3b1b-manim/hilbert/curves.py

from mobject import Mobject, Point, Mobject1D
from scene import Scene
from animation.transform import Transform
from animation.simple_animations import ShowCreation
from topics.geometry import Line

from helpers import *

def rotate(points, angle = np.pi, axis = OUT):
    if axis is None:
        return points
    matrix = rotation_matrix(angle, axis)
    points = np.dot(points, np.transpose(matrix))
    return points


class SpaceFillingCurve(Mobject1D):
    DEFAULT_CONFIG = {
        "radius"      : 3,
        "order"       : 5,
        "start_color" : RED,
        "end_color"   : GREEN,
    }

    def generate_points(self):
        points = self.get_anchor_points()
        for pair in zip(points, points[1:]):
            self.add_line(*pair)
        self.gradient_highlight(self.start_color, self.end_color)

    def get_anchor_points(self):
        raise Exception("Not implemented")


class LindenmayerCurve(SpaceFillingCurve):
    DEFAULT_CONFIG = {
        "axiom"        : "A",
        "rule"         : {},
        "scale_factor" : 2,
        "radius"       : 3,
        "start_step"   : RIGHT,
        "angle"        : np.pi/2,
    }

    def expand_command_string(self, command):
        result = ""
        for letter in command:
            if letter in self.rule:
                result += self.rule[letter]
            else:
                result += letter
        return result

    def get_command_string(self):
        result = self.axiom
        for x in range(self.order):
            result = self.expand_command_string(result)
        return result

    def get_anchor_points(self):
        step = float(self.radius) * self.start_step 
        step /= (self.scale_factor**self.order)
        curr = np.zeros(3)
        result = [curr]
        for letter in self.get_command_string():
            if letter is "+":
                step = rotate(step, self.angle)
            elif letter is "-":
                step = rotate(step, -self.angle)
            else:
                curr = curr + step
                result.append(curr)
        return np.array(result) - center_of_mass(result)


class SelfSimilarSpaceFillingCurve(SpaceFillingCurve):
    DEFAULT_CONFIG = {
        "offsets" : [],
        #keys must awkwardly be in string form...
        "offset_to_rotation_axis" : {},
        "scale_factor" : 2,
        "radius_scale_factor" : 0.5,
    }
    def transform(self, points, offset):
        """
        How to transform the copy of points shifted by
        offset.  Generally meant to be extended in subclasses
        """
        copy = np.array(points)
        if str(offset) in self.offset_to_rotation_axis:
            copy = rotate(
                copy, 
                axis = self.offset_to_rotation_axis[str(offset)]
            )
        copy /= self.scale_factor,
        copy += offset*self.radius*self.radius_scale_factor
        return copy

    def refine_into_subparts(self, points):
        transformed_copies = [
            self.transform(points, offset)
            for offset in self.offsets
        ]
        return reduce(
            lambda a, b : np.append(a, b, axis = 0),
            transformed_copies
        )


    def get_anchor_points(self):
        points = np.zeros((1, 3))
        for count in range(self.order):
            points = self.refine_into_subparts(points)
        return points

    def generate_grid(self):
        raise Exception("Not implemented")



class HilbertCurve(SelfSimilarSpaceFillingCurve):
    DEFAULT_CONFIG = {
        "offsets" : [
            LEFT+DOWN,
            LEFT+UP,
            RIGHT+UP,
            RIGHT+DOWN,
        ],
        "offset_to_rotation_axis" : {
            str(LEFT+DOWN)  : RIGHT+UP,
            str(RIGHT+DOWN) : RIGHT+DOWN,
        },
     }


class HilbertCurve3D(SelfSimilarSpaceFillingCurve):
    DEFAULT_CONFIG = {
        "offsets" : [ 
           LEFT+DOWN+OUT,
           LEFT+UP+OUT,
           LEFT+UP+IN,
           LEFT+DOWN+IN,
           RIGHT+DOWN+IN,                                               
           RIGHT+UP+IN,
           RIGHT+UP+OUT,
           RIGHT+DOWN+OUT,
        ],
        "offset_to_rotation_axis" : {}#TODO
    }

class PeanoCurve(SelfSimilarSpaceFillingCurve):
    DEFAULT_CONFIG = {
        "start_color" : PURPLE,
        "end_color"   : TEAL,
        "offsets" : [
            LEFT+DOWN,
            LEFT,
            LEFT+UP,
            UP,
            ORIGIN,
            DOWN,
            RIGHT+DOWN,
            RIGHT,
            RIGHT+UP,
        ],
        "offset_to_rotation_axis" : {
            str(LEFT)   : UP,       
            str(UP)     : RIGHT,    
            str(ORIGIN) : LEFT+UP,  
            str(DOWN)   : RIGHT, 
            str(RIGHT)  : UP,   
        },
        "scale_factor" : 3,
        "radius_scale_factor" : 2.0/3,
    }

class TriangleFillingCurve(SelfSimilarSpaceFillingCurve):
    DEFAULT_CONFIG = {
        "start_color" : MAROON,
        "end_color"   : YELLOW,
        "offsets" : [
            LEFT/4.+DOWN/6.,
            ORIGIN,
            RIGHT/4.+DOWN/6.,
            UP/3.,
        ],
        "offset_to_rotation_axis" : {
            str(ORIGIN): RIGHT,
            str(UP/3.) : UP,
        },
        "scale_factor" : 2,
        "radius_scale_factor" : 1.5,
    }

# class HexagonFillingCurve(SelfSimilarSpaceFillingCurve):
#     DEFAULT_CONFIG = {
#         "start_color" : WHITE,
#         "end_color"   : BLUE_D,
#         "axis_offset_pairs" : [
#             (None,                1.5*DOWN + 0.5*np.sqrt(3)*LEFT),
#             (UP+np.sqrt(3)*RIGHT, 1.5*DOWN + 0.5*np.sqrt(3)*RIGHT),
#             (np.sqrt(3)*UP+RIGHT, ORIGIN),            
#             ((UP, RIGHT),         np.sqrt(3)*LEFT),
#             (None,                1.5*UP + 0.5*np.sqrt(3)*LEFT),
#             (None,                1.5*UP + 0.5*np.sqrt(3)*RIGHT),
#             (RIGHT,               np.sqrt(3)*RIGHT),
#         ],
#         "scale_factor" : 3,
#         "radius_scale_factor" : 2/(3*np.sqrt(3)),
#     }

#     def refine_into_subparts(self, points):
#         return SelfSimilarSpaceFillingCurve.refine_into_subparts(
#             self,
#             rotate(points, np.pi/6, IN)
#         )


class UtahFillingCurve(SelfSimilarSpaceFillingCurve):
    DEFAULT_CONFIG = {
        "start_color" : WHITE,
        "end_color"   : BLUE_D,
        "axis_offset_pairs" : [

        ],
        "scale_factor" : 3,
        "radius_scale_factor" : 2/(3*np.sqrt(3)),
    }


class FlowSnake(LindenmayerCurve):
    DEFAULT_CONFIG = {
        "start_color" : YELLOW,
        "end_color"   : GREEN,
        "axiom"       : "A",
        "rule" : {
            "A" : "A-B--B+A++AA+B-",
            "B" : "+A-BB--B-A++A+B",
        },
        "radius"       : 6, #TODO, this is innaccurate
        "scale_factor" : np.sqrt(7),
        "start_step"   : RIGHT,
        "angle"        : -np.pi/3,
    }
    def __init__(self, **kwargs):
        LindenmayerCurve.__init__(self, **kwargs)
        self.rotate(-self.order*np.pi/9)

class Sierpinski(LindenmayerCurve):
    DEFAULT_CONFIG = {
        "start_color" : RED,
        "end_color"   : WHITE,
        "axiom"       : "A",
        "rule" : {
            "A" : "+B-A-B+",
            "B" : "-A+B+A-",
        },
        "radius"       : 6, #TODO, this is innaccurate
        "scale_factor" : 2,
        "start_step"   : RIGHT,
        "angle"        : -np.pi/3,
    }


class SnakeCurve(SpaceFillingCurve):
    DEFAULT_CONFIG = {
        "start_color" : BLUE,
        "end_color"   : YELLOW,
    }
    def get_anchor_points(self):
        result = []
        resolution = 2**self.order
        step = 2.0*self.radius / resolution
        lower_left = ORIGIN + \
                     LEFT*(self.radius - step/2) + \
                     DOWN*(self.radius - step/2)

        for y in range(resolution):
            x_range = range(resolution)
            if y%2 == 0:
                x_range.reverse()
            for x in x_range:
                result.append(
                    lower_left + x*step*RIGHT + y*step*UP
                )
        return result



class SpaceFillingCurveScene(Scene):
    @staticmethod
    def args_to_string(CurveClass, order):
        return CurveClass.__name__ + "Order" + str(order)

    @staticmethod
    def string_to_args(arg_str):
        curve_class_name, order_str = arg_str.split()
        space_filling_curves = dict([
            (Class.__name__, Class)
            for Class in get_all_descendent_classes(SpaceFillingCurve)
        ])
        if curve_class_name not in space_filling_curves:
            raise Exception(
                "%s is not a space filling curve"%curve_class_name
            )
        CurveClass = space_filling_curves[curve_class_name]
        return CurveClass, int(order_str)

class TransformOverIncreasingOrders(SpaceFillingCurveScene):
    def construct(self, CurveClass, max_order):
        sample = CurveClass(order = 1)
        self.curve = Line(sample.radius*LEFT, sample.radius*RIGHT)
        self.curve.gradient_highlight(
            sample.start_color, 
            sample.end_color
        )
        for order in range(1, max_order):
            new_curve = CurveClass(order = order)
            self.play(
                Transform(self.curve, new_curve),
                run_time = 3/np.sqrt(order),
            )
        self.dither()


class DrawSpaceFillingCurve(SpaceFillingCurveScene):
    def construct(self, CurveClass, order):
        curve = CurveClass(order = order)
        self.play(ShowCreation(curve), run_time = 10)
        self.dither()