from manim import * import numpy as np import itertools config.frame_width = 16 config.frame_height = 9 config.pixel_width = 1920 config.pixel_height = 1080 config.background_color = "#0B0E14" SAFE_X = config.frame_width / 2 - 0.35 SAFE_Y = config.frame_height / 2 - 0.35 SAFE_Y_BOTTOM = config.frame_height / 2 - 0.7 S = 2.3 CUBE_SHIFT = np.array([-1.0, -0.5, 0.0]) VERTS = [np.array(v, dtype=float) for v in itertools.product([0, 1], repeat=3)] EDGES = [(a, b) for a in range(8) for b in range(a + 1, 8) if abs(np.sum(np.abs(VERTS[a] - VERTS[b])) - 1) < 1e-9] def to_scene(p): return np.array(p) * S - np.array([S / 2, S / 2, S / 2]) def section_points(k): pts = [] for a, b in EDGES: va, vb = VERTS[a], VERTS[b] sa, sb = va.sum() - k, vb.sum() - k if sa * sb < 0: t = sa / (sa - sb) pts.append(va + t * (vb - va)) else: if abs(sa) < 1e-9: pts.append(va.copy()) if abs(sb) < 1e-9: pts.append(vb.copy()) if len(pts) < 3: return [] P = np.unique(np.round(np.array(pts), 9), axis=0) if len(P) < 3: return [] c = P.mean(axis=0) n = np.array([1.0, 1.0, 1.0]) / np.sqrt(3) u = P[0] - c if np.linalg.norm(u) < 1e-9: return [] u = u / np.linalg.norm(u) v = np.cross(n, u) ang = np.arctan2((P - c) @ v, (P - c) @ u) return [P[i] for i in np.argsort(ang)] def density(k): if k <= 0 or k >= 3: return 0.0 if k < 1: return 0.5 * k * k if k < 2: return 0.5 * (-2 * k * k + 6 * k - 3) return 0.5 * (3 - k) ** 2 def required_zoom_to_fit(points, phi, theta, gamma=0.0, focal_distance=20.0, safe_half_width=None, safe_half_height=None): if safe_half_width is None: safe_half_width = config.frame_width / 2 - 0.35 if safe_half_height is None: safe_half_height = config.frame_height / 2 - 0.35 def rot_z(a): c, s = np.cos(a), np.sin(a) return np.array([[c, -s, 0], [s, c, 0], [0, 0, 1]]) def rot_x(a): c, s = np.cos(a), np.sin(a) return np.array([[1, 0, 0], [0, c, -s], [0, s, c]]) rot = rot_z(gamma) @ rot_x(-phi) @ rot_z(-theta - PI / 2) worst = 0.0 for p in points: x, y, z = rot @ np.array(p) denom = focal_distance - z factor = 1e6 if denom < 0 else focal_distance / denom worst = max(worst, abs(x * factor) / safe_half_width, abs(y * factor) / safe_half_height) return None if worst <= 1.0 else 1.0 / worst class PlaneSweep(ThreeDScene): def construct(self): k = ValueTracker(0.001) cube_edges = VGroup(*[ Line(to_scene(VERTS[a]) + CUBE_SHIFT, to_scene(VERTS[b]) + CUBE_SHIFT, color="#5A6B85", stroke_width=2.5) for a, b in EDGES ]) cube_faces = Cube(side_length=S, fill_opacity=0.06, fill_color="#7FB2FF", stroke_width=0).shift(CUBE_SHIFT) origin_dot = Dot3D(to_scene([0, 0, 0]) + CUBE_SHIFT, radius=0.055, color="#FFD166") far_dot = Dot3D(to_scene([1, 1, 1]) + CUBE_SHIFT, radius=0.055, color="#FF6B9A") def make_section(): P = section_points(k.get_value()) if len(P) < 3: return VGroup() pts = [to_scene(p) + CUBE_SHIFT for p in P] return VGroup(Polygon(*pts, fill_color="#4CE0B3", fill_opacity=0.65, stroke_color="#B8FFE9", stroke_width=4)) section = always_redraw(make_section) phi, theta = 68 * DEGREES, -55 * DEGREES pts_all = [to_scene(v) + CUBE_SHIFT for v in VERTS] zoom = required_zoom_to_fit(pts_all, phi=phi, theta=theta, safe_half_width=3.0, safe_half_height=SAFE_Y - 1.0) self.set_camera_orientation(phi=phi, theta=theta, zoom=zoom or 1.0) title = Tex(r"Slicing the unit cube by $x+y+z=k$", font_size=38) title.to_edge(UP, buff=0.4) self.add_fixed_in_frame_mobjects(title) readout = MathTex(r"k = 0.00", font_size=34, color="#FFD166") readout.to_corner(DL, buff=0.8) self.add_fixed_in_frame_mobjects(readout) shape_lab = MathTex(r"\text{sides} = 3", font_size=32, color="#4CE0B3") shape_lab.next_to(readout, UP, buff=0.4).align_to(readout, LEFT) self.add_fixed_in_frame_mobjects(shape_lab) def update_shape(m): n = len(section_points(k.get_value())) digit = 6 if n >= 5 else 3 new = MathTex(rf"\text{{sides}} = {digit}", font_size=32, color="#4CE0B3") new.next_to(readout, UP, buff=0.4).align_to(readout, LEFT) m.become(new) ax = Axes( x_range=[0, 3, 1], y_range=[0, 0.8, 0.25], x_length=3.8, y_length=2.4, axis_config={"include_tip": False, "stroke_width": 2.5, "color": "#8899AA", "font_size": 22}, ) ax.to_corner(DR, buff=0.95) xlab = MathTex("k", font_size=28, color="#8899AA").next_to(ax.x_axis, RIGHT, buff=0.2) ylab = Tex("area", font_size=26, color="#8899AA").next_to(ax.y_axis, UP, buff=0.2) self.add_fixed_in_frame_mobjects(ax, xlab, ylab) full_curve = ax.plot(density, x_range=[0, 3, 0.01], color="#3A5568", stroke_width=3) traced = always_redraw(lambda: ax.plot( density, x_range=[0.0, max(0.005, min(3.0, k.get_value())), 0.01], color="#4CE0B3", stroke_width=5)) moving_dot = always_redraw(lambda: Dot( ax.c2p(min(3.0, k.get_value()), density(min(3.0, k.get_value()))), radius=0.06, color="#FFD166")) self.add_fixed_in_frame_mobjects(full_curve, traced, moving_dot) self.play(Create(cube_edges, lag_ratio=0.05), FadeIn(cube_faces), run_time=2) self.play(FadeIn(origin_dot), FadeIn(far_dot), Write(title), run_time=1.2) self.play(FadeIn(readout), FadeIn(shape_lab), Create(full_curve), run_time=1.2) self.add(section) def update_readout(m): val = min(3.0, k.get_value()) new = MathTex(f"k = {val:.2f}", font_size=34, color="#FFD166") new.to_corner(DL, buff=0.8) m.become(new) readout.add_updater(update_readout) shape_lab.add_updater(update_shape) self.begin_ambient_camera_rotation(rate=0.10) self.play(k.animate.set_value(1.0), run_time=4, rate_func=linear) self.wait(0.4) self.play(k.animate.set_value(2.0), run_time=4, rate_func=linear) self.wait(0.4) self.play(k.animate.set_value(2.999), run_time=4, rate_func=linear) self.stop_ambient_camera_rotation() self.wait(0.5) readout.clear_updaters() shape_lab.clear_updaters() self.remove(section) closing = Tex(r"Three quadratic arcs: the Irwin--Hall density", font_size=34, color="#B8FFE9") closing.next_to(title, DOWN, buff=0.5) self.add_fixed_in_frame_mobjects(closing) self.play(FadeOut(readout), FadeOut(shape_lab), Write(closing), run_time=1.5) self.wait(1.5)

Plane Intersection