import numpy as np import itertools as it from helpers import * from mobject.tex_mobject import TexMobject, TextMobject, Brace from mobject import Mobject, Mobject1D from mobject.image_mobject import \ ImageMobject, MobjectFromPixelArray from topics.three_dimensions import Stars from animation import Animation from animation.transform import \ Transform, CounterclockwiseTransform, ApplyPointwiseFunction,\ FadeIn, FadeOut, GrowFromCenter, ApplyFunction, ApplyMethod, \ ShimmerIn from animation.simple_animations import \ ShowCreation, Homotopy, PhaseFlow, ApplyToCenters, DelayByOrder, \ ShowPassingFlash from animation.playground import TurnInsideOut, Vibrate from topics.geometry import \ Line, Circle, Square, Grid, Rectangle, Arrow, Dot, Point, \ Arc, FilledRectangle from topics.characters import Randolph, Mathematician from topics.functions import ParametricFunction, FunctionGraph from topics.number_line import NumberPlane from mobject.region import Region, region_from_polygon_vertices from scene import Scene from scene.zoomed_scene import ZoomedScene from brachistochrone.curves import Cycloid class Lens(Arc): CONFIG = { "radius" : 2, "angle" : np.pi/2, "color" : BLUE_B, } def __init__(self, **kwargs): digest_config(self, kwargs) Arc.__init__(self, self.angle, **kwargs) def generate_points(self): Arc.generate_points(self) self.rotate(-np.pi/4) self.shift(-self.get_left()) self.add_points(self.copy().rotate(np.pi).points) class PhotonScene(Scene): def wavify(self, mobject): result = mobject.copy() result.ingest_sub_mobjects() tangent_vectors = result.points[1:]-result.points[:-1] lengths = np.apply_along_axis( np.linalg.norm, 1, tangent_vectors ) thick_lengths = lengths.repeat(3).reshape((len(lengths), 3)) unit_tangent_vectors = tangent_vectors/thick_lengths rot_matrix = np.transpose(rotation_matrix(np.pi/2, OUT)) normal_vectors = np.dot(unit_tangent_vectors, rot_matrix) # total_length = np.sum(lengths) times = np.cumsum(lengths) nudge_sizes = 0.1*np.sin(2*np.pi*times) thick_nudge_sizes = nudge_sizes.repeat(3).reshape((len(nudge_sizes), 3)) nudges = thick_nudge_sizes*normal_vectors result.points[1:] += nudges return result def photon_run_along_path(self, path, color = YELLOW, **kwargs): photon = self.wavify(path) photon.highlight(color) return ShowPassingFlash(photon, **kwargs) class SimplePhoton(PhotonScene): def construct(self): text = TextMobject("Light") text.to_edge(UP) self.play(ShimmerIn(text)) self.play(self.photon_run_along_path( Cycloid(), rate_func = None )) self.dither() class MultipathPhotonScene(PhotonScene): CONFIG = { "num_paths" : 5 } def run_along_paths(self, **kwargs): paths = self.get_paths() colors = Color(YELLOW).range_to(WHITE, len(paths)) for path, color in zip(paths, colors): path.highlight(color) photon_runs = [ self.photon_run_along_path(path) for path in paths ] for photon_run, path in zip(photon_runs, paths): self.play( photon_run, ShowCreation( path, rate_func = lambda t : 0.9*smooth(t) ), **kwargs ) self.dither() def generate_paths(self): raise Exception("Not Implemented") class PhotonThroughLens(MultipathPhotonScene): def construct(self): self.lens = Lens() self.add(self.lens) self.run_along_paths() def get_paths(self): interval_values = np.arange(self.num_paths).astype('float') interval_values /= (self.num_paths-1.) first_contact = [ self.lens.point_from_proportion(0.4*v+0.55) for v in reversed(interval_values) ] second_contact = [ self.lens.point_from_proportion(0.3*v + 0.1) for v in interval_values ] focal_point = 2*RIGHT return [ Mobject( Line(SPACE_WIDTH*LEFT + fc[1]*UP, fc), Line(fc, sc), Line(sc, focal_point), Line(focal_point, 6*focal_point-5*sc) ).ingest_sub_mobjects() for fc, sc in zip(first_contact, second_contact) ] class PhotonOffMirror(MultipathPhotonScene): def construct(self): self.mirror = Line(*SPACE_HEIGHT*np.array([DOWN, UP])) self.mirror.highlight(GREY) self.add(self.mirror) self.run_along_paths() def get_paths(self): interval_values = np.arange(self.num_paths).astype('float') interval_values /= (self.num_paths-1) anchor_points = [ self.mirror.point_from_proportion(0.6*v+0.3) for v in interval_values ] start_point = 5*LEFT+3*UP end_points = [] for point in anchor_points: vect = start_point-point vect[1] *= -1 end_points.append(point+2*vect) return [ Mobject( Line(start_point, anchor_point), Line(anchor_point, end_point) ).ingest_sub_mobjects() for anchor_point, end_point in zip(anchor_points, end_points) ] class PhotonsInGlass(MultipathPhotonScene): def construct(self): glass = Region(lambda x, y : y < 0) self.highlight_region(glass, BLUE_E) self.run_along_paths() def get_paths(self): x, y = -3, 3 start_point = x*RIGHT + y*UP angles = np.arange(np.pi/18, np.pi/3, np.pi/18) midpoints = y*np.arctan(angles) end_points = midpoints + SPACE_HEIGHT*np.arctan(2*angles) return [ Mobject( Line(start_point, [midpoint, 0, 0]), Line([midpoint, 0, 0], [end_point, -SPACE_HEIGHT, 0]) ).ingest_sub_mobjects() for midpoint, end_point in zip(midpoints, end_points) ] class ShowMultiplePathsScene(PhotonScene): def construct(self): text = TextMobject("Which path minimizes travel time?") text.to_edge(UP) self.generate_start_and_end_points() point_a = Dot(self.start_point) point_b = Dot(self.end_point) A = TextMobject("A").next_to(point_a, UP) B = TextMobject("B").next_to(point_b, DOWN) paths = self.get_paths() for point, letter in [(point_a, A), (point_b, B)]: self.play( ShowCreation(point), ShimmerIn(letter) ) self.play(ShimmerIn(text)) curr_path = paths[0].copy() curr_path_copy = curr_path.copy().ingest_sub_mobjects() self.play( self.photon_run_along_path(curr_path), ShowCreation(curr_path_copy, rate_func = rush_into) ) self.remove(curr_path_copy) for path in paths[1:] + [paths[0]]: self.play(Transform(curr_path, path, run_time = 4)) self.dither() def generate_start_and_end_points(self): raise Exception("Not Implemented") def get_paths(self): raise Exception("Not implemented") class ShowMultiplePathsThroughLens(ShowMultiplePathsScene): def construct(self): self.lens = Lens() self.add(self.lens) ShowMultiplePathsScene.construct(self) def generate_start_and_end_points(self): self.start_point = 3*LEFT + UP self.end_point = 2*RIGHT def get_paths(self): alphas = [0.25, 0.4, 0.58, 0.75] lower_right, upper_right, upper_left, lower_left = map( self.lens.point_from_proportion, alphas ) return [ Mobject( Line(self.start_point, a), Line(a, b), Line(b, self.end_point) ).highlight(color) for (a, b), color in zip( [ (upper_left, upper_right), (upper_left, lower_right), (lower_left, lower_right), (lower_left, upper_right), ], Color(YELLOW).range_to(WHITE, 4) ) ] class ShowMultiplePathsOffMirror(ShowMultiplePathsScene): def construct(self): mirror = Line(*SPACE_HEIGHT*np.array([DOWN, UP])) mirror.highlight(GREY) self.add(mirror) ShowMultiplePathsScene.construct(self) def generate_start_and_end_points(self): self.start_point = 4*LEFT + 2*UP self.end_point = 4*LEFT + 2*DOWN def get_paths(self): return [ Mobject( Line(self.start_point, midpoint), Line(midpoint, self.end_point) ).highlight(color) for midpoint, color in zip( [2*UP, 2*DOWN], Color(YELLOW).range_to(WHITE, 2) ) ] class ShowMultiplePathsInGlass(ShowMultiplePathsScene): def construct(self): glass = Region(lambda x, y : y < 0) self.highlight_region(glass, BLUE_E) ShowMultiplePathsScene.construct(self) def generate_start_and_end_points(self): self.start_point = 3*LEFT + 2*UP self.end_point = 3*RIGHT + 2*DOWN def get_paths(self): return [ Mobject( Line(self.start_point, midpoint), Line(midpoint, self.end_point) ).highlight(color) for midpoint, color in zip( [3*LEFT, 3*RIGHT], Color(YELLOW).range_to(WHITE, 2) ) ] class MultilayeredGlass(PhotonScene, ZoomedScene): CONFIG = { "num_discrete_layers" : 5, "num_variables" : 3, "top_color" : BLUE_E, "bottom_color" : BLUE_A, "zoomed_canvas_space_shape" : (5, 5), "square_color" : GREEN_B, } def construct(self): self.cycloid = Cycloid(end_theta = np.pi) self.cycloid.highlight(YELLOW) self.top = self.cycloid.get_top()[1] self.bottom = self.cycloid.get_bottom()[1]-1 self.generate_layers() self.generate_discrete_path() photon_run = self.photon_run_along_path( self.discrete_path, run_time = 1, rate_func = rush_into ) self.continuous_to_smooth() self.add(*self.layers) self.show_layer_variables() self.play(photon_run) self.play(ShowCreation(self.discrete_path)) self.isolate_bend_points() self.clear() self.add(*self.layers) self.show_main_equation() self.ask_continuous_question() def continuous_to_smooth(self): self.add(*self.layers) continuous = self.get_continuous_background() self.add(continuous) self.dither() self.play(ShowCreation( continuous, rate_func = lambda t : smooth(1-t) )) self.remove(continuous) self.dither() def get_continuous_background(self): glass = FilledRectangle( height = self.top-self.bottom, width = 2*SPACE_WIDTH, ) glass.sort_points(lambda p : -p[1]) glass.shift((self.top-glass.get_top()[1])*UP) glass.gradient_highlight(self.top_color, self.bottom_color) return glass def generate_layer_info(self): self.layer_thickness = float(self.top-self.bottom)/self.num_discrete_layers self.layer_tops = np.arange( self.top, self.bottom, -self.layer_thickness ) top_rgb, bottom_rgb = [ np.array(Color(color).get_rgb()) for color in self.top_color, self.bottom_color ] epsilon = 1./(self.num_discrete_layers-1) self.layer_colors = [ Color(rgb = interpolate(top_rgb, bottom_rgb, alpha)) for alpha in np.arange(0, 1+epsilon, epsilon) ] def generate_layers(self): self.generate_layer_info() def create_region(top, color): return Region( lambda x, y : (y < top) & (y > top-self.layer_thickness), color = color ) self.layers = [ create_region(top, color) for top, color in zip(self.layer_tops, self.layer_colors) ] def generate_discrete_path(self): points = self.cycloid.points tops = list(self.layer_tops) tops.append(tops[-1]-self.layer_thickness) indices = [ np.argmin(np.abs(points[:, 1]-top)) for top in tops ] self.bend_points = points[indices[1:-1]] self.path_angles = [] self.discrete_path = Mobject1D( color = YELLOW, density = 3*DEFAULT_POINT_DENSITY_1D ) for start, end in zip(indices, indices[1:]): start_point, end_point = points[start], points[end] self.discrete_path.add_line( start_point, end_point ) self.path_angles.append( angle_of_vector(start_point-end_point)-np.pi/2 ) self.discrete_path.add_line( points[end], SPACE_WIDTH*RIGHT+(self.layer_tops[-1]-1)*UP ) def show_layer_variables(self): layer_top_pairs = zip( self.layer_tops[:self.num_variables], self.layer_tops[1:] ) v_equations = [] start_ys = [] end_ys = [] center_paths = [] braces = [] for (top1, top2), x in zip(layer_top_pairs, it.count(1)): eq_mob = TexMobject( ["v_%d"%x, "=", "\sqrt{\phantom{y_1}}"], size = "\\Large" ) midpoint = UP*(top1+top2)/2 eq_mob.shift(midpoint) v_eq = eq_mob.split() center_paths.append(Line( midpoint+SPACE_WIDTH*LEFT, midpoint+SPACE_WIDTH*RIGHT )) brace_endpoints = Mobject( Point(self.top*UP+x*RIGHT), Point(top2*UP+x*RIGHT) ) brace = Brace(brace_endpoints, RIGHT) start_y = TexMobject("y_%d"%x, size = "\\Large") end_y = start_y.copy() start_y.next_to(brace, RIGHT) end_y.shift(v_eq[-1].get_center()) end_y.shift(0.2*RIGHT) v_equations.append(v_eq) start_ys.append(start_y) end_ys.append(end_y) braces.append(brace) for v_eq, path, time in zip(v_equations, center_paths, [2, 1, 0.5]): photon_run = self.photon_run_along_path( path, rate_func = None ) self.play( ShimmerIn(v_eq[0]), photon_run, run_time = time ) self.dither() for start_y, brace in zip(start_ys, braces): self.add(start_y) self.play(GrowFromCenter(brace)) self.dither() quads = zip(v_equations, start_ys, end_ys, braces) self.equations = [] for v_eq, start_y, end_y, brace in quads: self.remove(brace) self.play( ShowCreation(v_eq[1]), ShowCreation(v_eq[2]), Transform(start_y, end_y) ) v_eq.append(start_y) self.equations.append(Mobject(*v_eq)) def isolate_bend_points(self): arc_radius = 0.1 self.activate_zooming() little_square = self.get_zoomed_camera_mobject() for index in range(3): bend_point = self.bend_points[index] line = Line( bend_point+DOWN, bend_point+UP, color = WHITE, density = self.zoom_factor*DEFAULT_POINT_DENSITY_1D ) angle_arcs = [] for i, rotation in [(index, np.pi/2), (index+1, -np.pi/2)]: arc = Arc(angle = self.path_angles[i]) arc.scale(arc_radius) arc.rotate(rotation) arc.shift(bend_point) angle_arcs.append(arc) thetas = [] for i in [index+1, index+2]: theta = TexMobject("\\theta_%d"%i) theta.scale(0.5/self.zoom_factor) vert = UP if i == index+1 else DOWN horiz = rotate_vector(vert, np.pi/2) theta.next_to( Point(bend_point), horiz, buff = 0.01 ) theta.shift(1.5*arc_radius*vert) thetas.append(theta) figure_marks = [line] + angle_arcs + thetas self.play(ApplyMethod( little_square.shift, bend_point - little_square.get_center(), run_time = 2 )) self.play(*map(ShowCreation, figure_marks)) self.dither() equation_frame = little_square.copy() equation_frame.scale(0.5) equation_frame.shift( little_square.get_corner(UP+RIGHT) - \ equation_frame.get_corner(UP+RIGHT) ) equation_frame.scale_in_place(0.9) self.show_snells(index+1, equation_frame) self.remove(*figure_marks) self.disactivate_zooming() def show_snells(self, index, frame): left_text, right_text = [ "\\dfrac{\\sin(\\theta_%d)}{\\phantom{\\sqrt{y_1}}}"%x for x in index, index+1 ] left, equals, right = TexMobject( [left_text, "=", right_text] ).split() vs = [] sqrt_ys = [] for x, numerator in [(index, left), (index+1, right)]: v, sqrt_y = [ TexMobject( text, size = "\\Large" ).next_to(numerator, DOWN) for text in "v_%d"%x, "\\sqrt{y_%d}"%x ] vs.append(v) sqrt_ys.append(sqrt_y) start, end = [ Mobject( left.copy(), mobs[0], equals.copy(), right.copy(), mobs[1] ).replace(frame) for mobs in vs, sqrt_ys ] self.add(start) self.dither(2) self.play(Transform( start, end, path_func = counterclockwise_path() )) self.dither(2) self.remove(start, end) def show_main_equation(self): self.equation = TexMobject(""" \\dfrac{\\sin(\\theta)}{\\sqrt{y}} = \\text{constant} """) self.equation.shift(LEFT) self.equation.shift( (self.layer_tops[0]-self.equation.get_top())*UP ) self.add(self.equation) self.dither() def ask_continuous_question(self): continuous = self.get_continuous_background() line = Line( UP, DOWN, density = self.zoom_factor*DEFAULT_POINT_DENSITY_1D ) theta = TexMobject("\\theta") theta.scale(0.5/self.zoom_factor) self.play( ShowCreation(continuous), Animation(self.equation) ) self.remove(*self.layers) self.play(ShowCreation(self.cycloid)) self.activate_zooming() little_square = self.get_zoomed_camera_mobject() self.add(line) indices = np.arange( 0, self.cycloid.get_num_points()-1, 10 ) for index in indices: point = self.cycloid.points[index] next_point = self.cycloid.points[index+1] angle = angle_of_vector(point - next_point) for mob in little_square, line: mob.shift(point - mob.get_center()) arc = Arc(angle-np.pi/2, start_angle = np.pi/2) arc.scale(0.1) arc.shift(point) self.add(arc) if angle > np.pi/2 + np.pi/6: vect_angle = interpolate(np.pi/2, angle, 0.5) vect = rotate_vector(RIGHT, vect_angle) theta.center() theta.shift(point) theta.shift(0.15*vect) self.add(theta) self.dither(self.frame_duration) self.remove(arc) class StraightLinesFastestInConstantMedium(PhotonScene): def construct(self): kwargs = {"size" : "\\Large"} left = TextMobject("Speed of light is constant", **kwargs) arrow = TexMobject("\\Rightarrow", **kwargs) right = TextMobject("Staight path is fastest", **kwargs) left.next_to(arrow, LEFT) right.next_to(arrow, RIGHT) squaggle, line = self.get_paths() self.play(*map(ShimmerIn, [left, arrow, right])) self.play(ShowCreation(squaggle)) self.play(Transform( squaggle, line, path_func = path_along_arc(np.pi) )) self.play(self.photon_run_along_path(line)) self.dither() def get_paths(self): squaggle = ParametricFunction( lambda t : (0.5*t+np.cos(t))*RIGHT+np.sin(t)*UP, start = -np.pi, end = 2*np.pi ) squaggle.shift(2*UP) start, end = squaggle.points[0], squaggle.points[-1] line = Line(start, end) result = [squaggle, line] for mob in result: mob.highlight(BLUE_D) return result class GlassAndAir(PhotonScene): def construct(self): pass