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animation.py
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from itertools import product
import numpy as np
from manim import Circle, Dot, Create, FadeIn, Line, \
Transform, MovingCameraScene, Flash, YELLOW, Text, UP, Write, \
DOWN, Tex, WHITE, PURPLE, GREY, Uncreate, AnimationGroup, Unwrite, ImageMobject, LEFT, RIGHT, \
MarkupText, Polygon, PI, DecimalNumber, ValueTracker, Arrow, VGroup, FadeOut, Indicate, \
ReplacementTransform, BLACK, VMobject, GREEN_B, ORANGE, DARK_GREY, RED, LIGHT_GRAY
from euclidean_hexagon import EuclideanHexagon, get_diagonals
from geometry_util import polar_to_point, \
tf_klein_to_poincare, get_intersections_of_n_tangent_circles, get_intersections_of_circles_with_unit_circle, \
get_intersection_from_angles, get_parallel_to_line_through_point, tf_poincare_to_klein, \
get_both_intersections_line_with_unit_circle, get_intersection, hyperbolic_distance_function
from hexagon import HexagonCircles, HexagonMainDiagonals, HyperbolicArcBetweenPoints, HexagonAngles
from hyperbolic_polygon import HyperbolicPolygon
class SevenCircles(MovingCameraScene):
def construct(self):
self.camera.frame.width = 9
OUTER_CIRCLE_COLOR = WHITE
INNER_CIRCLE_COLOR = GREEN_B
INNER_INTERSECTION_COLOR = PURPLE
OUTER_INTERSECTION_COLOR = DARK_GREY
HEXAGON_COLOR = GREY
DIAGONAL_COLOR = ORANGE
DIAGONAL_INTERSECTION_COLOR = YELLOW
title = Text("The Seven Circles Theorem").scale(0.8)
theorem_text_white = Tex(r"Let $C$ be a circle ", "containing six smaller circles, ",
"such that each inner circle is tangent to $C$ ",
"and any two adjacent inner circles are also tangent to each other. ",
"Let there also be a hexagon formed by the intersection points of each inner circle with $C$. ",
"Then the three main diagonals of the hexagon will always meet in a single point.", "",
stroke_width=.05).scale(
0.5).move_to([0, -2, 0])
theorem_text_colored = Tex(r"Let $C$ be a circle ", "containing six smaller circles, ",
"such that each inner circle is tangent to $C$ ",
"and any two adjacent inner circles are also tangent to each other. ",
"Let there also be a hexagon formed by the intersection points of each inner circle with $C$. ",
"Then the three main diagonals of the hexagon will always meet in a single point.",
"",
substrings_to_isolate=[r"$C$", "six smaller circles", "tangent to $C$",
"adjacent inner circles are also tangent",
"hexagon", "main diagonals", "single point"],
stroke_width=.05).set_color_by_tex(r"$C$",
OUTER_CIRCLE_COLOR).set_color_by_tex(
"six smaller circles",
INNER_CIRCLE_COLOR).set_color_by_tex(
"tangent to $C$",
OUTER_INTERSECTION_COLOR).set_color_by_tex(
"adjacent inner circles are also tangent",
INNER_INTERSECTION_COLOR).set_color_by_tex(
"hexagon", HEXAGON_COLOR).set_color_by_tex(
"main diagonals", DIAGONAL_COLOR).set_color_by_tex(
"single point", DIAGONAL_INTERSECTION_COLOR).scale(
0.5).move_to([0, -2, 0])
circle = Circle(color=OUTER_CIRCLE_COLOR, stroke_width=2)
# phis = create_phis(min_dist=.9, max_dist=1.2)
phis = HexagonAngles(np.array([.3, 1.6, 2.2, 3, 4.3]))
first_circle_radius = .4
hexagon = EuclideanHexagon(phis, color=HEXAGON_COLOR, stroke_width=2)
hexagon_circles = HexagonCircles(hexagon, first_circle_radius, stroke_width=2, color=INNER_CIRCLE_COLOR)
inner_intersections = get_intersections_of_n_tangent_circles(hexagon_circles.circles,
color=INNER_INTERSECTION_COLOR)
outer_intersections = get_intersections_of_circles_with_unit_circle(hexagon_circles.circles,
color=OUTER_INTERSECTION_COLOR)
diagonals = get_diagonals(hexagon, color=DIAGONAL_COLOR, stroke_width=2)
diagonal_intersection = Dot(get_intersection_from_angles(phis[0], phis[3], phis[1], phis[4]),
color=DIAGONAL_INTERSECTION_COLOR, radius=.05)
self.add_subcaption("Wir beschäftigen uns heute mit dem Sieben-Kreise-Satz", duration=3)
self.play(Write(title))
self.wait(2)
self.add_subcaption("Der Sieben-Kreise-Satz sagt Folgendes aus:", duration=2)
self.play(title.animate.shift(1.4 * UP).scale(0.7), self.camera.frame.animate.shift(0.8 * DOWN))
self.wait(1)
self.add_subcaption("Sei C_0 ein Kreis", duration=2)
self.play(Write(theorem_text_white[0], run_time=1))
self.play(FadeIn(circle), FadeIn(theorem_text_colored[1]))
self.add_foreground_mobject(circle)
self.add_subcaption("und C_1,...,C_6 in C_0 enthaltene Kreise", duration=4)
self.play(Write(theorem_text_white[1], run_time=3))
self.play(Create(hexagon_circles, run_time=5), FadeIn(theorem_text_colored[4], run_time=1))
self.remove_foreground_mobjects(circle)
self.add_subcaption("sodass jeder innere Kreis zum äußeren Kreis tangential ist", duration=5)
self.play(Write(theorem_text_white[2], run_time=3))
self.play(FadeIn(theorem_text_colored[7]))
for i in range(6):
self.play(Create(outer_intersections[i], run_time=.5))
self.add_foreground_mobjects(*outer_intersections)
self.add_subcaption("und je zwei nebeneinanderliegende innere Kreise ebenfalls zueinander tangential sind.",
duration=8)
self.play(Write(theorem_text_white[3]), run_time=6)
self.play(FadeIn(theorem_text_colored[10]))
for i in range(6):
self.play(Create(inner_intersections[i], run_time=.5))
self.add_subcaption(
"Dann treffen sich die drei Diagonalen des Hexagons, das von den Schnittpunkten der inneren Kreise mit dem äußeren Kreis gebildet wird,",
duration=8)
self.play(Write(theorem_text_white[4]), run_time=6)
self.play(Create(hexagon, run_time=5), FadeIn(theorem_text_colored[13]))
self.add_subcaption("in einem Punkt", duration=7)
self.play(Write(theorem_text_white[5]), run_time=3)
self.play(FadeIn(theorem_text_colored[18], run_time=1))
for x in diagonals:
self.play(Create(x), run_time=1)
self.play(FadeIn(theorem_text_colored[22], run_time=1))
self.play(Create(diagonal_intersection))
self.wait(1)
self.play(Flash(diagonal_intersection))
self.wait(1)
self.add_subcaption("Unser Ziel ist es jetzt diesen Satz zu beweisen.", duration=4)
self.wait(4)
self.add_subcaption(
"Das wird uns mithilfe einer Einbettung des Problems in einen hyperbolischen Rahmen gelingen.", duration=5)
self.wait(5)
self.add_subcaption("Zuerst geben wir dafür eine kleine Einführung in die hyperbolische Geometrie,", duration=4)
self.wait(4)
self.add_subcaption(
"um das Problem mit unserem neugewonnenen Wissen umzuformulieren und weiter zu bearbeiten.", duration=4)
self.wait(3)
everything = VGroup()
for mob in self.mobjects:
if isinstance(mob, VMobject):
everything.add(mob)
endDot = Dot(color=BLACK)
self.add(endDot)
self.play(Transform(everything, endDot), run_time=2)
self.wait(2)
class ParallelAxiom(MovingCameraScene):
def construct(self):
center = [0, 2, 0]
left = -7
right = 25
title = Text("Parallel axiom").scale(0.7).shift(5.4 * UP)
g_fun = lambda x: -0.7 * x
g_points = [[left, g_fun(left), 0], [right, g_fun(right), 0]]
g = Line(g_points[0], g_points[1], name="g")
g_text = Tex("$g$", stroke_width=0.05).move_to([-3.7, 3, 0])
p_array = polar_to_point(1)
p = Dot(p_array, name="P", color=YELLOW)
p_text = Tex("$P$", stroke_width=0.05).next_to(p, UP)
h_fun = get_parallel_to_line_through_point(g_points, p_array)
h_points = [[left, h_fun(left), 0], [right, h_fun(right), 0]]
h = Line(h_points[0], h_points[1], name="h")
h_text = Tex("$h$", stroke_width=0.05).move_to([-1.7, 3, 0])
similar_to_g = [lambda x: x * -.65, lambda x: x * -.8, lambda x: x * -.57, lambda x: x * -.77]
similar_points = [[[left, _(left), 0.], [right, _(right), 0.]] for _ in similar_to_g]
similar_funcs = [get_parallel_to_line_through_point(sim_points, p_array) for sim_points in similar_points]
sim_h_points = [[[left, _(left), 0.], [right, _(right), 0.]] for _ in similar_funcs]
similar_lines = [Line(_[0], _[1], stroke_width=1) for _ in sim_h_points]
intersection_points = [get_intersection(g_points[0], g_points[1],
[left, _(left), 0.], [right, _(right), 0.]) for _ in similar_funcs]
intersection_dots = [Dot(point=p, color=RED) for p in intersection_points]
elements = [title, g, g_text, h, h_text, p, p_text, similar_lines[0], similar_lines[1],
similar_lines[2], similar_lines[3], intersection_dots[0],
intersection_dots[1], intersection_dots[2], intersection_dots[3]]
uncreate = AnimationGroup(*[Uncreate(i, run_time=1) for i in elements])
camera_shift = 1.7 * np.array([10, -7, 0])
question = Text("How can we represent the hyperbolic in the Euclidean plane?").scale(0.65).move_to(center)
self.camera.frame.move_to(center)
self.add_subcaption("Axiomatisch unterscheidet sich der hyperbolische Raum vom Euklidischen", duration=4)
self.wait(4)
self.add_subcaption("dadurch, dass das Parallelenaxiom nicht gilt.", duration=4)
self.wait(2)
self.play(Write(title), run_time=2)
self.add_subcaption("Das Parallelenaxiom sagt aus, dass zu jeder Geraden g", duration=3)
self.play(Create(g))
self.play(Write(g_text))
self.wait(2)
self.add_subcaption("und jedem Punkt P, der nicht auf g liegt", duration=3)
self.play(Create(p))
self.play(Write(p_text))
self.add_foreground_mobject(p)
self.wait(2)
self.add_subcaption("genau eine Gerade h existiert, die durch P verläuft und zu g parallel ist.", duration=4)
self.play(Create(h))
self.play(Write(h_text))
self.wait(2)
self.add_subcaption(
"Parallel heißt hier einfach, dass sich die beiden Geraden nicht schneiden.", duration=4)
self.wait(5)
self.add_subcaption("In hyperbolischer Geometrie ist diese Eigenschaft nicht gegeben.", duration=3)
self.wait(4)
self.add_subcaption(
"Dort gibt es für jede solche g und P mehrere - sogar unendlich viele - Geraden durch P, die zu g parallel sind also g nicht schneiden.",
duration=10)
self.play(Create(similar_lines[0]))
self.play(Create(similar_lines[1]))
self.play(Create(similar_lines[2]))
self.play(Create(similar_lines[3]))
for intersection_dot in intersection_dots:
self.add(intersection_dot)
self.wait(6)
self.add_subcaption(
"Unsere Darstellung hier ist aber irreführend, denn natürlich wird g von allen außer der ursprünglichen Geraden h geschnitten, wenn wir ihnen nur lange genug folgen.",
duration=10)
self.play(self.camera.frame.animate.shift(camera_shift),
title.animate.shift(camera_shift),
run_time=5)
self.wait(5)
self.add_subcaption(
"Allgemein haben wir das Problem, dass wir einen hyperbolischen Raum in der euklidischen Ebene - diesem Video - darstellen wollen.",
duration=9)
self.wait(9)
self.add_subcaption(
"Die Frage die wir jetzt angehen ist also: Wie stellt man die hyperbolische Ebene im Euklidischen dar?",
duration=7)
self.play(uncreate)
self.play(self.camera.frame.animate.move_to(center))
self.play(Write(question))
self.wait(3)
self.play(Unwrite(question))
class HyperbolicModelsPoincare(MovingCameraScene):
def construct(self):
center = np.array([0, 0, 0])
klein_origin = np.array([3.5, -1, 0])
poincare_origin = np.array([-3.5, -1, 0])
MY_BLUE = "#22c1dd"
title = MarkupText(
"Models for the hyperbolic plane").scale(
0.8).shift(3.5 * UP)
klein_model = ImageMobject("tessellation_klein.png").scale(0.7).move_to(klein_origin)
klein_text = Text("Klein model").scale(0.6).move_to(3.5 * RIGHT + 2.2 * UP)
poincare_model = ImageMobject("tessellation_poincare.png").scale(0.7).move_to(poincare_origin)
poincare_text = Text("Poincaré model").scale(0.6).move_to(3.5 * LEFT + 2.2 * UP)
p1 = np.array([[0., 0., 0.], [1.25, 0, 0], [0.78, 0.98, 0]])
p2 = np.array([[-.28, 1.23, 0], [-1.12, 0.55, 0], [-1.12, 1.42, 0]])
scale_front = 0.4
scale_back = 2.5
p1 = scale_front * p1
p2 = scale_front * p2
pcircle = Circle(color=MY_BLUE, stroke_width=1).scale(scale_back).move_to(poincare_origin)
ptri1 = HyperbolicPolygon(p1,
stroke_width=1.5, dot_radius=.02, color=YELLOW, dot_color=YELLOW).scale(
scale_back)
ptri1.move_to(ptri1.get_center() * scale_back).shift(poincare_origin)
ptri2 = HyperbolicPolygon(p2,
stroke_width=1.5, dot_radius=.02, color=YELLOW, dot_color=YELLOW).scale(
scale_back)
ptri2.move_to(ptri2.get_center() * scale_back).shift(poincare_origin)
# self.add(ktri1, ktri2, ptri1, ptri2)
ptri1_text = Tex(r"$\Delta_1$").next_to(ptri1).scale(.6).shift(0.5 * LEFT)
ptri2_text = Tex(r"$\Delta_2$").next_to(ptri2).scale(.6).shift(0.3 * LEFT)
ptri_size_text = Tex(r"$A(\Delta_1) = A(\Delta_2)$", font_size=35).next_to(pcircle, buff=.4)
phis = [[0.4, 2], [1.3, 6], [3.5, 5], [4, 1.5]]
phis_as_points = [[polar_to_point(x), polar_to_point(y)] for [x, y] in phis]
p_geodesics_raw = [HyperbolicArcBetweenPoints(x, y) for [x, y] in phis_as_points]
p_geodesics = [geo.scale(scale_back).move_to(geo.get_center() * scale_back).shift(poincare_origin) for geo in
p_geodesics_raw]
angle_color = LIGHT_GRAY
right_angles = [[[Line(.925 * polar_to_point(x), .925 * polar_to_point(x - .08), color=angle_color),
Dot(.925 * polar_to_point(x - .08), color=angle_color, radius=.008),
Line(.92 * polar_to_point(x - .08), polar_to_point(x - .08), color=angle_color)],
[Line(.925 * polar_to_point(y), .925 * polar_to_point(y - .08), color=angle_color),
Dot(.925 * polar_to_point(y - .08), color=angle_color, radius=.008),
Line(.92 * polar_to_point(y - .08), polar_to_point(y - .08), color=angle_color)]] for [x, y]
in phis]
right_angles_shifted = [[[l1.scale(scale_back).move_to(l1.get_center() * scale_back).shift(poincare_origin),
p.scale(scale_back).move_to(p.get_center() * scale_back).shift(poincare_origin),
l2.scale(scale_back).move_to(l2.get_center() * scale_back).shift(poincare_origin)],
[m1.scale(scale_back).move_to(m1.get_center() * scale_back).shift(poincare_origin),
q.scale(scale_back).move_to(q.get_center() * scale_back).shift(poincare_origin),
m2.scale(scale_back).move_to(m2.get_center() * scale_back).shift(poincare_origin)]]
for [[l1, p, l2], [m1, q, m2]] in right_angles]
point_coords = [0, -0.3, 0]
p_point = Dot(point_coords)
p_point.move_to(p_point.get_center() * scale_back).shift(poincare_origin)
p_point_text = Tex("$P$").scale(0.8).next_to(p_point, direction=UP + 0.2 * RIGHT)
point_geodesic_point_phis = [0, 2.3, 2.9, 3.15, 3.4, 3.8, 4.1, 4.5, 5]
p_point_geodesic_points = [polar_to_point(x) for x in point_geodesic_point_phis]
p_point_geodesics_both_points = [
get_both_intersections_line_with_unit_circle(x, tf_poincare_to_klein(point_coords)) for x in
p_point_geodesic_points]
p_point_geodesics = [HyperbolicArcBetweenPoints(points[0], points[1], color=MY_BLUE) for points in
p_point_geodesics_both_points]
p_point_moved_geodesics = [geo.scale(scale_back).move_to(geo.get_center() * scale_back).shift(poincare_origin)
for geo in
p_point_geodesics]
p_moving_dot = Dot(poincare_origin)
p_moving_text = Tex("$P$", font_size=25).next_to(p_moving_dot, 0.5 * RIGHT + 0.6 * UP, buff=0)
p_moving_dot_phi = 5 * PI / 3
norm_factor = 6 / pow(PI, 2)
p_origin = Dot(poincare_origin)
p_origin_text = Tex("0", font_size=25).next_to(p_origin, LEFT, buff=.1)
p_current_point = np.array(center)
p_distance_text = Tex(r"$\mathrm{dist}(0,P) = $ ", font_size=30).next_to(pcircle, buff=.2)
p_distance_number = DecimalNumber(0.0,
num_decimal_places=2, show_ellipsis=True, group_with_commas=False,
font_size=35).next_to(p_distance_text, buff=.15)
label = VGroup(p_distance_text, p_distance_number)
p_distance_infty = Tex(r"$\infty$", font_size=30).next_to(p_distance_text, buff=.15)
p_distance_tracker = ValueTracker(0.0)
self.add_subcaption("Es gibt verschiedene Modelle für die hyperbolische Ebene.", duration=3)
self.play(Write(title), run_time=2)
self.add_foreground_mobject(title)
self.wait(1)
self.add_subcaption("Solche Modelle sind einfach Abbildungen von der hyperbolischen in die euklidische Ebene",
duration=5)
self.wait(5)
self.add_subcaption("Wir beschäftigen uns genauer mit dem Poincare-", duration=3)
self.add_foreground_mobject(poincare_text)
self.play(Write(poincare_text), FadeIn(poincare_model), run_time=3)
self.add_subcaption("und dem Klein-Modell.", duration=3)
self.add_foreground_mobject(klein_text)
self.play(Write(klein_text), FadeIn(klein_model), run_time=3)
self.wait(3)
self.add_subcaption("Zuerst schauen wir uns das Poincare-Modell etwas genauer an.", duration=4)
self.add(pcircle)
self.play(self.camera.frame.animate.scale(0.8).move_to(np.add(poincare_origin, [0, 0.4, 0])),
FadeOut(klein_model), FadeOut(klein_text), FadeOut(poincare_model), run_time=1.5)
self.wait(2.5)
self.add_subcaption(
"Die Darstellung der hyperbolischen Ebene im Poincare-Modell beschränkt sich auf die Einheitsscheibe",
duration=6)
self.wait(3)
self.play(Indicate(pcircle, color=MY_BLUE, scale_factor=1.1))
self.wait(2)
self.add_subcaption(
"Die Geodätischen des Poincare-Modells sind - im Gegensatz zu Geraden im Euklidischen - ",
duration=5)
self.play(Create(p_geodesics[0]))
self.play(Create(p_geodesics[1]))
self.play(Create(p_geodesics[2]))
self.play(Create(p_geodesics[3]))
self.add_foreground_mobjects(*p_geodesics)
self.play(*[Create(right_angles_shifted[i][j][0]) for i, j in product(range(4), range(2))],
run_time=0.5,
rate_func=lambda x: x)
self.add(*[right_angles_shifted[i][j][1] for i, j in product(range(4), range(2))])
self.play(*[Create(right_angles_shifted[i][j][2]) for i, j in product(range(4), range(2))],
run_time=0.5,
rate_func=lambda x: x)
self.add_subcaption("Kreissegmente, die den Einheitskreis im rechten Winkel schneiden",
duration=4)
self.wait(4)
self.add_subcaption("Damit wird klar, dass das Parallelenaxiom hier nicht gilt:", duration=4)
self.play(*[Uncreate(right_angles_shifted[i][j][2]) for i, j in product(range(4), range(2))],
run_time=0.5,
rate_func=lambda x: 1 - x) # inverse rate_func
self.remove(*[right_angles_shifted[i][j][1] for i, j in product(range(4), range(2))])
self.play(*[Uncreate(right_angles_shifted[i][j][0]) for i, j in product(range(4), range(2))],
run_time=0.5,
rate_func=lambda x: 1 - x)
self.play(Uncreate(p_geodesics[1], run_time=2), Uncreate(p_geodesics[2], run_time=2),
Uncreate(p_geodesics[3], run_time=2), Create(p_point, run_time=1),
Write(p_point_text, run_time=1))
self.wait(2)
self.add_subcaption("Für einen Punkt P und eine Geodätische findet man", duration=3)
self.add_foreground_mobject(p_point)
self.add_foreground_mobject(p_point_text)
self.wait(3)
self.add_subcaption(
"beliebig viele andere Geodätische, die P, aber nicht die ursprüngliche Geodätische schneiden.", duration=6)
self.play(Create(p_point_moved_geodesics[0]))
self.play(Create(p_point_moved_geodesics[1]))
self.play(Create(p_point_moved_geodesics[2]))
self.play(Create(p_point_moved_geodesics[3]))
self.play(Create(p_point_moved_geodesics[4]))
self.play(Create(p_point_moved_geodesics[5]))
self.play(Create(p_point_moved_geodesics[6]))
self.add_subcaption(
"Auch wenn es auf den ersten Blick so scheint, ist die hyperbolische Ebene keineswegs beschränkt.",
duration=6)
self.wait(2)
self.play(Uncreate(p_geodesics[0]),
Uncreate(p_point_moved_geodesics[0]),
Uncreate(p_point_moved_geodesics[1]),
Uncreate(p_point_moved_geodesics[2]),
Uncreate(p_point_moved_geodesics[3]),
Uncreate(p_point_moved_geodesics[4]),
Uncreate(p_point_moved_geodesics[5]),
Uncreate(p_point_moved_geodesics[6]),
Uncreate(p_point), Unwrite(p_point_text))
self.wait(3)
self.add_subcaption(
"Lassen wir einen Punkt in der hyperbolischen Ebene mit konstanter Geschwindigkeit in eine Richtung laufen, so sehen wir, dass dieser den Rand des Poincare-Modell nie erreicht.",
duration=12)
self.play(Create(p_moving_dot), Create(p_origin), Write(p_distance_text), Write(p_origin_text),
Write(p_moving_text))
self.play(Write(p_distance_number))
self.wait(4)
p_distance_number.add_updater(lambda d: d.set_value(p_distance_tracker.get_value()))
steps = 30
frames = 420
duration = 7
frame_rate = duration / frames
batch_size = frames / steps
offset = 0
for t in range(1, frames):
if batch_size == 1:
r = t
else:
r = 1 + int(t / batch_size)
if t % batch_size == 1 or batch_size == 1:
offset = np.array(
polar_to_point(p_moving_dot_phi, norm_factor / (batch_size * pow(r, 2))))
p_current_point = p_current_point + offset
pos = p_current_point * scale_back + np.array(poincare_origin)
moving_dot = Dot(pos, radius=0.08)
p_distance_number.font_size = 35
p_distance_tracker.set_value(np.exp(
hyperbolic_distance_function(center, p_current_point)))
label.arrange(buff=.15)
label.next_to(pcircle, buff=.2)
p_moving_text.next_to(p_moving_dot, 0.5 * RIGHT + 0.6 * UP, buff=0)
self.play(Transform(p_moving_dot, moving_dot), run_time=frame_rate,
rate_func=lambda a: a)
self.remove(p_moving_dot)
p_moving_dot = moving_dot
label.arrange(buff=.05)
label.next_to(pcircle, buff=.2)
self.add_subcaption("Der Einheitskreis liegt im Unendlichen und gehört nicht zum Poincare-Modell",
duration=5)
moving_dot = Dot(polar_to_point(p_moving_dot_phi) * scale_back + np.array(poincare_origin))
self.play(Transform(p_moving_dot, moving_dot), FadeOut(p_distance_number), FadeIn(p_distance_infty),
FadeOut(p_moving_text),
run_time=.25)
self.wait(4)
self.play(FadeOut(p_moving_dot), FadeOut(p_distance_infty), FadeOut(p_distance_text))
self.add_subcaption("Zum weiteren Verständnis ist folgende Darstellung der Verzerrung hilfreich", duration=5)
self.wait(3)
self.play(FadeIn(poincare_model), FadeOut(pcircle), run_time=2)
self.add_subcaption(
"Dies ist eine Parkettierung der hyperbolischen Ebene. All diese Dreiecke haben die gleiche Form und Fläche im Hyperbolischen.",
duration=8)
self.play(Create(ptri1), run_time=2)
self.play(Write(ptri1_text.shift(0.6 * LEFT + 0.15 * DOWN)))
self.wait(1)
self.play(Create(ptri2), run_time=2)
self.play(Write(ptri2_text.shift(0.8 * LEFT + 0.07 * UP)))
self.wait(1)
self.play(Write(ptri_size_text), run_time=2)
self.add_subcaption(
"Das Poincare-Modell liefert also eine stark verzerrte Darstellung des hyperbolischen Raumes.", duration=6)
self.wait(4)
self.play(Uncreate(ptri1), Uncreate(ptri2), Unwrite(ptri1_text), Unwrite(ptri2_text),
Unwrite(ptri_size_text), run_time=2)
self.play(self.camera.frame.animate.scale(1.25).move_to(center), FadeIn(klein_model), FadeIn(klein_text))
self.wait(2)
class HyperbolicModelsKlein(MovingCameraScene):
def construct(self):
center = np.array([0, 0, 0])
klein_origin = np.array([3.5, -1, 0])
poincare_origin = np.array([-3.5, -1, 0])
MY_BLUE = "#22c1dd"
title = MarkupText("Models for the hyperbolic plane")\
.scale(0.8)\
.shift(3.5 * UP)
klein_model = ImageMobject("tessellation_klein.png").scale(0.7).move_to(klein_origin)
klein_text = Text("Klein model").scale(0.6).move_to(3.5 * RIGHT + 2.2 * UP)
poincare_model = ImageMobject("tessellation_poincare.png").scale(0.7).move_to(poincare_origin)
poincare_text = Text("Poincaré model").scale(0.6).move_to(3.5 * LEFT + 2.2 * UP)
title2 = Text("Transformations between hyperbolic models").scale(.75).shift(3.5 * UP)
scale_back = 2.5
phis = [[0.4, 2], [1.3, 6], [3.5, 5], [4, 1.5]]
k1 = np.array([[0, 0, 0], [2, 0, 0], [1.24, 1.58, 0]])
k2 = np.array([[-0.46, 1.95, 0], [-1.79, 0.89, 0], [-1.49, 1.86, 0]])
raw_ktri1 = Polygon(*k1, color=YELLOW, stroke_width=1.5).shift(klein_origin)
kpoints1 = [Dot(np.add(p, klein_origin), color=YELLOW, radius=.06) for p in k1]
raw_ktri2 = Polygon(*k2, color=YELLOW, stroke_width=1.5).shift(klein_origin)
kpoints2 = [Dot(np.add(p, klein_origin), color=YELLOW, radius=.06) for p in k2]
kcircle = Circle(color=MY_BLUE, stroke_width=1).scale(scale_back).move_to(klein_origin)
pcircle = Circle(color=MY_BLUE, stroke_width=1).scale(scale_back).move_to(poincare_origin)
p_geodesics_raw = [HyperbolicArcBetweenPoints(polar_to_point(x), polar_to_point(y)) for [x, y] in phis]
p_geodesics = [geo.scale(scale_back).move_to(geo.get_center() * scale_back).shift(poincare_origin) for geo in
p_geodesics_raw]
k_geodesics_raw = [Line(polar_to_point(x), polar_to_point(y)) for [x, y] in phis]
k_geodesics = [geo.scale(scale_back).move_to(geo.get_center() * scale_back).shift(klein_origin) for geo in
k_geodesics_raw]
point_coords = [0, -0.3, 0]
point_geodesic_point_phis = [0, 2.3, 2.9, 3.15, 3.4, 3.8, 4.1, 4.5, 5]
k_point = Dot(point_coords)
k_point.move_to(k_point.get_center() * scale_back).shift(klein_origin)
k_point_text = Tex("$P$").scale(0.8).next_to(k_point, direction=UP + 0.2 * RIGHT)
k_point_geodesics_intersections = [
get_both_intersections_line_with_unit_circle(point_coords, polar_to_point(phi))
for phi in point_geodesic_point_phis]
k_point_geodesics_raw = [Line(x, y, color=MY_BLUE) for [x, y] in k_point_geodesics_intersections]
k_point_geodesics = [l.scale(scale_back).move_to(l.get_center() * scale_back).shift(klein_origin) for l in
k_point_geodesics_raw]
ktri1_text = Tex(r"$\Delta_1$").next_to(raw_ktri1, LEFT).scale(.6).shift(1.6 * RIGHT + 0.2 * DOWN)
ktri2_text = Tex(r"$\Delta_2$").next_to(raw_ktri2, LEFT).scale(.6).shift(1.05 * RIGHT + .2 * UP)
ktri_size_text = Tex(r"$A(\Delta_1) = A(\Delta_2)$", font_size=37).next_to(kcircle, LEFT, buff=.35)
arrow_lr = Arrow(start=2.5 * LEFT, end=2.5 * RIGHT, stroke_width=4, max_tip_length_to_length_ratio=.5)
arrow_rl = Arrow(start=2.5 * RIGHT, end=2.5 * LEFT, stroke_width=4, max_tip_length_to_length_ratio=.5)
arrow_group = VGroup(arrow_lr, arrow_rl).arrange(DOWN)
arrow_group.shift(DOWN)
f_text = Tex(r"$f$", font_size=40).next_to(arrow_lr, UP, buff=0)
f_inv_text = Tex(r"$f^{-1}$", font_size=40).next_to(arrow_rl, DOWN, buff=0)
f_formula = Tex(r"$f(x,y) = \frac{1}{1+\sqrt{1-x^2-y^2}}(x,y)$", font_size=40).move_to(.5 * UP)
f_inv_formula = Tex(r"$f^{-1}(x,y) = \frac{1}{1+x^2+y^2}(2x,2y)$", font_size=40).move_to(2.5 * DOWN)
p_geo0 = p_geodesics[0].copy()
p_geo0 = p_geo0.shift(-poincare_origin).scale(0.7).move_to(p_geo0.get_center() * 0.7).shift(
poincare_origin + 1.5 * LEFT)
p_geo0_copy = p_geo0.copy()
p_geo1 = p_geodesics[1].copy()
p_geo1 = p_geo1.shift(-poincare_origin).scale(0.7).move_to(p_geo1.get_center() * 0.7).shift(
poincare_origin + 1.5 * LEFT)
k_geo0 = k_geodesics[0].copy()
k_geo0 = k_geo0.shift(-klein_origin).scale(0.7).move_to(k_geo0.get_center() * 0.7).shift(
klein_origin + 1.5 * RIGHT)
k_geo1 = k_geodesics[1].copy()
k_geo1 = k_geo1.shift(-klein_origin).scale(0.7).move_to(k_geo1.get_center() * 0.7).shift(
klein_origin + 1.5 * RIGHT)
k_geo1_copy = k_geo1.copy()
k_geo_intersect_coords = get_intersection_from_angles(phis[0][0], phis[0][1], phis[1][0], phis[1][1])
p_geo_intersect_coords = tf_klein_to_poincare(k_geo_intersect_coords)
k_geo_intersect_raw = Dot(k_geo_intersect_coords, color=YELLOW, radius=.075)
k_geo_intersect_moved = k_geo_intersect_raw.move_to(k_geo_intersect_raw.get_center() * 1.75).shift(
klein_origin + 1.5 * RIGHT)
p_geo_intersect_raw = Dot(p_geo_intersect_coords, color=YELLOW, radius=.075)
p_geo_intersect_moved = p_geo_intersect_raw.move_to(p_geo_intersect_raw.get_center() * 1.75).shift(
poincare_origin + 1.5 * LEFT)
self.add(title, poincare_text, poincare_model, klein_text, klein_model)
self.add_subcaption(
"Nun betrachten wir das Klein-Modell. Wir werden starke Ähnlichkeiten zum Poincare-Modell erkennen.",
duration=7)
self.wait(4)
self.add(kcircle)
self.play(self.camera.frame.animate.scale(0.8).move_to(np.add(klein_origin, [0, 0.4, 0])),
FadeOut(poincare_model), FadeOut(poincare_text), FadeOut(klein_model))
self.wait(2)
self.add_subcaption(
"Der größte Unterschied ist, dass Geodätische hier tatsächlich euklidischen Geraden entsprechen.",
duration=6)
self.play(Create(k_geodesics[0]))
self.play(Create(k_geodesics[1]))
self.play(Create(k_geodesics[2]))
self.play(Create(k_geodesics[3]))
self.wait(2)
self.play(Uncreate(k_geodesics[1]), Uncreate(k_geodesics[2]), Uncreate(k_geodesics[3]), Create(k_point),
Write(k_point_text))
self.add_subcaption(
"Da diese aber wie im Poincare-Modell durch den Einheitskreis beschränkt werden, "
"gilt das Parallelenaxiom auch hier nicht.",
duration=7)
self.add_foreground_mobject(k_point)
self.add_foreground_mobject(k_point_text)
self.wait(2)
self.play(Create(k_point_geodesics[0]))
self.play(Create(k_point_geodesics[1]))
self.play(Create(k_point_geodesics[2]))
self.play(Create(k_point_geodesics[3]))
self.play(Create(k_point_geodesics[4]))
self.play(Create(k_point_geodesics[5]))
self.wait(2)
self.play(Uncreate(k_geodesics[0]),
Uncreate(k_point_geodesics[0]),
Uncreate(k_point_geodesics[1]),
Uncreate(k_point_geodesics[2]),
Uncreate(k_point_geodesics[3]),
Uncreate(k_point_geodesics[4]),
Uncreate(k_point_geodesics[5]),
Uncreate(k_point), Unwrite(k_point_text))
self.add_subcaption("Ebenso gibt es im Klein-Modell eine starke Verzerrung des Raumes", duration=4)
self.wait(3)
self.play(FadeIn(klein_model), FadeOut(kcircle))
self.add_subcaption("Hier sehen wir die gleiche Parkettierung "
"der hyperbolischen Ebene im Klein-Modell.",
duration=6)
self.play(Create(kpoints1[0]),
Create(kpoints1[1]),
Create(kpoints1[2]))
self.play(Create(raw_ktri1), run_time=2)
self.play(Write(ktri1_text))
self.wait(1)
self.play(Create(kpoints2[0]),
Create(kpoints2[1]),
Create(kpoints2[2]))
self.play(Create(raw_ktri2), run_time=2)
self.play(Write(ktri2_text))
self.wait(1)
self.add_subcaption("Zum Rand hin werden Distanzen auch hier immer größer", duration=4)
self.play(Write(ktri_size_text), run_time=2)
self.wait(4)
self.play(Uncreate(raw_ktri1),
Uncreate(kpoints1[0]),
Uncreate(kpoints1[1]),
Uncreate(kpoints1[2]),
Uncreate(raw_ktri2),
Uncreate(kpoints2[0]),
Uncreate(kpoints2[1]),
Uncreate(kpoints2[2]),
Unwrite(ktri1_text), Unwrite(ktri2_text),
Unwrite(ktri_size_text), run_time=2)
self.wait(2)
self.play(self.camera.frame.animate.scale(1.25).move_to(center), FadeIn(poincare_model), FadeIn(poincare_text))
self.add_subcaption(
"Da beide Modelle den gleichen Raum darstellen, ist es naheliegend, dass es Abbildungen gibt, "
"die die Punkte der Modelle miteinander identifizieren.",
duration=8)
self.wait(1)
self.play(Transform(title, title2))
self.wait(2)
self.play(poincare_model.animate.scale(0.7).shift(1.5 * LEFT),
klein_model.animate.scale(0.7).shift(1.5 * RIGHT),
poincare_text.animate.shift(1.5 * LEFT + .5 * DOWN),
klein_text.animate.shift(1.5 * RIGHT + .5 * DOWN))
self.add(pcircle.scale(0.7).shift(1.5 * LEFT), kcircle.scale(0.7).shift(1.5 * RIGHT))
self.wait(3)
self.add_subcaption("Tatsächlich gibt es diese, wir nennen sie f und f^-1. ", duration=4)
self.play(Write(arrow_group))
self.wait(3)
self.play(Write(f_text), Write(f_inv_text))
self.add_subcaption("Wie wir hier sehen, sind die Formeln für diese Abbildungen ziemlich einfach.", duration=4)
self.wait(2)
self.play(Write(f_formula))
self.wait(1)
self.play(Write(f_inv_formula))
self.wait(5)
self.add_subcaption(
"Mit f und f^-1 können wir jetzt Objekte aus "
"dem einen Modell ins andere übertragen. Insbesondere Geodätische.",
duration=7)
self.play(Create(p_geo0), FadeOut(poincare_model))
self.add(p_geo0_copy)
self.wait(2)
self.play(Transform(p_geo0, k_geo0), FadeOut(klein_model), Indicate(f_text))
self.wait(1)
self.play(Create(k_geo1))
self.add(k_geo1_copy)
self.wait(1)
self.play(Transform(k_geo1, p_geo1), Indicate(f_inv_text))
self.add_subcaption(
"Da die Abbildungen punktweise agieren, "
"bleiben Schnittpunkte zwischen Geodätischen erhalten.",
duration=6)
self.wait(2)
self.play(Create(k_geo_intersect_moved), Create(p_geo_intersect_moved),
Flash(k_geo_intersect_moved), Flash(p_geo_intersect_moved))
self.wait(3)
self.add_subcaption("Das liefert uns jetzt die folgende Umformulierung des Sieben-Kreise-Satzes.", duration=4)
everything = VGroup(*filter(lambda x: isinstance(x, VMobject), self.mobjects))
self.wait(3)
endDot = Dot(color=BLACK)
self.play(Transform(everything, endDot))
self.wait(3)
class SevenCirclesHyperbolic(MovingCameraScene):
def construct(self):
self.camera.frame.width = 9
OUTER_CIRCLE_COLOR = WHITE
INNER_CIRCLE_COLOR = GREEN_B
INNER_INTERSECTION_COLOR = PURPLE
OUTER_INTERSECTION_COLOR = DARK_GREY
HEXAGON_COLOR = GREY
DIAGONAL_COLOR = ORANGE
DIAGONAL_INTERSECTION_COLOR = YELLOW
title = Text("The hyperbolic Seven Circles Theorem").scale(0.8)
theorem_text_colored = Tex(r"Let $C\ $be a circle ", r"containing six smaller circles, ",
r"such that each inner circle is tangent to $C$ ",
"and any two adjacent inner circles are tangent. ",
r"Let there also be a hyperbolic hexagon formed by the intersection points of each inner cicle and $C$. ",
"Then the three hyperbolic diagonals will always meet at a single point.", "",
substrings_to_isolate=[r"$C\ $", r"six smaller circles", r"tangent to $C$",
"adjacent inner circles are tangent",
"hyperbolic hexagon",
"hyperbolic diagonals",
"single point"],
stroke_width=.05).set_color_by_tex(r"$C\ $",
OUTER_CIRCLE_COLOR).set_color_by_tex(
r"six smaller circles",
INNER_CIRCLE_COLOR).set_color_by_tex(
r"tangent to $C$",
OUTER_INTERSECTION_COLOR).set_color_by_tex(
"adjacent inner circles are tangent",
INNER_INTERSECTION_COLOR).set_color_by_tex(
"hyperbolic hexagon", HEXAGON_COLOR).set_color_by_tex(
"hyperbolic diagonals", DIAGONAL_COLOR).set_color_by_tex(
"single point", DIAGONAL_INTERSECTION_COLOR).scale(
0.5).move_to([0, -2, 0])
theorem_text_white = Tex(r"Let $C\ $be a circle ", r"containing six smaller circles, ",
r"such that each inner circle is tangent to $C$ ",
"and any two adjacent inner circles are tangent. ",
r"Let there also be a hyperbolic hexagon formed by the intersection points of each inner cicle and $C$. ",
"Then the three hyperbolic diagonals will always meet at a single point.", "",
stroke_width=.05).scale(
0.5).move_to([0, -2, 0])
circle = Circle(color=OUTER_CIRCLE_COLOR, stroke_width=2)
phis = HexagonAngles(np.array([.3, 1.6, 2.2, 3, 4.3]))
# phis = create_phis(min_dist=.9, max_dist=1.2)
first_circle_radius = .4
hexagon = HyperbolicPolygon([polar_to_point(phi) for phi in phis], color=HEXAGON_COLOR,
dot_color=OUTER_INTERSECTION_COLOR, stroke_width=2)
hexagon_circles = HexagonCircles(hexagon, first_circle_radius, stroke_width=2, color=INNER_CIRCLE_COLOR)
inner_intersections = get_intersections_of_n_tangent_circles(hexagon_circles.circles,
color=INNER_INTERSECTION_COLOR)
outer_intersections = get_intersections_of_circles_with_unit_circle(hexagon_circles.circles,
color=OUTER_INTERSECTION_COLOR)
diagonals = HexagonMainDiagonals(hexagon, color=DIAGONAL_COLOR, stroke_width=2)
diagonal_intersection = Dot(
tf_klein_to_poincare(get_intersection_from_angles(phis[0], phis[3], phis[1], phis[4])),
color=DIAGONAL_INTERSECTION_COLOR, radius=.05)
eucl_hexagon = EuclideanHexagon(phis, color=HEXAGON_COLOR, stroke_width=2)
eucl_diagonals = get_diagonals(hexagon, color=DIAGONAL_COLOR, stroke_width=2)
eucl_intersection = Dot(get_intersection_from_angles(phis[0], phis[3], phis[1], phis[4]),
color=DIAGONAL_INTERSECTION_COLOR, radius=.05)
self.camera.frame.shift(0.8 * DOWN)
self.play(Write(title.shift(1.4 * UP).scale(.7)))
self.wait(2)
self.add_subcaption("Sei C_0 ein Kreis", duration=2)
self.play(Write(theorem_text_white[0], run_time=2))
self.play(FadeIn(circle), FadeIn(theorem_text_colored[1]))
self.add_foreground_mobject(circle)
self.add_subcaption("und C_1,...,C_6 in C_0 enthaltene Kreise", duration=4)
self.play(Write(theorem_text_white[1], run_time=3))
self.play(Create(hexagon_circles, run_time=4), FadeIn(theorem_text_colored[4], run_time=1))
self.remove_foreground_mobjects(circle)
self.add_subcaption("sodass jeder innere Kreis zum äußeren Kreis tangential ist", duration=5)
self.play(Write(theorem_text_white[2], run_time=3))
self.play(FadeIn(theorem_text_colored[7]))
self.play(Create(outer_intersections[0]),
Create(outer_intersections[1]),
Create(outer_intersections[2]),
Create(outer_intersections[3]),
Create(outer_intersections[4]),
Create(outer_intersections[5]),
run_time=1)
self.add_foreground_mobjects(*outer_intersections)
self.add_subcaption("und je zwei nebeneinanderliegende innere Kreise ebenfalls zueinander tangential sind.",
duration=7)
self.play(Write(theorem_text_white[3]), run_time=6)
self.play(FadeIn(theorem_text_colored[10]))
self.play(Create(inner_intersections[0]),
Create(inner_intersections[1]),
Create(inner_intersections[2]),
Create(inner_intersections[3]),
Create(inner_intersections[4]),
Create(inner_intersections[5]),
run_time=1)
for i in range(6):
self.add_foreground_mobject(inner_intersections[i])
self.add_subcaption(
"Dann treffen sich die drei hyperbolischen Diagonalen des hyperbolischen Hexagons, das von den Schnittpunkten der inneren Kreise mit dem äußeren Kreis gebildet wird,",
duration=10)
self.play(Write(theorem_text_white[4]), run_time=8)
self.play(Create(hexagon, run_time=5), FadeIn(theorem_text_colored[13], run_time=1))