Optimized get_unit_normal() and replaced np.cross() with custom cross() in manim.utils.space_ops (#3494)

* Added cross and optimized get_unit_normal in manim.utils.space_ops

* Added missing border case to new get_unit_normal where one vector is nonzero

* Updated test_threed.py::test_Sphere test data

Author: chopan050

manim/utils/space_ops.py

@@ -2,7 +2,7 @@
 
 from __future__ import annotations
 
-from manim.typing import Point3D_Array, Vector
+from manim.typing import Point3D_Array, Vector, Vector3
 
 __all__ = [
     "quaternion_mult",
@@ -53,6 +53,16 @@ def norm_squared(v: float) -> float:
     return np.dot(v, v)
 
 
+def cross(v1: Vector3, v2: Vector3) -> Vector3:
+    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],
+        ]
+    )
+
+
 # Quaternions
 # TODO, implement quaternion type
 
@@ -273,12 +283,12 @@ def z_to_vector(vector: np.ndarray) -> np.ndarray:
     (normalized) vector provided as an argument
     """
     axis_z = normalize(vector)
-    axis_y = normalize(np.cross(axis_z, RIGHT))
-    axis_x = np.cross(axis_y, axis_z)
+    axis_y = normalize(cross(axis_z, RIGHT))
+    axis_x = cross(axis_y, axis_z)
     if np.linalg.norm(axis_y) == 0:
         # the vector passed just so happened to be in the x direction.
-        axis_x = normalize(np.cross(UP, axis_z))
-        axis_y = -np.cross(axis_x, axis_z)
+        axis_x = normalize(cross(UP, axis_z))
+        axis_y = -cross(axis_x, axis_z)
 
     return np.array([axis_x, axis_y, axis_z]).T
 
@@ -359,7 +369,7 @@ def normalize_along_axis(array: np.ndarray, axis: np.ndarray) -> np.ndarray:
     return array
 
 
-def get_unit_normal(v1: np.ndarray, v2: np.ndarray, tol: float = 1e-6) -> np.ndarray:
+def get_unit_normal(v1: Vector3, v2: Vector3, tol: float = 1e-6) -> Vector3:
     """Gets the unit normal of the vectors.
 
     Parameters
@@ -376,16 +386,37 @@ def get_unit_normal(v1: np.ndarray, v2: np.ndarray, tol: float = 1e-6) -> np.nda
     np.ndarray
         The normal of the two vectors.
     """
-    v1, v2 = (normalize(i) for i in (v1, v2))
-    cp = np.cross(v1, v2)
-    cp_norm = np.linalg.norm(cp)
-    if cp_norm < tol:
-        # Vectors align, so find a normal to them in the plane shared with the z-axis
-        cp = np.cross(np.cross(v1, OUT), v1)
-        cp_norm = np.linalg.norm(cp)
-        if cp_norm < tol:
+    # Instead of normalizing v1 and v2, just divide by the greatest
+    # of all their absolute components, which is just enough
+    div1, div2 = max(np.abs(v1)), max(np.abs(v2))
+    if div1 == 0.0:
+        if div2 == 0.0:
             return DOWN
-    return normalize(cp)
+        u = v2 / div2
+    elif div2 == 0.0:
+        u = v1 / div1
+    else:
+        # Normal scenario: v1 and v2 are both non-null
+        u1, u2 = v1 / div1, v2 / div2
+        cp = cross(u1, u2)
+        cp_norm = np.sqrt(norm_squared(cp))
+        if cp_norm > tol:
+            return cp / cp_norm
+        # Otherwise, v1 and v2 were aligned
+        u = u1
+
+    # If you are here, you have an "unique", non-zero, unit-ish vector u
+    # If it's also too aligned to the Z axis, just return DOWN
+    if abs(u[0]) < tol and abs(u[1]) < tol:
+        return DOWN
+    # Otherwise rotate u in the plane it shares with the Z axis,
+    # 90° TOWARDS the Z axis. This is done via (u x [0, 0, 1]) x u,
+    # which gives [-xz, -yz, x²+y²] (slightly scaled as well)
+    cp = np.array([-u[0] * u[2], -u[1] * u[2], u[0] * u[0] + u[1] * u[1]])
+    cp_norm = np.sqrt(norm_squared(cp))
+    # Because the norm(u) == 0 case was filtered in the beginning,
+    # there is no need to check if the norm of cp is 0
+    return cp / cp_norm
 
 
 ###
@@ -529,8 +560,8 @@ def line_intersection(
         np.pad(np.array(i)[:, :2], ((0, 0), (0, 1)), constant_values=1)
         for i in (line1, line2)
     )
-    line1, line2 = (np.cross(*i) for i in padded)
-    x, y, z = np.cross(line1, line2)
+    line1, line2 = (cross(*i) for i in padded)
+    x, y, z = cross(line1, line2)
 
     if z == 0:
         raise ValueError(
@@ -558,7 +589,7 @@ def find_intersection(
     result = []
 
     for p0, v0, p1, v1 in zip(*[p0s, v0s, p1s, v1s]):
-        normal = np.cross(v1, np.cross(v0, v1))
+        normal = cross(v1, cross(v0, v1))
         denom = max(np.dot(v0, normal), threshold)
         result += [p0 + np.dot(p1 - p0, normal) / denom * v0]
     return result
@@ -814,6 +845,6 @@ def perpendicular_bisector(
     """
     p1 = line[0]
     p2 = line[1]
-    direction = np.cross(p1 - p2, norm_vector)
+    direction = cross(p1 - p2, norm_vector)
     m = midpoint(p1, p2)
     return [m + direction, m - direction]