From f58a15aaaaaae5fdf8be1dba4d47198157156eb6 Mon Sep 17 00:00:00 2001 From: Sujay Garlanka Date: Mon, 2 Jun 2025 16:16:38 -0400 Subject: [PATCH] Add marching cubes --- Geometry.ipynb | 4 +- animations/code/modeling/marching_cubes.py | 728 ++++++++++++++++++++ animations/media/modeling/MarchingCubes.mp4 | Bin 0 -> 12010377 bytes 3 files changed, 731 insertions(+), 1 deletion(-) create mode 100644 animations/code/modeling/marching_cubes.py create mode 100644 animations/media/modeling/MarchingCubes.mp4 diff --git a/Geometry.ipynb b/Geometry.ipynb index f6492fe..c3da16a 100644 --- a/Geometry.ipynb +++ b/Geometry.ipynb @@ -6006,7 +6006,9 @@ "Occupancy grids are often converted to triangle meshes by outputting a\n", "rectangle when an occupied cell borders an unoccupied one. This has the\n", "effect of creating a jagged \"stair step\" appearance because each face is\n", - "axis-aligned.\n" + "axis-aligned. See the animation below for an example.\n", + "\n", + "\n" ] }, { diff --git a/animations/code/modeling/marching_cubes.py b/animations/code/modeling/marching_cubes.py new file mode 100644 index 0000000..b9ed073 --- /dev/null +++ b/animations/code/modeling/marching_cubes.py @@ -0,0 +1,728 @@ +from manimlib import * +from klampt.math import so3 + +manim_config.camera.background_color = WHITE + +class Cube(Group): + # https://sketchfab.com/3d-models/15-cases-marching-cubes-representation-78927bfd96694fccbc064bb5fe820f3d + # https://www.cs.carleton.edu/cs_comps/0405/shape/marching_cubes.html + # v7_______e6_____________v6 + # /| /| + # / | / | + # e7/ | e5/ | + # /___|______e4_________/ | + # v4| | |v5 |e10 + # | | | | + # | |e11 |e9 | + # e8| | | | + # | |_________________|___| + # | / v3 e2 | /v2 + # | / | / + # | /e3 | /e1 + # |/_____________________|/ + # v0 e0 v1 + + CUBE_VERTICES = np.array([ + [-0.5, -0.5, -0.5], # 0 + [ 0.5, -0.5, -0.5], # 1 + [ 0.5, 0.5, -0.5], # 2 + [-0.5, 0.5, -0.5], # 3 + [-0.5, -0.5, 0.5], # 4 + [ 0.5, -0.5, 0.5], # 5 + [ 0.5, 0.5, 0.5], # 6 + [-0.5, 0.5, 0.5] # 7 + ]) + + # Edge → (start corner index, end corner index) + CUBE_EDGES = [ + (0, 1), (1, 2), (2, 3), (3, 0), # bottom face + (4, 5), (5, 6), (6, 7), (7, 4), # top face + (0, 4), (1, 5), (2, 6), (3, 7) # verticals + ] + + def __init__(self, scene, scale=1.0, vertex_idx=[], triangles=[]): + super().__init__() + self.scene = scene + self.edges = [] + self.vertices = [] + self.vertex_idx = [] + self.scale_val = scale + self.rotation_matrix = so3.identity() + + self.triangles = [] + self.triangles_idx = [] + self._create_cube() + self.add_triangles(triangles) + self.add_vertices(vertex_idx) + + + def _create_cube(self): + # Draw cube edges + for i, j in Cube.CUBE_EDGES: + edge = Line( + Cube.CUBE_VERTICES[i] * self.scale_val, + Cube.CUBE_VERTICES[j] * self.scale_val, + color=BLACK, + stroke_width=1.0 * self.scale_val, + ) + self.edges.append(edge) + self.add(edge) + + def clear_vertices(self, animation_speed=0.0): + # Clear previous vertices + for dot in self.vertices: + self.remove(dot) + + if animation_speed > 0.0: + self.scene.play( + *[FadeOut(dot) for dot in self.vertices], + run_time=animation_speed + ) + else: + self.scene.remove(*self.vertices) + + self.vertices = [] + self.vertex_idx = [] + + def add_vertices(self, vertex_idx, animation_speed=0.0): + self.vertex_idx = vertex_idx.copy() + + # Visualize corners + for idx, pos in enumerate(Cube.CUBE_VERTICES): + o = 1.0 if idx in self.vertex_idx else 0.0 + r = 0.05 * self.scale_val + dot = Sphere(radius=r, opacity=o, color="#62AFE0") + # Get the rotation frame of the group + dot_pos = np.array(so3.apply(self.rotation_matrix, pos)) * self.scale_val + self.get_center() + dot.move_to(dot_pos) + self.vertices.append(dot) + + if animation_speed > 0.0: + self.scene.play( + *[FadeIn(dot) for dot in self.vertices], + run_time=animation_speed + ) + self.add(*self.vertices) + else: + self.add(*self.vertices) + + def add_triangles(self, triangles): + self.triangles_idx = triangles.copy() + + for t in triangles: + triangle_points = [] + for edge in t: + e = Cube.CUBE_EDGES[edge] + v1 = Cube.CUBE_VERTICES[e[0]] * self.scale_val + v2 = Cube.CUBE_VERTICES[e[1]] * self.scale_val + triangle_points.append((v1 + v2)/2.0) + triangle = Polygon(triangle_points[0], triangle_points[1], triangle_points[2], fill_color=YELLOW, fill_opacity=1.0, stroke_color=BLACK, stroke_width=1.0) + self.triangles.append(triangle) + self.add(triangle) + + def get_vertex_locations(self): + return Cube.CUBE_VERTICES[self.vertex_idx] + + def rotate( + self, + angle, + axis, + **kwargs + ): + super().rotate(angle, axis=axis, **kwargs) + axis = np.array(axis)/np.linalg.norm(axis) + R = so3.rotation(axis, angle) + self.rotation_matrix = so3.mul(R, self.rotation_matrix) + + def scale(self, scale): + super().scale(scale) + self._scale(scale) + + def _scale(self, scale): + self.scale_val *= scale + + def custom_copy(self): + # Create a new cube with the same parameters + new_cube = Cube( + self.scene, + scale=self.scale_val, + ) + new_cube.rotation_matrix = self.rotation_matrix.copy() + new_cube.clear_vertices() + new_cube.add_vertices(self.vertex_idx) + new_cube.add_triangles(self.triangles_idx) + new_cube.move_to(self.get_center()) + + return new_cube + +class MarchingCubes(InteractiveScene): + def construct(self): + # Animation script + + # Show 256 combos + # Show 15 base cases rotating around + # Minimize to the top banner + # Show a few cubes intersecting + # - Find the cube in the table that shows in the top banner + # - Show the cube rotating + # - Show the cube moving to the location of the intersection + # Fast forward to the end without the intersection animations + + # Set up camera orientation + self.camera.frame.reorient(phi_degrees=90, theta_degrees=0) + + # Initial orientation + self.prev_ori_camera = self.camera.frame.get_orientation() + + self.lookup_table = self._create_lookup_table() + self._animate_different_corners() + self._show_base_cases() + + # Sphere + sphere = Sphere(radius=1, color=BLUE, opacity=0.2) + self.shape = sphere + + # Axes + axes = ThreeDAxes( + x_range=(-3, 3, 1.0), + y_range=(-3, 3, 1.0), + z_range=(-2, 2, 1.0) + ) + axes.set_color(GREY) + self.add(axes) + self.add(sphere) + + self._sample_marching_cubes() + self._write_info( + """ + The smaller the cubes, the more accurate the approximation as illustrated by the following example. + """, + {} + ) + self._full_marching_cubes(0.2) + + + def _animate_different_corners(self): + cube = Cube(self, 3.0) + self.add(cube) + + # Define an updater function + def rotate_cube(mob, dt): + mob.rotate(PI/4 * dt, axis=OUT) # rotate around z-axis + + # Add the updater + cube.add_updater(rotate_cube) + combos = [ + [0, 1, 2, 3], + [0, 3, 4, 7], + [0], + [0, 3, 5, 6], + [0, 1, 2], + [0, 5], + [0, 1, 2, 4], + [0, 1, 5, 6], + ] + self._write_info( + """ + When a surface intersects a cube, some vertices are inside the surface and others are outside. + There are 256 different ways this can happen, some of which are shown here. + """, + {"256": BLUE} + ) + + for combo in combos: + cube.clear_vertices(animation_speed=0.0) + cube.add_vertices(combo, animation_speed=0.0) + self.wait(1.5) + + # Remove the updater + cube.remove_updater(rotate_cube) + + self.play(FadeOut(cube), run_time=1) + self._delete_info() + + + def _show_base_cases(self): + # Show the base cases + rows = 3 + cols = 5 + spacing = 1.8 + + cubes = [] + # Arrange manually + for i, cube in enumerate(self.lookup_table): + row = i // cols + col = i % cols + x = (col - (cols - 1)/2) * spacing + z = - (row - (rows - 1)/2) * spacing + cube.move_to(np.array([x, 0, z])) + cubes.append(cube) + + self.play(*[FadeIn(cube) for cube in cubes]) + self._write_info( + """ + The 256 combinations can be reduced to 15 base cases, which are shown here. Each base case + also has a polygonal surface that can be used to approximate the surface that contains the + highlighted vertices. + """, + {"256": BLUE, "15": RED} + ) + + # Create target arrangement + target_group = Group(*cubes).copy() + target_group.arrange( + direction=RIGHT, + buff=1.0 # spacing between cubes + ).scale(0.4) # scale down the entire group + + target_group.move_to([0, 0, 3]) # move to top of screen + + # Animate each cube to its target position + self.play( + *[ + Transform(cube, target) + for cube, target in zip(cubes, target_group) + ], + run_time=2 + ) + + # Manually update scale. Hacky, but it works. + for cube in cubes: + cube._scale(0.4/cube.scale_val) + + self._delete_info() + + # Create and animate a large centered copy of the first cube + large_cube = (cubes[1]).custom_copy() + self.add(large_cube) + self.play( + large_cube.animate.scale(5).move_to(ORIGIN), + run_time=1.5 + ) + large_cube._scale(5) + + # Show rotation + self._write_info( + """ + The 15 base cases can be used for mirrored vertices and can be rotated to satisfy all 256 combinations. + """, + { "15": RED, "256": BLUE} + ) + self._delete_info() + + self._write_info( + """ + The base cube can be rotated around 6 different axes and 4 different angles. Some rotations are redundant. + The following animation shows how a cube rotated around different axes can match different vertex + combinations. + """, + {"6": BLUE, "4": RED} + ) + + + large_cube.add_updater(lambda m, dt: m.rotate((PI/2)/(1.0/dt) if dt > 0.0 else 0.0, axis=OUT) ) + self.wait(1.0) + large_cube.clear_updaters() + + self.wait(0.5) + + large_cube.add_updater(lambda m, dt: m.rotate((PI/2)/(1.0/dt) if dt > 0.0 else 0.0, axis=UP)) + self.wait(1.0) + large_cube.clear_updaters() + + self.wait(0.5) + + large_cube.add_updater(lambda m, dt: m.rotate((PI/2)/(1.0/dt) if dt > 0.0 else 0.0, axis=RIGHT)) + self.wait(1.0) + large_cube.clear_updaters() + + self.wait(0.5) + + large_cube.add_updater(lambda m, dt: m.rotate((2 * PI / 3)/(1.0/dt) if dt > 0.0 else 0.0, axis=np.array([1.0, 1.0, 1.0]))) + self.wait(1.0) + large_cube.clear_updaters() + + self.wait(0.5) + self._delete_info() + + # Show mirroring + self._write_info( + """ + When mirroring or swapping the inside and outside vertices, the base case can still be used since the approximated surface + is the same. + """, + {} + ) + mirror_idx = {0, 1, 2, 3, 4, 5, 6, 7} - set(large_cube.vertex_idx) + large_cube.clear_vertices(animation_speed=3.0) + large_cube.add_vertices(mirror_idx, animation_speed=3.0) + self._delete_info() + + self.play( + FadeOut(large_cube), + run_time=1 + ) + + + def _sample_marching_cubes(self): + scale = 0.5 + points = [ + [1, 0, 0], + [1, 0, 0.5], + [1, -0.5, 0.5], + [1, -0.5, 0], + ] + + # Move camera to a new view cinematically + self.play( + self.camera.frame.animate + .scale(0.23) + .move_to([1.3, -1, 0.5]) + .reorient( + phi_degrees=70, # Tilt down + theta_degrees=38 # Rotate around + ), # Pull back and shift perspective + run_time=2, + rate_func=smooth + ) + self._write_info(""" + Marching cubes works by finding the vertices of the cube inside the surface at a specific coordinate. + Then finding the base case that when rotated or having its vertices mirrored, matches the vertices + of the cube at the specific coordinate. Finally, the base case is placed at the specific coordinate. + """, + {} + ) + self._delete_info() + cubes = [] + for i, p in enumerate(points): + vertex_idx = self._find_cube_at_point(*p, scale) + cube = Cube(self, scale) + cube.move_to(p) + cube.clear_vertices() + cube.add_vertices(vertex_idx, animation_speed=0.0) + self.add(cube) + + self.play( + cube.animate.move_to([2, 0, 0]), + run_time=2 + ) + matching_cube, axis, angle = self._find_matching_cube(vertex_idx) + matching_cube.scale(cube.scale_val/matching_cube.scale_val) + self.add(matching_cube) + self.play( + matching_cube.animate.move_to([2, 0, 1]), + run_time=2 + ) + self.play( + Rotate(matching_cube, angle, axis), + run_time=2 + ) + self.play( + FadeOut(cube), + run_time=1 + ) + self.play( + matching_cube.animate.move_to(p), + run_time=2 + ) + cubes.append(matching_cube) + if i == len(points)-2: + self.play( + self.camera.frame.animate + .scale(1.0/self.camera.frame.get_scale()) + .move_to([0.0, 0.0, 0.0]) + .reorient( + phi_degrees=90, # Tilt down + theta_degrees=0 # Rotate around + ), # Pull back and shift perspective + run_time=2, + rate_func=smooth + ) + self.play( + *[FadeOut(cube) for cube in cubes], + run_time=1 + ) + self._write_info( + """ + The final product of applying the marching cubes algorithm in a 3D grid that covers the surface is the following. + """, + {} + ) + self._full_marching_cubes(0.5) + self._delete_info() + + + def _full_marching_cubes(self, scale): + cubes = [] + # Create a 3D grid of points + x = np.arange(-1, 2, scale) + y = np.arange(-1, 2, scale) + z = np.arange(-1, 2, scale) + X, Y, Z = np.meshgrid(x, y, z) + + for i in range(X.shape[0]): + for j in range(X.shape[1]): + for k in range(X.shape[2]): + vertex_idx = self._find_cube_at_point(X[i, j, k], Y[i, j, k], Z[i, j, k], scale) + if len(vertex_idx) == 0 or len(vertex_idx) == 8: + continue + matching_cube, axis, angle = self._find_matching_cube(vertex_idx) + + matching_cube.scale(scale/matching_cube.scale_val) + matching_cube.rotate(angle, axis=axis) + matching_cube.move_to([X[i, j, k], Y[i, j, k], Z[i, j, k]]) + cubes.append(matching_cube) + + self.play(*[FadeIn(cube) for cube in cubes], run_time=1) + # Fade out the edges and vertices of the cubes + for cube in cubes: + cube.clear_vertices() + self.play(*[FadeOut(e) for cube in cubes for e in cube.edges], run_time=1) + for cube in cubes: + for edge in cube.edges: + cube.remove(edge) + self.play( + self.camera.frame.animate + .scale(0.5/self.camera.frame.get_scale()) + .reorient( + phi_degrees=70, + theta_degrees=0 + ), + run_time=2, + rate_func=linear + ) + self.camera.frame.add_updater(lambda m, dt: m.increment_theta(2.0*PI/(5.0/dt)) if dt > 0.0 else 0.0) + self.wait(5) + self.camera.frame.clear_updaters() + self.play(*[FadeOut(cube) for cube in cubes], run_time=1) + + + def _find_cube_at_point(self, x, y, z, scale): + vertex_locations_scaled = Cube.CUBE_VERTICES * scale + # Check if the point is inside the sphere + vertex_idx = [] + for index, v in enumerate(vertex_locations_scaled): + if ((x + v[0])**2 + (y + v[1])**2 + (z + v[2])**2)**0.5 < self.shape.radius: + vertex_idx.append(index) + if len(vertex_idx) > 4: + # mirror the vertices + vertex_idx = {0, 1, 2, 3, 4, 5, 6, 7} - set(vertex_idx) + return vertex_idx + + + def _find_matching_cube(self, vertex_idx): + # 24 possible rotations + cube_rotations = [ + # Identity + (np.array([0, 0, 1]), 0), + + # 90°, 180°, 270° about X + (np.array([1, 0, 0]), PI / 2), + (np.array([1, 0, 0]), PI), + (np.array([1, 0, 0]), 3 * PI / 2), + + # 90°, 180°, 270° about Y + (np.array([0, 1, 0]), PI / 2), + (np.array([0, 1, 0]), PI), + (np.array([0, 1, 0]), 3 * PI / 2), + + # 90°, 180°, 270° about Z + (np.array([0, 0, 1]), PI / 2), + (np.array([0, 0, 1]), PI), + (np.array([0, 0, 1]), 3 * PI / 2), + + # 120°, 240° about body diagonals + (np.array([1, 1, 1]), 2 * PI / 3), + (np.array([1, 1, 1]), 4 * PI / 3), + (np.array([-1, 1, 1]), 2 * PI / 3), + (np.array([-1, 1, 1]), 4 * PI / 3), + (np.array([1, -1, 1]), 2 * PI / 3), + (np.array([1, -1, 1]), 4 * PI / 3), + (np.array([1, 1, -1]), 2 * PI / 3), + (np.array([1, 1, -1]), 4 * PI / 3), + + # 180° about face diagonals (edge centers) + (np.array([0, 1, 1]), PI), + (np.array([0, -1, 1]), PI), + (np.array([1, 0, 1]), PI), + (np.array([-1, 0, 1]), PI), + (np.array([1, 1, 0]), PI), + (np.array([-1, 1, 0]), PI), + ] + + # Find the cube in the lookup table that matches the given vertices + vertices = [tuple(Cube.CUBE_VERTICES[vert]) for vert in vertex_idx] + for cube in self.lookup_table: + for axis, angle in cube_rotations: + rotated_cubeVs = self._rotate_points(cube.get_vertex_locations(), axis, angle) + if set(vertices) == set(rotated_cubeVs): + return (cube.custom_copy(), axis, angle) + + def _rotate_points(self, points, axis, angle): + matrix = rotation_matrix(angle, axis) + return [tuple(np.round(matrix.dot(p), 2)) for p in points] + + def _create_lookup_table(self): + baseCubes = [ + { + "vertices": [0, 2, 3, 6], + "triangles": [ + [0, 8, 11], + [0, 1, 5], + [0, 11, 5], + [11, 5, 6] + ] + }, + { + "vertices": [0, 4, 6, 2], + "triangles": [ + [0, 3, 7], + [7, 4, 0], + [1, 2, 6], + [6, 5, 1] + ] + }, + { + "vertices": [1, 4, 6], + "triangles": [ + [8, 4, 7], + [0, 1, 9], + [10, 5, 6] + ] + }, + { + "vertices": [0, 1, 6], + "triangles": [ + [3, 1, 8], + [1, 8, 9], + [10, 5, 6] + ] + }, + { + "vertices": [0, 6], + "triangles": [ + [8, 0, 3], + [10, 5, 6], + ] + }, + { + "vertices": [1, 2, 3, 7], + "triangles": [ + [3, 0, 7], + [0, 10, 9], + [0, 10, 7], + [7, 6, 10] + ] + }, + { + "vertices": [0, 2, 3, 7], + "triangles": [ + [8, 7, 6], + [8, 6, 0], + [0, 6, 10], + [0, 1, 10] + ] + }, + { + "vertices": [0, 2, 5, 7], + "triangles": [ + [0, 3, 8], + [6, 7, 11], + [4, 9, 6], + [1, 2, 10] + ] + }, + { + "vertices": [1, 2, 3, 4], + "triangles": [ + [8, 7, 4], + [0, 3, 9], + [3, 11, 9], + [9, 10, 11] + ] + }, + { + "vertices": [0, 1, 2, 3], + "triangles": [ + [8, 9, 11], + [9, 10, 11] + ] + }, + { + "vertices": [1, 2, 3], + "triangles": [ + [9, 10, 11], + [3, 11, 9], + [0, 3, 9] + ] + }, + { + "vertices": [0, 5], + "triangles": [ + [0, 3, 8], + [4, 5, 9] + ] + }, + { + "vertices": [0, 1], + "triangles":[ + [3, 1, 8], + [8, 9, 1] + ] + }, + { + "vertices": [0], + "triangles":[ + [3, 0, 8], + ] + }, + { + "vertices": [], + "triangles":[] + }, + ] + table = [] + for c in baseCubes: + cube = Cube(self, 1.0, c["vertices"], c["triangles"]) + table.append(cube) + return table + + def _write_info(self, text, coloring): + # Create a Text object with the given text + words = text.split() + lines = [] + current_line = "" + max_line_length = 90 # Maximum length of each line + + for word in words: + if len(current_line) + len(word) + 1 <= max_line_length: + current_line += word + " " + else: + lines.append(current_line.rstrip()) + current_line = word + " " + + if current_line: + lines.append(current_line.rstrip()) + + # Join lines with LaTeX line break + final_text = "\n".join(lines) + text_obj = Text( + final_text, + font="Arial", + font_size=30, + ) + text_obj.set_color(BLACK) + + text_obj.move_to(DOWN*3.3 + LEFT*6, aligned_edge=LEFT) + text_obj.fix_in_frame(True) + for word, color in coloring.items(): + text_obj.set_color_by_text(word, color) + + self.text_obj = text_obj + # Add the text to the scene + for letter in self.text_obj: + self.play(FadeIn(letter), run_time=0.05) + + def _delete_info(self): + # Remove the text from the scene + self.play(FadeOut(self.text_obj)) \ No newline at end of file diff --git a/animations/media/modeling/MarchingCubes.mp4 b/animations/media/modeling/MarchingCubes.mp4 new file mode 100644 index 0000000000000000000000000000000000000000..d864476c393550a779621110e67f412f9be2f4c7 GIT binary patch literal 12010377 zcmZ^~1yEf+aRPzUplN003z2>g8zV;$Q~=Kmz_`DrV(~# zAlDC_@31iM?^Wwk$K403vaRWiE5KEX^9xUQZcZQ*(A2>h3}pKtojKT8d4U{kU^YH; zz7K&c^M?Y9qKcFZ6DLqiOX5S*3~c%#kZ^GHvIU#F0@+zvIhfd4**QL#maeXj{46XU 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