower/upper vertex, then the
+                        // other edge's opposite vertex.
+                        //
+                        // If we're seeing two of the same vertices in a row,
+                        // we're not actually passing through the polygon
+                        // boundary. Then we're just touching a vertex without
+                        // passing through anything.
+                        true
+                    }
+                    (Some(RaySegmentHit::Parallel), _) => {
+                        // A parallel edge must be completely ignored. Its
+                        // presence won't change anything, so we can treat it as
+                        // if it wasn't there, and its neighbors were connected
+                        // to each other.
+                        continue;
+                    }
+                    _ => {
+                        // Any other case is not a valid hit.
+                        false
+                    }
+                };
+
+                if count_hit {
+                    num_hits += 1;
+                }
+
+                previous_hit = hit;
+            }
+        }
+
+        num_hits % 2 == 1
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use fj_math::Point;
+
+    use crate::{
+        algorithms::Contains,
+        objects::{Face, Surface},
+    };
+
+    #[test]
+    fn ray_hits_vertex_while_passing_outside() {
+        let face = Face::build(Surface::xy_plane()).polygon_from_points([
+            [0., 0.],
+            [2., 1.],
+            [0., 2.],
+        ]);
+
+        assert_contains_point(face, [1., 1.]);
+    }
+
+    #[test]
+    fn ray_hits_vertex_at_cycle_seam() {
+        let face = Face::build(Surface::xy_plane())
+            .polygon_from_points([[4., 2.], [0., 4.], [0., 0.]])
+            .with_hole([[1., 1.], [2., 1.], [1., 3.]]);
+
+        assert_contains_point(face, [1., 2.]);
+    }
+
+    #[test]
+    fn ray_hits_vertex_while_staying_inside() {
+        let face = Face::build(Surface::xy_plane()).polygon_from_points([
+            [0., 0.],
+            [2., 1.],
+            [3., 0.],
+            [3., 4.],
+        ]);
+
+        assert_contains_point(face, [1., 1.]);
+    }
+
+    #[test]
+    fn ray_hits_parallel_edge() {
+        // Ray passes face boundary at a vertex.
+        let face = Face::build(Surface::xy_plane()).polygon_from_points([
+            [0., 0.],
+            [2., 1.],
+            [3., 1.],
+            [0., 2.],
+        ]);
+        assert_contains_point(face, [1., 1.]);
+
+        // Ray hits a vertex, but doesn't pass face boundary there.
+        let face = Face::build(Surface::xy_plane()).polygon_from_points([
+            [0., 0.],
+            [2., 1.],
+            [3., 1.],
+            [4., 0.],
+            [4., 5.],
+        ]);
+        assert_contains_point(face, [1., 1.]);
+    }
+
+    fn assert_contains_point(
+        face: impl Into<Face>,
+        point: impl Into<Point<2>>,
+    ) {
+        let face = face.into();
+        let point = point.into();
+
+        assert!(face.contains(&point));
+    }
+}
diff --git a/crates/fj-kernel/src/algorithms/contains/mod.rs b/crates/fj-kernel/src/algorithms/contains/mod.rs
index 81c219f08..e7ed67a27 100644
--- a/crates/fj-kernel/src/algorithms/contains/mod.rs
+++ b/crates/fj-kernel/src/algorithms/contains/mod.rs
@@ -1,3 +1,5 @@
+mod face_point;
+
 /// Test whether an object or shape contains another
 pub trait Contains<T> {
     /// Test whether an object or shape contains another