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space_ops.py
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space_ops.py
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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],
out: np.ndarray | None = None
) -> Vect3 | Vect3Array:
is2d = isinstance(v1, np.ndarray) and len(v1.shape) == 2
if is2d:
x1, y1, z1 = v1[:, 0], v1[:, 1], v1[:, 2]
x2, y2, z2 = v2[:, 0], v2[:, 1], v2[:, 2]
else:
x1, y1, z1 = v1
x2, y2, z2 = v2
if out is None:
out = np.empty(np.shape(v1))
out.T[:] = [
y1 * z2 - z1 * y2,
z1 * x2 - x1 * z2,
x1 * y2 - y1 * x2,
]
return out
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:
atol = 1e-8
if get_norm(v1 - v2) < atol:
return np.identity(3)
axis = cross(v1, v2)
if get_norm(axis) < atol:
# v1 and v2 align
axis = cross(v1, RIGHT)
if get_norm(axis) < atol:
# v1 and v2 _and_ RIGHT all align
axis = 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 math.atan2(vector[1], vector[0])
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
return (array.T / norms).T
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 | Vect3Array,
v0: Vect3 | Vect3Array,
p1: Vect3 | Vect3Array,
v1: Vect3 | Vect3Array,
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
"""
d = len(p0.shape)
if d == 1:
is_3d = any(arr[2] for arr in (p0, v0, p1, v1))
else:
is_3d = any(z for arr in (p0, v0, p1, v1) for z in arr.T[2])
if not is_3d:
numer = np.array(cross2d(v1, p1 - p0))
denom = np.array(cross2d(v1, v0))
else:
cp1 = cross(v1, p1 - p0)
cp2 = cross(v1, v0)
numer = np.array((cp1 * cp1).sum(d - 1))
denom = np.array((cp1 * cp2).sum(d - 1))
denom[abs(denom) < threshold] = np.inf
ratio = numer / denom
return p0 + (ratio * v0.T).T
def line_intersects_path(
start: Vect2 | Vect3,
end: Vect2 | Vect3,
path: Vect2Array | Vect3Array,
) -> bool:
"""
Tests whether the line (start, end) intersects
a polygonal path defined by its vertices
"""
n = len(path) - 1
p1 = np.empty((n, 2))
q1 = np.empty((n, 2))
p1[:] = start[:2]
q1[:] = end[:2]
p2 = path[:-1, :2]
q2 = path[1:, :2]
v1 = q1 - p1
v2 = q2 - p2
mis1 = cross2d(v1, p2 - p1) * cross2d(v1, q2 - p1) < 0
mis2 = cross2d(v2, p1 - p2) * cross2d(v2, q1 - p2) < 0
return bool((mis1 * mis2).any())
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:
if len(i) < 2:
continue
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