diff --git a/active_projects/wallis_g.py b/active_projects/wallis_g.py index cf90cefa..b825000f 100644 --- a/active_projects/wallis_g.py +++ b/active_projects/wallis_g.py @@ -4,23 +4,238 @@ from once_useful_constructs.light import Lighthouse from once_useful_constructs.light import SwitchOn # from once_useful_constructs.light import LightSource +PRODUCT_COLOR = BLUE +CHEAP_AMBIENT_LIGHT_CONFIG = { + "num_levels": 5, + "radius": 1, +} -class IntroduceDistanceProduct(MovingCameraScene): + +class DistanceProductScene(MovingCameraScene): CONFIG = { "ambient_light_config": { - "opacity_function": inverse_quadratic(1, 1.1, 1), + "opacity_function": inverse_power_law(1, 1.5, 1, 4), "num_levels": 100, + "light_radius": 5, + "max_opacity": 0.8, + "color": PRODUCT_COLOR, + }, + "circle_color": BLUE, + "circle_radius": 3, + "num_lighthouses": 6, + "lighthouse_height": 0.5, + "ignored_lighthouse_indices": [], + "observer_config": { + "color": MAROON_B, + "mode": "pondering", + "height": 0.25, + "flip_at_start": True, + }, + "observer_fraction": 1.0 / 3, + "d_label_height": 0.35, + "numeric_distance_label_height": 0.25, + "default_product_column_top": FRAME_WIDTH * RIGHT / 4 + 1.5 * UP, + } + + def setup(self): + super(DistanceProductScene, self).setup() + self.circle = Circle( + color=self.circle_color, + radius=self.circle_radius, + ) + + def get_circle_point_at_proportion(self, alpha): + radius = self.get_radius() + center = self.circle.get_center() + angle = alpha * TAU + unit_circle_point = np.cos(angle) * RIGHT + np.sin(angle) * UP + return radius * unit_circle_point + center + + def get_lh_points(self): + return np.array([ + self.get_circle_point_at_proportion(fdiv(i, self.num_lighthouses)) + for i in range(self.num_lighthouses) + if i not in self.ignored_lighthouse_indices + ]) + + def get_observer_point(self, fraction=None): + if fraction is None: + fraction = self.observer_fraction + return self.get_circle_point_at_proportion(fraction / self.num_lighthouses) + + def get_observer(self): + observer = self.observer = PiCreature(**self.observer_config) + observer.next_to(self.get_observer_point(), RIGHT, buff=SMALL_BUFF) + return observer + + def get_observer_dot(self): + self.observer_dot = Dot( + self.get_observer_point(), + color=self.observer_config["color"] + ) + return self.observer_dot + + def get_lighthouses(self): + self.lighthouses = VGroup() + for point in self.get_lh_points(): + lighthouse = Lighthouse() + lighthouse.scale_to_fit_height(self.lighthouse_height) + lighthouse.move_to(point) + self.lighthouses.add(lighthouse) + return self.lighthouses + + def get_lights(self): + self.lights = VGroup() + for point in self.get_lh_points(): + light = AmbientLight( + source_point=VectorizedPoint(point), + **self.ambient_light_config + ) + self.lights.add(light) + return self.lights + + def get_distance_lines(self): + self.distance_lines = VGroup(*[ + Line(self.get_observer_point(), point) + for point in self.get_lh_points() + ]) + return self.distance_lines + + def get_symbolic_distance_labels(self): + if not hasattr(self, "distance_lines"): + self.get_distance_lines() + self.d_labels = VGroup() + for i, line in enumerate(self.distance_lines): + d_label = TexMobject("d_%d" % i) + d_label.scale_to_fit_height(self.d_label_height) + vect = rotate_vector(line.get_vector(), 90 * DEGREES) + vect *= 2.5 * SMALL_BUFF / np.linalg.norm(vect) + d_label.move_to(line.get_center() + vect) + self.d_labels.add(d_label) + return self.d_labels + + def get_numeric_distance_labels(self, num_decimal_points=3, show_ellipsis=True): + radius = self.circle.get_width() / 2 + if not hasattr(self, "distance_lines"): + self.get_distance_lines() + labels = self.numeric_distance_labels = VGroup() + for line in self.distance_lines: + label = DecimalNumber( + line.get_length() / radius, + num_decimal_points=num_decimal_points, + show_ellipsis=show_ellipsis, + ) + label.scale_to_fit_height(self.numeric_distance_label_height) + max_width = 0.5 * line.get_length() + if label.get_width() > max_width: + label.scale_to_fit_width(max_width) + angle = (line.get_angle() % TAU) - TAU / 2 + if np.abs(angle) > TAU / 4: + angle += np.sign(angle) * np.pi + label.angle = angle + label.next_to(line.get_center(), UP, SMALL_BUFF) + label.rotate(angle, about_point=line.get_center()) + labels.add(label) + return labels + + def get_circle_group(self): + group = VGroup(self.circle) + if not hasattr(self, "observer_dot"): + self.get_observer_dot() + if not hasattr(self, "observer"): + self.get_observer() + if not hasattr(self, "lighthouses"): + self.get_lighthouses() + if not hasattr(self, "lights"): + self.get_lights() + group.add( + self.observer_dot, + self.observer, + self.lighthouses, + self.lights, + ) + return group + + def setup_lighthouses_and_observer(self): + self.add(*self.get_circle_group()) + + # Numerical results + + def get_radius(self): + return self.circle.get_width() / 2.0 + + def get_distance_product(self, fraction=None): + radius = self.get_radius() + observer_point = self.get_observer_point(fraction) + distances = [ + np.linalg.norm(point - observer_point) / radius + for point in self.get_lh_points() + ] + return reduce(op.mul, distances, 1.0) + + # Animating methods + + def add_numeric_distance_labels(self, show_line_creation=True): + anims = [] + if not hasattr(self, "distance_lines"): + self.get_distance_lines() + if not hasattr(self, "numeric_distance_labels"): + self.get_numeric_distance_labels() + if show_line_creation: + anims.append(LaggedStart(ShowCreation, self.distance_lines)) + anims.append(LaggedStart(FadeIn, self.numeric_distance_labels)) + + self.play(*anims) + + def show_distance_product_in_column(self, column_top=None): + if not hasattr(self, "numeric_distance_labels"): + self.get_numeric_distance_labels() + if column_top is None: + column_top = self.default_product_column_top + labels = self.numeric_distance_labels + stacked_labels = labels.copy() + for label in stacked_labels: + label.rotate(-label.angle) + label.scale_to_fit_height(self.numeric_distance_label_height) + stacked_labels.arrange_submobjects(DOWN) + stacked_labels.move_to(column_top, UP) + + h_line = Line(LEFT, RIGHT) + h_line.scale_to_fit_width(1.5 * stacked_labels.get_width()) + h_line.next_to(stacked_labels, DOWN, aligned_edge=RIGHT) + times = TexMobject("\\times") + times.next_to(h_line, UP, SMALL_BUFF, aligned_edge=LEFT) + + product_decimal = DecimalNumber( + self.get_distance_product(), + num_decimal_points=3, + show_ellipsis=True + ) + product_decimal.scale_to_fit_height(self.numeric_distance_label_height) + product_decimal.next_to(h_line, DOWN) + product_decimal.align_to(stacked_labels, RIGHT) + product_decimal.set_color(BLUE) + + self.play(ReplacementTransform(labels.copy(), stacked_labels)) + self.play( + ShowCreation(h_line), + Write(times) + ) + self.play( + ReplacementTransform( + stacked_labels.copy(), + VGroup(product_decimal) + ) + ) + + +class IntroduceDistanceProduct(DistanceProductScene): + CONFIG = { + "ambient_light_config": { # "num_levels": 10, - "light_radius": 10, # "radius": 1, - "max_opacity": 0.4, "color": YELLOW, }, - "num_lighthouses": 6, - "circle_radius": 3, - "observer_color": MAROON_B, - "lighthouse_height": 0.5, - "camera_class": MovingCamera, } def construct(self): @@ -29,12 +244,10 @@ class IntroduceDistanceProduct(MovingCameraScene): self.show_sum_of_inverse_squares() def draw_circle_with_points(self): - circle = Circle(color=BLUE) - circle.scale(self.circle_radius) + circle = self.circle lh_dots = self.lh_dots = VGroup(*[ - Dot().move_to(circle.point_from_proportion(alpha)) - for alpha in np.arange(0, 1, 1.0 / self.num_lighthouses) + Dot(point) for point in self.get_lh_points() ]) lh_dot_arrows = VGroup(*[ Arrow(*[ @@ -47,8 +260,7 @@ class IntroduceDistanceProduct(MovingCameraScene): evenly_space_dots_label.scale_to_fit_width(0.5 * circle.get_width()) evenly_space_dots_label.move_to(circle) - special_dot = self.special_dot = Dot(color=self.observer_color) - special_dot.move_to(circle.point_from_proportion(0.04)) + special_dot = self.special_dot = self.get_observer_dot() special_dot_arrow = Vector(DL) special_dot_arrow.next_to(special_dot, UR, SMALL_BUFF) special_dot_arrow.match_color(special_dot) @@ -77,24 +289,13 @@ class IntroduceDistanceProduct(MovingCameraScene): self.play(FadeOut(VGroup(special_dot_arrow, special_dot_label))) def turn_into_lighthouses_and_observer(self): - lighthouses = self.lighthouses = VGroup() - lights = self.lights = VGroup() - for dot in self.lh_dots: - point = dot.get_center() - lighthouse = Lighthouse() - lighthouse.scale_to_fit_height(self.lighthouse_height) - lighthouse.move_to(point) - lighthouses.add(lighthouse) + lighthouses = self.get_lighthouses() + lights = self.get_lights() - light = AmbientLight( - source_point=VectorizedPoint(point), - **self.ambient_light_config - ) - lights.add(light) - - observer = self.observer = PiCreature(color=self.observer_color) - observer.flip() - observer.move_to(self.special_dot) + observer = self.get_observer() + observer.save_state() + observer.scale_to_fit_height(2) + observer.change_mode("happy") observer.to_edge(RIGHT) self.play( @@ -104,23 +305,12 @@ class IntroduceDistanceProduct(MovingCameraScene): ) self.wait() self.play(FadeIn(observer)) - self.play( - observer.scale_to_fit_height, 0.25, - observer.next_to, self.special_dot, RIGHT, 0.5 * SMALL_BUFF, - ) + self.play(observer.restore) self.wait() def show_sum_of_inverse_squares(self): - lines = VGroup(*[ - Line(self.special_dot.get_center(), dot.get_center()) - for dot in self.lh_dots - ]) - labels = VGroup(*[TexMobject("d_%d" % i) for i in range(len(lines))]) - for label, line in zip(labels, lines): - label.scale(0.75) - vect = rotate_vector(line.get_vector(), TAU / 4) - vect *= 2 * SMALL_BUFF / np.linalg.norm(vect) - label.move_to(line.get_center() + vect) + lines = self.get_distance_lines() + labels = self.get_symbolic_distance_labels() sum_of_inverse_squares = TexMobject(*it.chain(*[ ["{1", "\\over", "(", "d_%d" % i, ")", "^2}", "+"] @@ -182,7 +372,7 @@ class IntroduceDistanceProduct(MovingCameraScene): labels[:-1].copy(), d_terms[:-1], ), - circle_group.scale, 0.8, {"about_edge": DOWN} + circle_group.scale, 0.8, {"about_point": FRAME_Y_RADIUS * DOWN} ) self.wait() self.play(LaggedStart( @@ -195,9 +385,10 @@ class IntroduceDistanceProduct(MovingCameraScene): # Mention useful just to basel problem circle_group.save_state() + v_point = VectorizedPoint(FRAME_X_RADIUS * LEFT + FRAME_Y_RADIUS * DOWN) self.play( - circle_group.scale, 0.5, - circle_group.to_corner, DL, + circle_group.next_to, v_point, UR, {"submobject_to_align": self.circle}, + circle_group.scale, 0.5, {"about_point": v_point.get_center()}, ) self.play( GrowFromCenter(brace), @@ -230,7 +421,9 @@ class IntroduceDistanceProduct(MovingCameraScene): wallis_product.next_to, basel_sum, UP, {"aligned_edge": RIGHT}, ) self.play( - d_terms.shift, d_terms.get_height() * UP / 2, + d_terms.shift, 0.75 * d_terms.get_height() * UP, + d_terms.set_color, PRODUCT_COLOR, + light_rings.set_fill, PRODUCT_COLOR, *[ FadeOut(mob) for mob in sum_of_inverse_squares @@ -240,7 +433,7 @@ class IntroduceDistanceProduct(MovingCameraScene): self.wait() self.play( FadeOut(plusses), - d_terms.arrange_submobjects, RIGHT, 0.5 * SMALL_BUFF, + d_terms.arrange_submobjects, RIGHT, 0.25 * SMALL_BUFF, d_terms.move_to, sum_of_inverse_squares, DOWN, ) self.wait() @@ -268,6 +461,182 @@ class IntroduceDistanceProduct(MovingCameraScene): self.wait() +class Lemma1(DistanceProductScene): + CONFIG = { + "circle_radius": 2.5, + "observer_fraction": 0.5, + # "ambient_light_config": { + # "num_levels": 5, + # "radius": 1, + # }, + "lighthouse_height": 0.25, + "lemma_text": "distance product = 2", + } + + def construct(self): + self.add_title() + self.add_circle_group() + self.state_lemma_premise() + self.show_product() + + def add_title(self): + title = self.title = TextMobject("Two lemmas:") + title.set_color(YELLOW) + title.to_edge(UP, buff=MED_SMALL_BUFF) + self.add(title) + + def add_circle_group(self): + self.circle.to_corner(DL) + circle_group = self.get_circle_group() + self.play(LaggedStart(FadeIn, VGroup(*circle_group.family_members_with_points()))) + + def state_lemma_premise(self): + premise = TextMobject("Lemma 1: If observer is halfway between lighthouses,") + self.premise = premise + premise.next_to(self.title, DOWN) + + frac = 1.0 / self.num_lighthouses + arc1, arc2 = arcs = VGroup(VMobject(), VMobject()) + arc1.pointwise_become_partial(self.circle, 0, frac / 2) + arc2.pointwise_become_partial(self.circle, frac / 2, frac) + arc1.reverse_points() + arcs.set_stroke(YELLOW, 5) + show_arcs = ShowCreationThenDestruction( + arcs, + submobject_mode="all_at_once", + run_time=2, + ) + + self.play(Write(premise), show_arcs, run_time=2) + self.wait() + self.play(show_arcs) + self.wait() + + def show_product(self): + lemma = TextMobject(self.lemma_text) + lemma.set_color(BLUE) + lemma.next_to(self.premise, DOWN) + self.add_numeric_distance_labels() + self.play(Write(lemma, run_time=1)) + self.show_distance_product_in_column() + self.wait() + + +class Lemma1With7Lighthouses(Lemma1): + CONFIG = { + "num_lighthouses": 7, + } + + +class Lemma1With8Lighthouses(Lemma1): + CONFIG = { + "num_lighthouses": 8, + } + + +class Lemma1With9Lighthouses(Lemma1): + CONFIG = { + "num_lighthouses": 9, + } + + +class Lemma2(Lemma1): + CONFIG = { + # "ambient_light_config": CHEAP_AMBIENT_LIGHT_CONFIG, + "lemma_text": "distance product = \\# Initial lighthouses" + } + + def construct(self): + self.add_title() + self.add_circle_group() + self.state_lemma_premise() + self.replace_first_lighthouse() + self.show_product() + + def state_lemma_premise(self): + premise = self.premise = TextMobject( + "If the observer replaces a lighthouse," + ) + premise.next_to(self.title, DOWN) + + self.play(Write(premise, run_time=1)) + + def replace_first_lighthouse(self): + dot = self.observer_dot + observer_anim = MaintainPositionRelativeTo(self.observer, dot) + lighthouse_group = VGroup(self.lighthouses[0], self.lights[0]) + point = self.get_lh_points()[0] + + self.play( + lighthouse_group.shift, 5 * RIGHT, + lighthouse_group.fade, 1, + run_time=1.5, + rate_func=running_start, + remover=True, + ) + self.play( + dot.move_to, point, + observer_anim, + path_arc=(-120 * DEGREES), + ) + self.wait() + + self.ignored_lighthouse_indices = [0] + self.observer_fraction = 0 + for group in self.lighthouses, self.lights: + self.lighthouses.submobjects.pop(0) + + +class Lemma2With7Lighthouses(Lemma2): + CONFIG = { + "num_lighthouses": 7, + } + + +class Lemma2With8Lighthouses(Lemma2): + CONFIG = { + "num_lighthouses": 8, + } + + +class Lemma2With9Lighthouses(Lemma2): + CONFIG = { + "num_lighthouses": 9, + } + + +class FromGeometryToAlgebra(DistanceProductScene): + CONFIG = { + "num_lighthouses": 7, + "ambient_light_config": CHEAP_AMBIENT_LIGHT_CONFIG, + } + + def construct(self): + self.setup_lights() + self.point_out_evenly_spaced() + self.transition_to_complex_plane() + self.name_roots_of_unity() + + def setup_lights(self): + lights = self.get_lights() + circle = self.circle + + self.add(circle, lights) + + def point_out_evenly_spaced(self): + pass + + def transition_to_complex_plane(self): + pass + + def name_roots_of_unity(self): + pass + + + + + +