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AboutSpaceFillingCurves
This commit is contained in:
Grant Sanderson
parent
399852a994
commit
7ab9d6ab86
5 changed files with 466 additions and 0 deletions
0
hilbert/__init__.py
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0
hilbert/__init__.py
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339
hilbert/curves.py
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339
hilbert/curves.py
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from mobject import Mobject, Point, Mobject1D
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from scene import Scene
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from animation.transform import Transform
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from animation.simple_animations import ShowCreation
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from topics.geometry import Line
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from helpers import *
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def rotate(points, angle = np.pi, axis = OUT):
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if axis is None:
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return points
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matrix = rotation_matrix(angle, axis)
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points = np.dot(points, np.transpose(matrix))
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return points
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class SpaceFillingCurve(Mobject1D):
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DEFAULT_CONFIG = {
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"radius" : 3,
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"order" : 5,
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"start_color" : RED,
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"end_color" : GREEN,
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}
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def generate_points(self):
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points = self.get_anchor_points()
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for pair in zip(points, points[1:]):
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self.add_line(*pair)
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self.gradient_highlight(self.start_color, self.end_color)
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def get_anchor_points(self):
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raise Exception("Not implemented")
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class LindenmayerCurve(SpaceFillingCurve):
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DEFAULT_CONFIG = {
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"axiom" : "A",
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"rule" : {},
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"scale_factor" : 2,
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"radius" : 3,
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"start_step" : RIGHT,
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"angle" : np.pi/2,
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}
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def expand_command_string(self, command):
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result = ""
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for letter in command:
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if letter in self.rule:
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result += self.rule[letter]
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else:
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result += letter
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return result
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def get_command_string(self):
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result = self.axiom
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for x in range(self.order):
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result = self.expand_command_string(result)
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return result
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def get_anchor_points(self):
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step = float(self.radius) * self.start_step
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step /= (self.scale_factor**self.order)
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curr = np.zeros(3)
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result = [curr]
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for letter in self.get_command_string():
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if letter is "+":
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step = rotate(step, self.angle)
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elif letter is "-":
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step = rotate(step, -self.angle)
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else:
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curr = curr + step
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result.append(curr)
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return np.array(result) - center_of_mass(result)
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class SelfSimilarSpaceFillingCurve(SpaceFillingCurve):
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DEFAULT_CONFIG = {
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"offsets" : [],
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#keys must awkwardly be in string form...
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"offset_to_rotation_axis" : {},
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"scale_factor" : 2,
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"radius_scale_factor" : 0.5,
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}
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def transform(self, points, offset):
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"""
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How to transform the copy of points shifted by
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offset. Generally meant to be extended in subclasses
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"""
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copy = np.array(points)
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if str(offset) in self.offset_to_rotation_axis:
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copy = rotate(
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copy,
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axis = self.offset_to_rotation_axis[str(offset)]
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)
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copy /= self.scale_factor,
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copy += offset*self.radius*self.radius_scale_factor
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return copy
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def refine_into_subparts(self, points):
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transformed_copies = [
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self.transform(points, offset)
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for offset in self.offsets
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]
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return reduce(
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lambda a, b : np.append(a, b, axis = 0),
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transformed_copies
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)
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def get_anchor_points(self):
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points = np.zeros((1, 3))
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for count in range(self.order):
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points = self.refine_into_subparts(points)
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return points
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class HilbertCurve(SelfSimilarSpaceFillingCurve):
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DEFAULT_CONFIG = {
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"offsets" : [
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LEFT+DOWN,
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LEFT+UP,
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RIGHT+UP,
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RIGHT+DOWN,
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],
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"offset_to_rotation_axis" : {
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str(LEFT+DOWN) : RIGHT+UP,
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str(RIGHT+DOWN) : RIGHT+DOWN,
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},
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}
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class HilbertCurve3D(SelfSimilarSpaceFillingCurve):
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DEFAULT_CONFIG = {
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"offsets" : [
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LEFT+DOWN+OUT,
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LEFT+UP+OUT,
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LEFT+UP+IN,
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LEFT+DOWN+IN,
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RIGHT+DOWN+IN,
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RIGHT+UP+IN,
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RIGHT+UP+OUT,
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RIGHT+DOWN+OUT,
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],
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"offset_to_rotation_axis" : {}#TODO
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}
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class PeanoCurve(SelfSimilarSpaceFillingCurve):
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DEFAULT_CONFIG = {
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"start_color" : PURPLE,
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"end_color" : TEAL,
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"offsets" : [
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LEFT+DOWN,
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LEFT,
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LEFT+UP,
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UP,
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ORIGIN,
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DOWN,
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RIGHT+DOWN,
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RIGHT,
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RIGHT+UP,
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],
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"offset_to_rotation_axis" : {
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str(LEFT) : UP,
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str(UP) : RIGHT,
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str(ORIGIN) : LEFT+UP,
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str(DOWN) : RIGHT,
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str(RIGHT) : UP,
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},
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"scale_factor" : 3,
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"radius_scale_factor" : 2.0/3,
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}
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class TriangleFillingCurve(SelfSimilarSpaceFillingCurve):
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DEFAULT_CONFIG = {
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"start_color" : MAROON,
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"end_color" : YELLOW,
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"offsets" : [
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LEFT/4.+DOWN/6.,
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ORIGIN,
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RIGHT/4.+DOWN/6.,
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UP/3.,
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],
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"offset_to_rotation_axis" : {
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str(ORIGIN): RIGHT,
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str(UP/3.) : UP,
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},
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"scale_factor" : 2,
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"radius_scale_factor" : 1.5,
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}
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# class HexagonFillingCurve(SelfSimilarSpaceFillingCurve):
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# DEFAULT_CONFIG = {
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# "start_color" : WHITE,
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# "end_color" : BLUE_D,
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# "axis_offset_pairs" : [
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# (None, 1.5*DOWN + 0.5*np.sqrt(3)*LEFT),
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# (UP+np.sqrt(3)*RIGHT, 1.5*DOWN + 0.5*np.sqrt(3)*RIGHT),
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# (np.sqrt(3)*UP+RIGHT, ORIGIN),
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# ((UP, RIGHT), np.sqrt(3)*LEFT),
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# (None, 1.5*UP + 0.5*np.sqrt(3)*LEFT),
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# (None, 1.5*UP + 0.5*np.sqrt(3)*RIGHT),
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# (RIGHT, np.sqrt(3)*RIGHT),
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# ],
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# "scale_factor" : 3,
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# "radius_scale_factor" : 2/(3*np.sqrt(3)),
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# }
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# def refine_into_subparts(self, points):
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# return SelfSimilarSpaceFillingCurve.refine_into_subparts(
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# self,
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# rotate(points, np.pi/6, IN)
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# )
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class UtahFillingCurve(SelfSimilarSpaceFillingCurve):
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DEFAULT_CONFIG = {
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"start_color" : WHITE,
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"end_color" : BLUE_D,
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"axis_offset_pairs" : [
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],
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"scale_factor" : 3,
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"radius_scale_factor" : 2/(3*np.sqrt(3)),
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}
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class FlowSnake(LindenmayerCurve):
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DEFAULT_CONFIG = {
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"start_color" : YELLOW,
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"end_color" : GREEN,
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"axiom" : "A",
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"rule" : {
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"A" : "A-B--B+A++AA+B-",
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"B" : "+A-BB--B-A++A+B",
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},
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"radius" : 6, #TODO, this is innaccurate
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"scale_factor" : np.sqrt(7),
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"start_step" : RIGHT,
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"angle" : -np.pi/3,
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}
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def __init__(self, **kwargs):
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LindenmayerCurve.__init__(self, **kwargs)
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self.rotate(-self.order*np.pi/9)
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class Sierpinski(LindenmayerCurve):
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DEFAULT_CONFIG = {
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"start_color" : RED,
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"end_color" : WHITE,
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"axiom" : "A",
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"rule" : {
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"A" : "+B-A-B+",
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"B" : "-A+B+A-",
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},
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"radius" : 6, #TODO, this is innaccurate
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"scale_factor" : 2,
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"start_step" : RIGHT,
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"angle" : -np.pi/3,
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}
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class SnakeCurve(SpaceFillingCurve):
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DEFAULT_CONFIG = {
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"start_color" : BLUE,
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"end_color" : YELLOW,
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}
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def get_anchor_points(self):
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result = []
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resolution = 2**self.order
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lower_left = ORIGIN + \
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LEFT*self.radius + \
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DOWN*self.radius
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step = 2.0*self.radius / (resolution-1)
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for y in range(resolution):
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x_range = range(resolution)
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if y%2 == 0:
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x_range.reverse()
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for x in x_range:
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result.append(
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lower_left + x*step*RIGHT + y*step*UP
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)
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return result
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class SpaceFillingCurveScene(Scene):
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@staticmethod
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def args_to_string(CurveClass, order):
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return CurveClass.__name__ + "Order" + str(order)
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@staticmethod
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def string_to_args(arg_str):
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curve_class_name, order_str = arg_str.split()
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space_filling_curves = dict([
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(Class.__name__, Class)
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for Class in get_all_descendent_classes(SpaceFillingCurve)
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])
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if curve_class_name not in space_filling_curves:
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raise Exception(
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"%s is not a space filling curve"%curve_class_name
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)
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CurveClass = space_filling_curves[curve_class_name]
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return CurveClass, int(order_str)
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class TransformOverIncreasingOrders(SpaceFillingCurveScene):
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def construct(self, CurveClass, max_order):
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sample = CurveClass(order = 1)
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self.curve = Line(sample.radius*LEFT, sample.radius*RIGHT)
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self.curve.gradient_highlight(
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sample.start_color,
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sample.end_color
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)
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for order in range(1, max_order):
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new_curve = CurveClass(order = order)
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self.play(
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Transform(self.curve, new_curve),
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run_time = 3/np.sqrt(order),
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)
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self.dither()
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class DrawSpaceFillingCurve(SpaceFillingCurveScene):
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def construct(self, CurveClass, order):
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curve = CurveClass(order = order)
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self.play(ShowCreation(curve), run_time = 10)
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self.dither()
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113
hilbert/section1.py
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113
hilbert/section1.py
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@ -0,0 +1,113 @@
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from mobject import Mobject, Point
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from mobject.tex_mobject import TexMobject, TextMobject
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from scene import Scene
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from animation.transform import Transform, CounterclockwiseTransform, ApplyMethod
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from animation.simple_animations import ShowCreation, ShimmerIn
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from animation.meta_animations import DelayByOrder
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from topics.geometry import Line
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from topics.characters import ThoughtBubble
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from helpers import *
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from hilbert.curves import TransformOverIncreasingOrders, FlowSnake
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class AboutSpaceFillingCurves(TransformOverIncreasingOrders):
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@staticmethod
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def args_to_string():
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return ""
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@staticmethod
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def string_to_args(arg_str):
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return ()
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def construct(self):
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self.bubble = ThoughtBubble().ingest_sub_mobjects()
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self.bubble.scale(1.5)
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TransformOverIncreasingOrders.construct(self, FlowSnake, 3)
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self.play(Transform(self.curve, self.bubble))
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self.show_infinite_objects()
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self.pose_question()
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self.dither()
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def show_infinite_objects(self):
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sigma, summand, equals, result = TexMobject([
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"\\sum_{n = 1}^{\\infty}",
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"\\dfrac{1}{n^2}",
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"=",
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"\\dfrac{\pi^2}{6}"
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]).split()
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alt_summand = TexMobject("n").replace(summand)
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alt_result = TexMobject("-\\dfrac{1}{12}").replace(result)
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rationals, other_equals, naturals = TexMobject([
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"|\\mathds{Q}|",
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"=",
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"|\\mathds{N}|"
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]).scale(2).split()
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infinity = TexMobject("\\infty").scale(2)
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local_mobjects = filter(
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lambda m : isinstance(m, Mobject),
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locals().values(),
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)
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for mob in local_mobjects:
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mob.sort_points(np.linalg.norm)
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self.play(ShimmerIn(infinity))
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self.dither()
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self.play(
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ShimmerIn(summand),
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ShimmerIn(equals),
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ShimmerIn(result),
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DelayByOrder(Transform(infinity, sigma))
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)
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self.dither()
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self.play(
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Transform(summand, alt_summand),
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Transform(result, alt_result),
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)
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self.dither()
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self.remove(infinity)
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self.play(*[
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CounterclockwiseTransform(
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Mobject(summand, equals, result, sigma),
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Mobject(rationals, other_equals, naturals)
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)
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])
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self.dither()
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self.clear()
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self.add(self.bubble)
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def pose_question(self):
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infinity, rightarrow, N = TexMobject([
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"\\infty", "\\rightarrow", "N"
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]).scale(2).split()
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question_mark = TextMobject("?").scale(2)
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self.add(question_mark)
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self.dither()
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self.play(*[
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ShimmerIn(mob)
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for mob in infinity, rightarrow, N
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] + [
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ApplyMethod(question_mark.next_to, rightarrow, UP),
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])
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self.dither()
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class PostponePhilosophizing(Scene):
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def construct(self):
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pass
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7
hilbert/section2.py
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7
hilbert/section2.py
Normal file
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@ -0,0 +1,7 @@
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from mobject import Mobject
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from scene import Scene
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from animation.transform import Transform
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from animation.simple_animations import ShowCreation
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from topics.geometry import Line, Point
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from helpers import *
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7
hilbert/section3.py
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7
hilbert/section3.py
Normal file
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@ -0,0 +1,7 @@
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from mobject import Mobject
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from scene import Scene
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from animation.transform import Transform
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from animation.simple_animations import ShowCreation
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from topics.geometry import Line, Point
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from helpers import *
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