BUG: spatial: error in Rotation.align_vectors() with an infinite weight

[scottshambaugh]

Describe your issue.

I found a degenerate case when aligning vectors exactly with Rotation.align_vectors and an infinite weight.

Reproducing Code Example

import numpy as np
from scipy.spatial.transform import Rotation

# Wrong:
Rotation.align_vectors([[0, 1, 0], [-1, 0, 0]], [[0, -1, 0], [-1, 0, 0]], weights=[np.inf, 1])[0].as_rotvec()
# array([0., 0., 0.])

# Right:
Rotation.align_vectors([[0, 1, 0], [-1, 0, 0]], [[0, -1, 0], [-1, 0, 0]], weights=[1e6, 1])[0].as_rotvec()
# array([3.14159265, 0., 0.])

SciPy/NumPy/Python version and system information

import sys, scipy, numpy; print(scipy.__version__, numpy.__version__, sys.version_info); scipy.show_config()
1.13.0 1.26.3 sys.version_info(major=3, minor=10, micro=12, releaselevel='final', serial=0)
Build Dependencies:
  blas:
    detection method: pkgconfig
    found: true
    include directory: /usr/local/include
    lib directory: /usr/local/lib
    name: openblas
    openblas configuration: USE_64BITINT=0 DYNAMIC_ARCH=1 DYNAMIC_OLDER= NO_CBLAS=
      NO_LAPACK= NO_LAPACKE= NO_AFFINITY=1 USE_OPENMP= ZEN MAX_THREADS=64
    pc file directory: /usr/local/lib/pkgconfig
    version: 0.3.26.dev
  lapack:
    detection method: pkgconfig
    found: true
    include directory: /usr/local/include
    lib directory: /usr/local/lib
    name: openblas
    openblas configuration: USE_64BITINT=0 DYNAMIC_ARCH=1 DYNAMIC_OLDER= NO_CBLAS=
      NO_LAPACK= NO_LAPACKE= NO_AFFINITY=1 USE_OPENMP= ZEN MAX_THREADS=64
    pc file directory: /usr/local/lib/pkgconfig
    version: 0.3.26.dev
  pybind11:
    detection method: config-tool
    include directory: unknown
    name: pybind11
    version: 2.12.0
Compilers:
  c:
    commands: cc
    linker: ld.bfd
    name: gcc
    version: 10.2.1
  c++:
    commands: c++
    linker: ld.bfd
    name: gcc
    version: 10.2.1
  cython:
    commands: cython
    linker: cython
    name: cython
    version: 3.0.10
  fortran:
    commands: gfortran
    linker: ld.bfd
    name: gcc
    version: 10.2.1
  pythran:
    include directory: ../../tmp/pip-build-env-0blqy1or/overlay/lib/python3.10/site-packages/pythran
    version: 0.15.0
Machine Information:
  build:
    cpu: x86_64
    endian: little
    family: x86_64
    system: linux
  cross-compiled: false
  host:
    cpu: x86_64
    endian: little
    family: x86_64
    system: linux
Python Information:
  path: /opt/python/cp310-cp310/bin/python
  version: '3.10'

[scottshambaugh]

I'm seeing the issue here, working on a fix. Problem is here:

scipy/scipy/spatial/transform/_rotation.pyx

Lines 3483 to 3492 in d55cb95

	# We first find the minimum angle rotation between the primary
	# vectors.
	cross = np.cross(b_pri[0], a_pri[0])
	theta = atan2(_norm3(cross), np.dot(a_pri[0], b_pri[0]))
	if theta < 1e-3: # small angle Taylor series approximation
	theta2 = theta * theta
	r = cross * (1 + theta2 / 6 + theta2 * theta2 * 7 / 360)
	else:
	r = cross * theta / np.sin(theta)
	R_pri = cls.from_rotvec(r)

There are two related problems:

• A perfect 180 deg rotation results in r = cross = [0, 0, 0] (this bug, an error)
• Potential numerical instabilities near a 180 deg rotation (not exactly this bug but will fix and add tests)