diff --git a/src/freeroute/geometry/__init__.py b/src/freeroute/geometry/__init__.py
index 30944c1..b0d45f7 100644
--- a/src/freeroute/geometry/__init__.py
+++ b/src/freeroute/geometry/__init__.py
@@ -22,6 +22,8 @@ from .float_point import FloatPoint
 from .limits import CRIT_DOUBLE, CRIT_INT
 from .line import Line
 from .point import IntPoint, Point, RationalPoint, point, rational_point
+from .polygon import Polygon
+from .polygon_shape import PolygonShape
 from .side import Side, Signum
 from .simplex import Simplex
 from .tile import TileShape
@@ -47,4 +49,6 @@ __all__ = [
     "IntBox",
     "TileShape",
     "Simplex",
+    "Polygon",
+    "PolygonShape",
 ]
diff --git a/src/freeroute/geometry/line.py b/src/freeroute/geometry/line.py
index 61777a9..41ae3bd 100644
--- a/src/freeroute/geometry/line.py
+++ b/src/freeroute/geometry/line.py
@@ -198,6 +198,28 @@ class Line:
             result = self.side_of(line_1.intersection(line_2))
         return result
 
+    def function_value_approx(self, x: float) -> float:
+        """Approximate ``y`` on this (non-vertical) line at abscissa ``x``."""
+        p1 = self.a.to_float()
+        p2 = self.b.to_float()
+        dx = p2.x - p1.x
+        if dx == 0:
+            return 0.0
+        dy = p2.y - p1.y
+        det = p1.x * p2.y - p2.x * p1.y
+        return (dy * x - det) / dx
+
+    def function_in_y_value_approx(self, y: float) -> float:
+        """Approximate ``x`` on this (non-horizontal) line at ordinate ``y``."""
+        p1 = self.a.to_float()
+        p2 = self.b.to_float()
+        dy = p2.y - p1.y
+        if dy == 0:
+            return 0.0
+        dx = p2.x - p1.x
+        det = p1.x * p2.y - p2.x * p1.y
+        return (dx * y + det) / dy
+
     # --- intersection (exact) ----------------------------------------------
 
     def intersection(self, other: Line) -> Point:
diff --git a/src/freeroute/geometry/polygon.py b/src/freeroute/geometry/polygon.py
new file mode 100644
index 0000000..adb0795
--- /dev/null
+++ b/src/freeroute/geometry/polygon.py
@@ -0,0 +1,77 @@
+"""``Polygon`` — a normalized corner list (no duplicate or collinear corners).
+
+Ports ``geometry/planar/Polygon.java``. Used as the corner normalizer behind
+:class:`~freeroute.geometry.polygon_shape.PolygonShape`.
+
+**Arithmetic model:** corner de-duplication and collinearity removal are
+**exact** (exact ``Point.side_of``); the winding-number test is approximate
+(sum of ``angle_approx``), exactly as upstream.
+"""
+
+from __future__ import annotations
+
+import math
+
+from .point import Point
+from .side import Side
+
+__all__ = ["Polygon"]
+
+
+class Polygon:
+    """A list of corners with no two consecutive equal and no three collinear."""
+
+    __slots__ = ("corners",)
+
+    def __init__(self, points) -> None:
+        corners: list[Point] = list(points)
+        corner_removed = True
+        while corner_removed:
+            corner_removed = False
+            if not corners:
+                break
+            # remove consecutive duplicate points
+            deduped: list[Point] = [corners[0]]
+            for pt in corners[1:]:
+                if pt != deduped[-1]:
+                    deduped.append(pt)
+            if len(deduped) != len(corners):
+                corner_removed = True
+            corners = deduped
+            # remove a point collinear with its previous and next point
+            if len(corners) >= 3:
+                for i in range(1, len(corners) - 1):
+                    if corners[i].side_of(corners[i - 1], corners[i + 1]) == Side.COLLINEAR:
+                        del corners[i]
+                        corner_removed = True
+                        break
+        self.corners: list[Point] = corners
+
+    def corner_array(self) -> list[Point]:
+        return list(self.corners)
+
+    def revert_corners(self) -> Polygon:
+        return Polygon(list(reversed(self.corners)))
+
+    def winding_number_after_closing(self) -> int:
+        """Winding number of the closed polygon: >0 counter-clockwise, <0 clockwise."""
+        arr = self.corners
+        if len(arr) < 2:
+            return 0
+        first_side = arr[1].difference_by(arr[0])
+        prev_side = first_side
+        corner_count = len(arr)
+        if arr[0] == arr[corner_count - 1]:
+            corner_count -= 1
+        angle_sum = 0.0
+        for i in range(1, corner_count + 1):
+            if i == corner_count - 1:
+                next_side = arr[0].difference_by(arr[i])
+            elif i == corner_count:
+                next_side = first_side
+            else:
+                next_side = arr[i + 1].difference_by(arr[i])
+            angle_sum += prev_side.angle_approx(next_side)
+            prev_side = next_side
+        angle_sum /= 2.0 * math.pi
+        return round(angle_sum)
diff --git a/src/freeroute/geometry/polygon_shape.py b/src/freeroute/geometry/polygon_shape.py
new file mode 100644
index 0000000..5ee36a6
--- /dev/null
+++ b/src/freeroute/geometry/polygon_shape.py
@@ -0,0 +1,262 @@
+"""``PolygonShape`` — a simple (possibly concave) polygon and ``split_to_convex``.
+
+Ports ``geometry/planar/PolygonShape.java``, in particular the recursive
+``split_to_convex`` that decomposes a simple polygon into convex
+:class:`~freeroute.geometry.simplex.Simplex` tiles — the decomposition the
+router's search tree relies on.
+
+**Arithmetic model:** corner orientation / convexity tests are **exact**
+(exact ``Point.side_of``); the division-point search is **approximate** (it uses
+``float`` line evaluations and rounds the split point to an integer corner, as
+upstream). Point containment is exact and delegated to the convex tiles.
+
+The recursion starts the concave-corner search at corner 0 (deterministic);
+upstream seeds a fixed PRNG to vary the start. The choice only affects *which*
+valid decomposition is produced, never its correctness.
+"""
+
+from __future__ import annotations
+
+from .float_point import FloatPoint
+from .line import Line
+from .point import Point
+from .polygon import Polygon
+from .side import Side
+from .simplex import Simplex
+
+__all__ = ["PolygonShape"]
+
+
+class PolygonShape:
+    """A simple polygon, normalized to a canonical counter-clockwise corner list."""
+
+    __slots__ = ("corners", "_convex_pieces")
+
+    def __init__(self, points) -> None:
+        poly = Polygon(points)
+        if poly.winding_number_after_closing() < 0:
+            poly = poly.revert_corners()  # make counter-clockwise
+        curr = poly.corner_array()
+        last = len(curr) - 1
+        if last > 0 and curr[0] == curr[last]:
+            last -= 1
+        if last >= 2 and curr[last].side_of(curr[last - 1], curr[0]) == Side.COLLINEAR:
+            last -= 1
+        first = 0
+        if last - first >= 2 and curr[0].side_of(curr[1], curr[last]) == Side.COLLINEAR:
+            first += 1
+        # rotate so the corner with the lowest y (then lowest x) comes first
+        start = first
+        start_c = curr[start].to_float()
+        for i in range(start + 1, last + 1):
+            c = curr[i].to_float()
+            if c.y < start_c.y or (c.y == start_c.y and c.x < start_c.x):
+                start = i
+                start_c = c
+        self.corners: list[Point] = curr[start : last + 1] + curr[first:start]
+        self._convex_pieces: list[Simplex] | None = None
+
+    def __repr__(self) -> str:
+        return f"PolygonShape(<{len(self.corners)} corners>)"
+
+    # --- accessors ----------------------------------------------------------
+
+    def corner(self, no: int) -> Point:
+        return self.corners[no]
+
+    def border_line_count(self) -> int:
+        return len(self.corners)
+
+    def border_line(self, no: int) -> Line:
+        nxt = self.corners[0] if no == len(self.corners) - 1 else self.corners[no + 1]
+        return Line(self.corners[no], nxt)
+
+    def dimension(self) -> int:
+        n = len(self.corners)
+        if n == 0:
+            return -1
+        if n == 1:
+            return 0
+        if n == 2:
+            return 1
+        return 2
+
+    def is_empty(self) -> bool:
+        return len(self.corners) == 0
+
+    def area(self) -> float:
+        if self.dimension() <= 2 and len(self.corners) < 3:
+            return 0.0
+        result = 0.0
+        corners = [c.to_float() for c in self.corners]
+        n = len(corners)
+        prev_corner = corners[n - 2]
+        curr_corner = corners[n - 1]
+        for i in range(n):
+            next_corner = corners[i]
+            result += curr_corner.x * (next_corner.y - prev_corner.y)
+            prev_corner = curr_corner
+            curr_corner = next_corner
+        return 0.5 * abs(result)
+
+    # --- convexity ----------------------------------------------------------
+
+    def is_convex(self) -> bool:
+        corners = self.corners
+        n = len(corners)
+        if n <= 2:
+            return True
+        prev_point = corners[n - 1]
+        curr_point = corners[0]
+        next_point = corners[1]
+        for ind in range(n):
+            if next_point.side_of(prev_point, curr_point) == Side.ON_THE_RIGHT:
+                return False
+            prev_point = curr_point
+            curr_point = next_point
+            next_point = corners[0] if ind == n - 2 else corners[(ind + 2) % n]
+        return True
+
+    # --- containment (via convex decomposition) ----------------------------
+
+    def is_outside(self, point: Point) -> bool:
+        return all(piece.is_outside(point) for piece in self.split_to_convex())
+
+    def contains(self, point: Point) -> bool:
+        return not self.is_outside(point)
+
+    def contains_inside(self, point: Point) -> bool:
+        # strictly inside some tile and not merely on a shared division edge
+        pieces = self.split_to_convex()
+        return any(p.contains_inside(point) for p in pieces)
+
+    # --- the key operation: split into convex tiles ------------------------
+
+    def split_to_convex(self) -> list[Simplex]:
+        if self._convex_pieces is None:
+            pieces = _split_recu(self.corners)
+            if pieces is None:
+                self._convex_pieces = []
+            else:
+                self._convex_pieces = [Simplex.from_corners(p) for p in pieces]
+        return self._convex_pieces
+
+
+def _split_recu(corners: list[Point]) -> list[list[Point]] | None:
+    """Recursively divide a simple polygon into convex corner lists."""
+    n = len(corners)
+    if n < 3:
+        return [corners]
+
+    # find the first concave corner
+    concave_no = -1
+    for start in range(n):
+        prev_corner = corners[start - 1]
+        curr_corner = corners[start]
+        next_corner = corners[(start + 1) % n]
+        if next_corner.side_of(prev_corner, curr_corner) == Side.ON_THE_RIGHT:
+            concave_no = start
+            break
+
+    if concave_no < 0:
+        return [corners]  # already convex
+
+    projection, corner_no_after = _division_point(corners, concave_no)
+    if projection is None:
+        return None  # self-intersecting or degenerate
+
+    split_pt = projection.round()
+
+    # first piece: concave_no .. corner_no_after (exclusive) + split point
+    count = corner_no_after - concave_no
+    if count < 0:
+        count += n
+    count += 1
+    first_arr: list[Point] = []
+    idx = concave_no
+    for _ in range(count - 1):
+        first_arr.append(corners[idx])
+        idx = (idx + 1) % n
+    first_arr.append(split_pt)
+
+    # last piece: split point + corner_no_after .. concave_no
+    count = concave_no - corner_no_after
+    if count < 0:
+        count += n
+    count += 2
+    last_arr: list[Point] = [split_pt]
+    idx = corner_no_after
+    for _ in range(1, count):
+        last_arr.append(corners[idx])
+        idx = (idx + 1) % n
+
+    c1 = _split_recu(PolygonShape(first_arr).corners)
+    if c1 is None:
+        return None
+    c2 = _split_recu(PolygonShape(last_arr).corners)
+    if c2 is None:
+        return None
+    return c1 + c2
+
+
+def _division_point(corners: list[Point], concave_no: int) -> tuple[FloatPoint | None, int]:
+    """Minimal axis-parallel division line from a concave corner (approximate)."""
+    n = len(corners)
+    concave = corners[concave_no].to_float()
+    before = corners[concave_no - 1].to_float()
+    after = corners[(concave_no + 1) % n].to_float()
+
+    search_right = before.y > concave.y or concave.y > after.y
+    search_left = before.y < concave.y or concave.y < after.y
+    search_up = before.x < concave.x or concave.x < after.x
+    search_down = before.x > concave.x or concave.x > after.x
+
+    min_dist = float("inf")
+    min_projection: FloatPoint | None = None
+    corner_no_after_min = 0
+
+    idx_after = (concave_no + 2) % n
+    corner_before = corners[idx_after - 1] if idx_after != 0 else corners[n - 1]
+    before_approx = corner_before.to_float()
+
+    for _ in range(n - 2):
+        corner_after = corners[idx_after]
+        after_approx = corner_after.to_float()
+
+        if before_approx.y != after_approx.y:  # horizontal division
+            lo_y = min(before_approx.y, after_approx.y)
+            hi_y = max(before_approx.y, after_approx.y)
+            if lo_y <= concave.y <= hi_y:
+                line = Line(corner_before, corner_after)
+                x_intersect = line.function_in_y_value_approx(concave.y)
+                dist = abs(x_intersect - concave.x)
+                ok = dist < min_dist and (
+                    (search_right and x_intersect > concave.x and concave.y <= after_approx.y)
+                    or (search_left and x_intersect < concave.x and concave.y >= after_approx.y)
+                )
+                if ok:
+                    min_dist = dist
+                    corner_no_after_min = idx_after
+                    min_projection = FloatPoint(x_intersect, concave.y)
+
+        if before_approx.x != after_approx.x:  # vertical division
+            lo_x = min(before_approx.x, after_approx.x)
+            hi_x = max(before_approx.x, after_approx.x)
+            if lo_x <= concave.x <= hi_x:
+                line = Line(corner_before, corner_after)
+                y_intersect = line.function_value_approx(concave.x)
+                dist = abs(y_intersect - concave.y)
+                ok = dist < min_dist and (
+                    (search_up and y_intersect > concave.y and concave.x >= after_approx.x)
+                    or (search_down and y_intersect < concave.y and concave.x <= after_approx.x)
+                )
+                if ok:
+                    min_dist = dist
+                    corner_no_after_min = idx_after
+                    min_projection = FloatPoint(concave.x, y_intersect)
+
+        corner_before = corner_after
+        before_approx = after_approx
+        idx_after = 0 if idx_after == n - 1 else idx_after + 1
+
+    return min_projection, corner_no_after_min
diff --git a/src/freeroute/geometry/simplex.py b/src/freeroute/geometry/simplex.py
index 6fba858..844387a 100644
--- a/src/freeroute/geometry/simplex.py
+++ b/src/freeroute/geometry/simplex.py
@@ -52,6 +52,18 @@ class Simplex(TileShape):
         arr.sort()
         return Simplex(arr)._remove_redundant_lines()
 
+    @staticmethod
+    def from_corners(corners) -> Simplex:
+        """Build a normalized simplex from the corners of a convex polygon.
+
+        Ports ``TileShape.get_instance(Point[])``: one directed border line per
+        edge (corner ``i`` to ``i+1``), then normalized.
+        """
+        corners = list(corners)
+        n = len(corners)
+        lines = [Line(corners[j], corners[(j + 1) % n]) for j in range(n)]
+        return Simplex.get_instance(lines)
+
     @staticmethod
     def empty() -> Simplex:
         return Simplex(())
diff --git a/tests/geometry/test_polygon.py b/tests/geometry/test_polygon.py
new file mode 100644
index 0000000..eccbf2d
--- /dev/null
+++ b/tests/geometry/test_polygon.py
@@ -0,0 +1,148 @@
+"""Invariant tests for Polygon / PolygonShape.split_to_convex.
+
+No external oracle exists, so these assert the decomposition invariants from the
+phase-2 brief for several concave polygons: the convex tiles' areas sum to the
+polygon area (no gaps, no overlap), a point is contained in the polygon iff it
+is contained in some tile, and no point is strictly inside more than one tile
+(interiors disjoint).
+
+Source: ``geometry/planar/{Polygon,PolygonShape}.java``.
+"""
+
+from __future__ import annotations
+
+import pytest
+
+from freeroute.geometry import IntPoint, Side
+from freeroute.geometry.polygon import Polygon
+from freeroute.geometry.polygon_shape import PolygonShape
+
+
+def pts(*coords):
+    return [IntPoint(x, y) for x, y in coords]
+
+
+# --- Polygon normalization --------------------------------------------------
+
+
+def test_polygon_removes_consecutive_duplicates():
+    poly = Polygon(pts((0, 0), (0, 0), (10, 0), (10, 10), (10, 10), (0, 10)))
+    assert len(poly.corners) == 4
+
+
+def test_polygon_removes_middle_collinear_corner():
+    # (5, 0) is collinear with its neighbours (0,0) and (10,0) -> dropped.
+    # Raw Polygon only removes *middle* collinear corners (endpoint-collinear
+    # removal is PolygonShape's job), so the trailing (0,5) is kept.
+    poly = Polygon(pts((0, 0), (5, 0), (10, 0), (10, 10), (0, 10), (0, 5)))
+    coords = [(c.x, c.y) for c in poly.corners]
+    assert (5, 0) not in coords
+    assert coords == [(0, 0), (10, 0), (10, 10), (0, 10), (0, 5)]
+
+
+def test_polygon_shape_removes_endpoint_collinear_corner():
+    # PolygonShape additionally drops the endpoint-collinear (0,5).
+    ps = PolygonShape(pts((0, 0), (5, 0), (10, 0), (10, 10), (0, 10), (0, 5)))
+    coords = {(c.x, c.y) for c in ps.corners}
+    assert coords == {(0, 0), (10, 0), (10, 10), (0, 10)}
+
+
+def test_winding_number_sign():
+    ccw = Polygon(pts((0, 0), (10, 0), (10, 10), (0, 10)))
+    cw = Polygon(pts((0, 0), (0, 10), (10, 10), (10, 0)))
+    assert ccw.winding_number_after_closing() > 0
+    assert cw.winding_number_after_closing() < 0
+
+
+# --- convexity --------------------------------------------------------------
+
+
+def test_is_convex():
+    square = PolygonShape(pts((0, 0), (10, 0), (10, 10), (0, 10)))
+    assert square.is_convex()
+    ell = PolygonShape(pts((0, 0), (60, 0), (60, 20), (20, 20), (20, 60), (0, 60)))
+    assert not ell.is_convex()
+
+
+# --- split_to_convex invariants ---------------------------------------------
+
+SHAPES = {
+    "L": pts((0, 0), (60, 0), (60, 20), (20, 20), (20, 60), (0, 60)),
+    "plus": pts(
+        (20, 0),
+        (40, 0),
+        (40, 20),
+        (60, 20),
+        (60, 40),
+        (40, 40),
+        (40, 60),
+        (20, 60),
+        (20, 40),
+        (0, 40),
+        (0, 20),
+        (20, 20),
+    ),
+    "staircase": pts((0, 0), (30, 0), (30, 10), (20, 10), (20, 20), (10, 20), (10, 30), (0, 30)),
+    "square": pts((0, 0), (40, 0), (40, 40), (0, 40)),
+}
+
+
+def _bounds(corners):
+    xs = [c.x for c in corners]
+    ys = [c.y for c in corners]
+    return min(xs) - 3, max(xs) + 3, min(ys) - 3, max(ys) + 3
+
+
+@pytest.mark.parametrize("name", list(SHAPES))
+def test_split_area_conserved(name):
+    ps = PolygonShape(SHAPES[name])
+    tiles = ps.split_to_convex()
+    assert tiles, "expected at least one convex tile"
+    assert sum(t.area() for t in tiles) == pytest.approx(ps.area())
+
+
+@pytest.mark.parametrize("name", list(SHAPES))
+def test_split_union_equals_polygon(name):
+    ps = PolygonShape(SHAPES[name])
+    tiles = ps.split_to_convex()
+    lo_x, hi_x, lo_y, hi_y = _bounds(ps.corners)
+    for x in range(lo_x, hi_x + 1):
+        for y in range(lo_y, hi_y + 1):
+            p = IntPoint(x, y)
+            in_poly = ps.contains(p)
+            in_some_tile = any(t.contains(p) for t in tiles)
+            assert in_poly == in_some_tile, (x, y)
+
+
+@pytest.mark.parametrize("name", list(SHAPES))
+def test_split_interiors_are_disjoint(name):
+    ps = PolygonShape(SHAPES[name])
+    tiles = ps.split_to_convex()
+    lo_x, hi_x, lo_y, hi_y = _bounds(ps.corners)
+    for x in range(lo_x, hi_x + 1):
+        for y in range(lo_y, hi_y + 1):
+            p = IntPoint(x, y)
+            strict = sum(1 for t in tiles if t.contains_inside(p))
+            assert strict <= 1, (x, y, strict)
+
+
+def test_convex_polygon_is_single_tile():
+    ps = PolygonShape(SHAPES["square"])
+    tiles = ps.split_to_convex()
+    assert len(tiles) == 1
+    assert tiles[0].dimension() == 2
+
+
+def test_concave_corner_detected_by_side_of():
+    # the reflex corner of the L is (20, 20): its next corner is on the right
+    ell = PolygonShape(SHAPES["L"])
+    corners = ell.corners
+    reflex_found = False
+    n = len(corners)
+    for i in range(n):
+        prev_c = corners[i - 1]
+        curr = corners[i]
+        nxt = corners[(i + 1) % n]
+        if nxt.side_of(prev_c, curr) == Side.ON_THE_RIGHT:
+            reflex_found = True
+    assert reflex_found