def show_sin_thetas(self): pc = Line(self.p_point, self.c_point) mob = Mobject(self.theta, self.d_mob).copy() mob.ingest_submobjects() triplets = [ (pc, "D\\sin(\\theta)", 0.5), (self.y_line, "D\\sin^2(\\theta)", 0.7), ] for line, tex, scale in triplets: trig_mob = TexMobject(tex) trig_mob.scale_to_fit_width( scale*line.get_length() ) trig_mob.shift(-1.2*trig_mob.get_top()) trig_mob.rotate(line.get_angle()) trig_mob.shift(line.get_center()) if line is self.y_line: trig_mob.shift(0.1*UP) self.play(Transform(mob, trig_mob)) self.add(trig_mob) self.dither() self.remove(mob) self.d_sin_squared_theta = trig_mob
def get_vector_label(self, vector, label, direction="left", rotate=False, color=None, label_scale_factor=VECTOR_LABEL_SCALE_FACTOR): if not isinstance(label, TexMobject): if len(label) == 1: label = "\\vec{\\textbf{%s}}" % label label = TexMobject(label) if color is None: color = vector.get_color() label.highlight(color) label.scale(label_scale_factor) label.add_background_rectangle() angle = vector.get_angle() if not rotate: label.rotate(-angle) if direction is "left": label.shift(-label.get_bottom() + 0.1 * UP) else: label.shift(-label.get_top() + 0.1 * DOWN) label.rotate(angle) label.shift((vector.get_end() - vector.get_start()) / 2) return label
def get_vector_label(self, vector, label, direction = "left", rotate = False, color = None, label_scale_factor = VECTOR_LABEL_SCALE_FACTOR): if not isinstance(label, TexMobject): if len(label) == 1: label = "\\vec{\\textbf{%s}}"%label label = TexMobject(label) if color is None: color = vector.get_color() label.highlight(color) label.scale(label_scale_factor) label.add_background_rectangle() angle = vector.get_angle() if not rotate: label.rotate(-angle) if direction is "left": label.shift(-label.get_bottom() + 0.1*UP) else: label.shift(-label.get_top() + 0.1*DOWN) label.rotate(angle) label.shift((vector.get_end() - vector.get_start())/2) return label
def show_sin_thetas(self): pc = Line(self.p_point, self.c_point) mob = Mobject(self.theta, self.d_mob).copy() mob.ingest_submobjects() triplets = [ (pc, "D\\sin(\\theta)", 0.5), (self.y_line, "D\\sin^2(\\theta)", 0.7), ] for line, tex, scale in triplets: trig_mob = TexMobject(tex) trig_mob.scale_to_fit_width( scale*line.get_length() ) trig_mob.shift(-1.2*trig_mob.get_top()) trig_mob.rotate(line.get_angle()) trig_mob.shift(line.get_center()) if line is self.y_line: trig_mob.shift(0.1*UP) self.play(Transform(mob, trig_mob)) self.add(trig_mob) self.wait() self.remove(mob) self.d_sin_squared_theta = trig_mob
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.wait() self.play(ShowCreation( continuous, rate_func = lambda t : smooth(1-t) )) self.remove(continuous) self.wait() 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.wait() for start_y, brace in zip(start_ys, braces): self.add(start_y) self.play(GrowFromCenter(brace)) self.wait() 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.wait() 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.wait(2) self.play(Transform( start, end, path_func = counterclockwise_path() )) self.wait(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.wait() 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.wait(self.frame_duration) self.remove(arc)
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)