mirror of
https://github.com/3b1b/manim.git
synced 2025-04-13 09:47:07 +00:00
461 lines
12 KiB
Python
461 lines
12 KiB
Python
from __future__ import annotations
|
|
|
|
from functools import reduce
|
|
import math
|
|
import operator as op
|
|
import platform
|
|
|
|
from mapbox_earcut import triangulate_float32 as earcut
|
|
import numpy as np
|
|
from scipy.spatial.transform import Rotation
|
|
from tqdm.auto import tqdm as ProgressDisplay
|
|
|
|
from manimlib.constants import DOWN, OUT, RIGHT, UP
|
|
from manimlib.constants import PI, TAU
|
|
from manimlib.utils.iterables import adjacent_pairs
|
|
from manimlib.utils.simple_functions import clip
|
|
|
|
from typing import TYPE_CHECKING
|
|
|
|
if TYPE_CHECKING:
|
|
from typing import Callable, Sequence, List, Tuple
|
|
from manimlib.typing import Vect2, Vect3, Vect4, VectN, Matrix3x3, Vect3Array, Vect2Array
|
|
|
|
|
|
def cross(v1: Vect3 | List[float], v2: Vect3 | List[float]) -> Vect3:
|
|
return np.array([
|
|
v1[1] * v2[2] - v1[2] * v2[1],
|
|
v1[2] * v2[0] - v1[0] * v2[2],
|
|
v1[0] * v2[1] - v1[1] * v2[0]
|
|
])
|
|
|
|
|
|
def get_norm(vect: VectN | List[float]) -> float:
|
|
return sum((x**2 for x in vect))**0.5
|
|
|
|
|
|
def normalize(
|
|
vect: VectN | List[float],
|
|
fall_back: VectN | List[float] | None = None
|
|
) -> VectN:
|
|
norm = get_norm(vect)
|
|
if norm > 0:
|
|
return np.array(vect) / norm
|
|
elif fall_back is not None:
|
|
return np.array(fall_back)
|
|
else:
|
|
return np.zeros(len(vect))
|
|
|
|
|
|
# Operations related to rotation
|
|
|
|
|
|
def quaternion_mult(*quats: Vect4) -> Vect4:
|
|
"""
|
|
Inputs are treated as quaternions, where the real part is the
|
|
last entry, so as to follow the scipy Rotation conventions.
|
|
"""
|
|
if len(quats) == 0:
|
|
return np.array([0, 0, 0, 1])
|
|
result = np.array(quats[0])
|
|
for next_quat in quats[1:]:
|
|
x1, y1, z1, w1 = result
|
|
x2, y2, z2, w2 = next_quat
|
|
result[:] = [
|
|
w1 * x2 + x1 * w2 + y1 * z2 - z1 * y2,
|
|
w1 * y2 + y1 * w2 + z1 * x2 - x1 * z2,
|
|
w1 * z2 + z1 * w2 + x1 * y2 - y1 * x2,
|
|
w1 * w2 - x1 * x2 - y1 * y2 - z1 * z2,
|
|
]
|
|
return result
|
|
|
|
|
|
def quaternion_from_angle_axis(
|
|
angle: float,
|
|
axis: Vect3,
|
|
) -> Vect4:
|
|
return Rotation.from_rotvec(angle * normalize(axis)).as_quat()
|
|
|
|
|
|
def angle_axis_from_quaternion(quat: Vect4) -> Tuple[float, Vect3]:
|
|
rot_vec = Rotation.from_quat(quat).as_rotvec()
|
|
norm = get_norm(rot_vec)
|
|
return norm, rot_vec / norm
|
|
|
|
|
|
def quaternion_conjugate(quaternion: Vect4) -> Vect4:
|
|
result = np.array(quaternion)
|
|
result[:3] *= -1
|
|
return result
|
|
|
|
|
|
def rotate_vector(
|
|
vector: Vect3,
|
|
angle: float,
|
|
axis: Vect3 = OUT
|
|
) -> Vect3:
|
|
rot = Rotation.from_rotvec(angle * normalize(axis))
|
|
return np.dot(vector, rot.as_matrix().T)
|
|
|
|
|
|
def rotate_vector_2d(vector: Vect2, angle: float) -> Vect2:
|
|
# Use complex numbers...because why not
|
|
z = complex(*vector) * np.exp(complex(0, angle))
|
|
return np.array([z.real, z.imag])
|
|
|
|
|
|
def rotation_matrix_transpose_from_quaternion(quat: Vect4) -> Matrix3x3:
|
|
return Rotation.from_quat(quat).as_matrix()
|
|
|
|
|
|
def rotation_matrix_from_quaternion(quat: Vect4) -> Matrix3x3:
|
|
return np.transpose(rotation_matrix_transpose_from_quaternion(quat))
|
|
|
|
|
|
def rotation_matrix(angle: float, axis: Vect3) -> Matrix3x3:
|
|
"""
|
|
Rotation in R^3 about a specified axis of rotation.
|
|
"""
|
|
return Rotation.from_rotvec(angle * normalize(axis)).as_matrix()
|
|
|
|
|
|
def rotation_matrix_transpose(angle: float, axis: Vect3) -> Matrix3x3:
|
|
return rotation_matrix(angle, axis).T
|
|
|
|
|
|
def rotation_about_z(angle: float) -> Matrix3x3:
|
|
cos_a = math.cos(angle)
|
|
sin_a = math.sin(angle)
|
|
return np.array([
|
|
[cos_a, -sin_a, 0],
|
|
[sin_a, cos_a, 0],
|
|
[0, 0, 1]
|
|
])
|
|
|
|
|
|
def rotation_between_vectors(v1: Vect3, v2: Vect3) -> Matrix3x3:
|
|
if np.isclose(v1, v2).all():
|
|
return np.identity(3)
|
|
axis = np.cross(v1, v2)
|
|
if np.isclose(axis, [0, 0, 0]).all():
|
|
# v1 and v2 align
|
|
axis = np.cross(v1, RIGHT)
|
|
if np.isclose(axis, [0, 0, 0]).all():
|
|
# v1 and v2 _and_ RIGHT all align
|
|
axis = np.cross(v1, UP)
|
|
return rotation_matrix(
|
|
angle=angle_between_vectors(v1, v2),
|
|
axis=axis,
|
|
)
|
|
|
|
|
|
def z_to_vector(vector: Vect3) -> Matrix3x3:
|
|
return rotation_between_vectors(OUT, vector)
|
|
|
|
|
|
def angle_of_vector(vector: Vect2 | Vect3) -> float:
|
|
"""
|
|
Returns polar coordinate theta when vector is project on xy plane
|
|
"""
|
|
return np.angle(complex(*vector[:2]))
|
|
|
|
|
|
def angle_between_vectors(v1: VectN, v2: VectN) -> float:
|
|
"""
|
|
Returns the angle between two 3D vectors.
|
|
This angle will always be btw 0 and pi
|
|
"""
|
|
n1 = get_norm(v1)
|
|
n2 = get_norm(v2)
|
|
if n1 == 0 or n2 == 0:
|
|
return 0
|
|
cos_angle = np.dot(v1, v2) / np.float64(n1 * n2)
|
|
return math.acos(clip(cos_angle, -1, 1))
|
|
|
|
|
|
def project_along_vector(point: Vect3, vector: Vect3) -> Vect3:
|
|
matrix = np.identity(3) - np.outer(vector, vector)
|
|
return np.dot(point, matrix.T)
|
|
|
|
|
|
def normalize_along_axis(
|
|
array: np.ndarray,
|
|
axis: int,
|
|
) -> np.ndarray:
|
|
norms = np.sqrt((array * array).sum(axis))
|
|
norms[norms == 0] = 1
|
|
buffed_norms = np.repeat(norms, array.shape[axis]).reshape(array.shape)
|
|
return array / buffed_norms
|
|
|
|
|
|
def get_unit_normal(
|
|
v1: Vect3,
|
|
v2: Vect3,
|
|
tol: float = 1e-6
|
|
) -> Vect3:
|
|
v1 = normalize(v1)
|
|
v2 = normalize(v2)
|
|
cp = cross(v1, v2)
|
|
cp_norm = get_norm(cp)
|
|
if cp_norm < tol:
|
|
# Vectors align, so find a normal to them in the plane shared with the z-axis
|
|
new_cp = cross(cross(v1, OUT), v1)
|
|
new_cp_norm = get_norm(new_cp)
|
|
if new_cp_norm < tol:
|
|
return DOWN
|
|
return new_cp / new_cp_norm
|
|
return cp / cp_norm
|
|
|
|
|
|
###
|
|
|
|
|
|
def thick_diagonal(dim: int, thickness: int = 2) -> np.ndarray:
|
|
row_indices = np.arange(dim).repeat(dim).reshape((dim, dim))
|
|
col_indices = np.transpose(row_indices)
|
|
return (np.abs(row_indices - col_indices) < thickness).astype('uint8')
|
|
|
|
|
|
def compass_directions(n: int = 4, start_vect: Vect3 = RIGHT) -> Vect3:
|
|
angle = TAU / n
|
|
return np.array([
|
|
rotate_vector(start_vect, k * angle)
|
|
for k in range(n)
|
|
])
|
|
|
|
|
|
def complex_to_R3(complex_num: complex) -> Vect3:
|
|
return np.array((complex_num.real, complex_num.imag, 0))
|
|
|
|
|
|
def R3_to_complex(point: Vect3) -> complex:
|
|
return complex(*point[:2])
|
|
|
|
|
|
def complex_func_to_R3_func(complex_func: Callable[[complex], complex]) -> Callable[[Vect3], Vect3]:
|
|
def result(p: Vect3):
|
|
return complex_to_R3(complex_func(R3_to_complex(p)))
|
|
return result
|
|
|
|
|
|
def center_of_mass(points: Sequence[Vect3]) -> Vect3:
|
|
return np.array(points).sum(0) / len(points)
|
|
|
|
|
|
def midpoint(point1: VectN, point2: VectN) -> VectN:
|
|
return center_of_mass([point1, point2])
|
|
|
|
|
|
def line_intersection(
|
|
line1: Tuple[Vect3, Vect3],
|
|
line2: Tuple[Vect3, Vect3]
|
|
) -> Vect3:
|
|
"""
|
|
return intersection point of two lines,
|
|
each defined with a pair of vectors determining
|
|
the end points
|
|
"""
|
|
x_diff = (line1[0][0] - line1[1][0], line2[0][0] - line2[1][0])
|
|
y_diff = (line1[0][1] - line1[1][1], line2[0][1] - line2[1][1])
|
|
|
|
def det(a, b):
|
|
return a[0] * b[1] - a[1] * b[0]
|
|
|
|
div = det(x_diff, y_diff)
|
|
if div == 0:
|
|
raise Exception("Lines do not intersect")
|
|
d = (det(*line1), det(*line2))
|
|
x = det(d, x_diff) / div
|
|
y = det(d, y_diff) / div
|
|
return np.array([x, y, 0])
|
|
|
|
|
|
def find_intersection(
|
|
p0: Vect3,
|
|
v0: Vect3,
|
|
p1: Vect3,
|
|
v1: Vect3,
|
|
threshold: float = 1e-5
|
|
) -> Vect3:
|
|
"""
|
|
Return the intersection of a line passing through p0 in direction v0
|
|
with one passing through p1 in direction v1. (Or array of intersections
|
|
from arrays of such points/directions).
|
|
For 3d values, it returns the point on the ray p0 + v0 * t closest to the
|
|
ray p1 + v1 * t
|
|
"""
|
|
p0 = np.array(p0, ndmin=2)
|
|
v0 = np.array(v0, ndmin=2)
|
|
p1 = np.array(p1, ndmin=2)
|
|
v1 = np.array(v1, ndmin=2)
|
|
m, n = np.shape(p0)
|
|
assert(n in [2, 3])
|
|
|
|
numer = np.cross(v1, p1 - p0)
|
|
denom = np.cross(v1, v0)
|
|
if n == 3:
|
|
d = len(np.shape(numer))
|
|
new_numer = np.multiply(numer, numer).sum(d - 1)
|
|
new_denom = np.multiply(denom, numer).sum(d - 1)
|
|
numer, denom = new_numer, new_denom
|
|
|
|
denom[abs(denom) < threshold] = np.inf # So that ratio goes to 0 there
|
|
ratio = numer / denom
|
|
ratio = np.repeat(ratio, n).reshape((m, n))
|
|
result = p0 + ratio * v0
|
|
if m == 1:
|
|
return result[0]
|
|
return result
|
|
|
|
|
|
def get_closest_point_on_line(a: VectN, b: VectN, p: VectN) -> VectN:
|
|
"""
|
|
It returns point x such that
|
|
x is on line ab and xp is perpendicular to ab.
|
|
If x lies beyond ab line, then it returns nearest edge(a or b).
|
|
"""
|
|
# x = b + t*(a-b) = t*a + (1-t)*b
|
|
t = np.dot(p - b, a - b) / np.dot(a - b, a - b)
|
|
if t < 0:
|
|
t = 0
|
|
if t > 1:
|
|
t = 1
|
|
return ((t * a) + ((1 - t) * b))
|
|
|
|
|
|
def get_winding_number(points: Sequence[Vect2 | Vect3]) -> float:
|
|
total_angle = 0
|
|
for p1, p2 in adjacent_pairs(points):
|
|
d_angle = angle_of_vector(p2) - angle_of_vector(p1)
|
|
d_angle = ((d_angle + PI) % TAU) - PI
|
|
total_angle += d_angle
|
|
return total_angle / TAU
|
|
|
|
|
|
##
|
|
|
|
def cross2d(a: Vect2 | Vect2Array, b: Vect2 | Vect2Array) -> Vect2 | Vect2Array:
|
|
if len(a.shape) == 2:
|
|
return a[:, 0] * b[:, 1] - a[:, 1] * b[:, 0]
|
|
else:
|
|
return a[0] * b[1] - b[0] * a[1]
|
|
|
|
|
|
def tri_area(
|
|
a: Vect2,
|
|
b: Vect2,
|
|
c: Vect2
|
|
) -> float:
|
|
return 0.5 * abs(
|
|
a[0] * (b[1] - c[1]) +
|
|
b[0] * (c[1] - a[1]) +
|
|
c[0] * (a[1] - b[1])
|
|
)
|
|
|
|
|
|
def is_inside_triangle(
|
|
p: Vect2,
|
|
a: Vect2,
|
|
b: Vect2,
|
|
c: Vect2
|
|
) -> bool:
|
|
"""
|
|
Test if point p is inside triangle abc
|
|
"""
|
|
crosses = np.array([
|
|
cross2d(p - a, b - p),
|
|
cross2d(p - b, c - p),
|
|
cross2d(p - c, a - p),
|
|
])
|
|
return bool(np.all(crosses > 0) or np.all(crosses < 0))
|
|
|
|
|
|
def norm_squared(v: VectN | List[float]) -> float:
|
|
return sum(x * x for x in v)
|
|
|
|
|
|
# TODO, fails for polygons drawn over themselves
|
|
def earclip_triangulation(verts: Vect3Array | Vect2Array, ring_ends: list[int]) -> list[int]:
|
|
"""
|
|
Returns a list of indices giving a triangulation
|
|
of a polygon, potentially with holes
|
|
|
|
- verts is a numpy array of points
|
|
|
|
- ring_ends is a list of indices indicating where
|
|
the ends of new paths are
|
|
"""
|
|
|
|
rings = [
|
|
list(range(e0, e1))
|
|
for e0, e1 in zip([0, *ring_ends], ring_ends)
|
|
]
|
|
epsilon = 1e-6
|
|
|
|
def is_in(point, ring_id):
|
|
return abs(abs(get_winding_number([i - point for i in verts[rings[ring_id]]])) - 1) < epsilon
|
|
|
|
def ring_area(ring_id):
|
|
ring = rings[ring_id]
|
|
s = 0
|
|
for i, j in zip(ring[1:], ring):
|
|
s += cross2d(verts[i], verts[j])
|
|
return abs(s) / 2
|
|
|
|
# Points at the same position may cause problems
|
|
for i in rings:
|
|
verts[i[0]] += (verts[i[1]] - verts[i[0]]) * epsilon
|
|
verts[i[-1]] += (verts[i[-2]] - verts[i[-1]]) * epsilon
|
|
|
|
# First, we should know which rings are directly contained in it for each ring
|
|
|
|
right = [max(verts[rings[i], 0]) for i in range(len(rings))]
|
|
left = [min(verts[rings[i], 0]) for i in range(len(rings))]
|
|
top = [max(verts[rings[i], 1]) for i in range(len(rings))]
|
|
bottom = [min(verts[rings[i], 1]) for i in range(len(rings))]
|
|
area = [ring_area(i) for i in range(len(rings))]
|
|
|
|
# The larger ring must be outside
|
|
rings_sorted = list(range(len(rings)))
|
|
rings_sorted.sort(key=lambda x: area[x], reverse=True)
|
|
|
|
def is_in_fast(ring_a, ring_b):
|
|
# Whether a is in b
|
|
return reduce(op.and_, (
|
|
left[ring_b] <= left[ring_a] <= right[ring_a] <= right[ring_b],
|
|
bottom[ring_b] <= bottom[ring_a] <= top[ring_a] <= top[ring_b],
|
|
is_in(verts[rings[ring_a][0]], ring_b)
|
|
))
|
|
|
|
chilren = [[] for i in rings]
|
|
ringenum = ProgressDisplay(
|
|
enumerate(rings_sorted),
|
|
total=len(rings),
|
|
leave=False,
|
|
ascii=True if platform.system() == 'Windows' else None,
|
|
dynamic_ncols=True,
|
|
desc="SVG Triangulation",
|
|
delay=3,
|
|
)
|
|
for idx, i in ringenum:
|
|
for j in rings_sorted[:idx][::-1]:
|
|
if is_in_fast(i, j):
|
|
chilren[j].append(i)
|
|
break
|
|
|
|
res = []
|
|
|
|
# Then, we can use earcut for each part
|
|
used = [False] * len(rings)
|
|
for i in rings_sorted:
|
|
if used[i]:
|
|
continue
|
|
v = rings[i]
|
|
ring_ends = [len(v)]
|
|
for j in chilren[i]:
|
|
used[j] = True
|
|
v += rings[j]
|
|
ring_ends.append(len(v))
|
|
res += [v[i] for i in earcut(verts[v, :2], ring_ends)]
|
|
|
|
return res
|