Source code for psychopy.tools.mathtools

```#!/usr/bin/env python
# -*- coding: utf-8 -*-

# Various math functions for working with vectors, matrices, and quaternions.
#

# Part of the PsychoPy library
# Copyright (C) 2002-2018 Jonathan Peirce (C) 2019-2024 Open Science Tools Ltd.

__all__ = ['normalize',
'lerp',
'slerp',
'multQuat',
'quatFromAxisAngle',
'quatToMatrix',
'scaleMatrix',
'rotationMatrix',
'transform',
'translationMatrix',
'concatenate',
'applyMatrix',
'invertQuat',
'quatToAxisAngle',
'posOriToMatrix',
'applyQuat',
'orthogonalize',
'reflect',
'cross',
'distance',
'dot',
'quatMagnitude',
'length',
'project',
'bisector',
'surfaceNormal',
'invertMatrix',
'angleTo',
'surfaceBitangent',
'surfaceTangent',
'vertexNormal',
'isOrthogonal',
'isAffine',
'perp',
'ortho3Dto2D',
'intersectRayPlane',
'matrixToQuat',
'lensCorrection',
'matrixFromEulerAngles',
'alignTo',
'quatYawPitchRoll',
'intersectRaySphere',
'intersectRayAABB',
'intersectRayOBB',
'intersectRayTriangle',
'scale',
'multMatrix',
'normalMatrix',
'fitBBox',
'computeBBoxCorners',
'zeroFix',
'accumQuat',
'fixTangentHandedness',
'articulate',
'forwardProject',
'reverseProject',
'lensCorrectionSpherical']

import numpy as np
import functools
import itertools

VEC_AXES = {'+x': (1, 0, 0), '-x': (-1, 0, 0),
'+y': (0, 1, 0), '-y': (0, -1, 0),
'+z': (0, 0, 1), '-z': (0, 0, -1)}

# ------------------------------------------------------------------------------
# Vector Operations
#

[docs]def length(v, squared=False, out=None, dtype=None):
"""Get the length of a vector.

Parameters
----------
v : array_like
Vector to normalize, can be Nx2, Nx3, or Nx4. If a 2D array is
specified, rows are treated as separate vectors.
squared : bool, optional
If ``True`` the squared length is returned. The default is ``False``.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
float or ndarray
Length of vector `v`.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

v = np.asarray(v, dtype=dtype)

if v.ndim == 2:
assert v.shape[1] <= 4
toReturn = np.zeros((v.shape[0],), dtype=dtype) if out is None else out
v2d, vr = np.atleast_2d(v, toReturn)  # 2d view of array
if squared:
vr[:, :] = np.sum(np.square(v2d), axis=1)
else:
vr[:, :] = np.sqrt(np.sum(np.square(v2d), axis=1))
elif v.ndim == 1:
assert v.shape[0] <= 4
if squared:
toReturn = np.sum(np.square(v))
else:
toReturn = np.sqrt(np.sum(np.square(v)))
else:
raise ValueError("Input arguments have invalid dimensions.")

[docs]def normalize(v, out=None, dtype=None):
"""Normalize a vector or quaternion.

v : array_like
Vector to normalize, can be Nx2, Nx3, or Nx4. If a 2D array is
specified, rows are treated as separate vectors. All vectors should have
nonzero length.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Normalized vector `v`.

Notes
-----
* If the vector has length is zero, a vector of all zeros is returned after
normalization.

Examples
--------
Normalize a vector::

v = [1., 2., 3., 4.]
vn = normalize(v)

The `normalize` function is vectorized. It's considerably faster to
normalize large arrays of vectors than to call `normalize` separately for
each one::

v = np.random.uniform(-1.0, 1.0, (1000, 4,))  # 1000 length 4 vectors
vn = np.zeros((1000, 4))  # place to write values
normalize(v, out=vn)  # very fast!

# don't do this!
for i in range(1000):
vn[i, :] = normalize(v[i, :])

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
toReturn = np.array(v, dtype=dtype)
else:
toReturn = out

v2d = np.atleast_2d(toReturn)  # 2d view of array
norm = np.linalg.norm(v2d, axis=1)
norm[norm == 0.0] = np.NaN  # make sure if length==0 division succeeds
v2d /= norm[:, np.newaxis]
np.nan_to_num(v2d, copy=False)  # fix NaNs

[docs]def orthogonalize(v, n, out=None, dtype=None):
"""Orthogonalize a vector relative to a normal vector.

This function ensures that `v` is perpendicular (or orthogonal) to `n`.

Parameters
----------
v : array_like
Vector to orthogonalize, can be Nx2, Nx3, or Nx4. If a 2D array is
specified, rows are treated as separate vectors.
n : array_like
Normal vector, must have same shape as `v`.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Orthogonalized vector `v` relative to normal vector `n`.

Warnings
--------
If `v` and `n` are the same, the direction of the perpendicular vector is
indeterminate. The resulting vector is degenerate (all zeros).

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

v = np.asarray(v, dtype=dtype)
n = np.asarray(n, dtype=dtype)

if out is None:
toReturn = np.zeros_like(v, dtype=dtype)
else:
toReturn = out
toReturn.fill(0.0)

v, n, vr = np.atleast_2d(v, n, toReturn)
vr[:, :] = v
vr[:, :] -= n * np.sum(n * v, axis=1)[:, np.newaxis]  # dot product
normalize(vr, out=vr)

[docs]def reflect(v, n, out=None, dtype=None):
"""Reflection of a vector.

Get the reflection of `v` relative to normal `n`.

Parameters
----------
v : array_like
Vector to reflect, can be Nx2, Nx3, or Nx4. If a 2D array is specified,
rows are treated as separate vectors.
n : array_like
Normal vector, must have same shape as `v`.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Reflected vector `v` off normal `n`.

"""
# based off https://github.com/glfw/glfw/blob/master/deps/linmath.h
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

v = np.asarray(v, dtype=dtype)
n = np.asarray(n, dtype=dtype)

if out is None:
toReturn = np.zeros_like(v, dtype=dtype)
else:
toReturn = out
toReturn.fill(0.0)

v, n, vr = np.atleast_2d(v, n, toReturn)

vr[:, :] = v
vr[:, :] -= (dtype(2.0) * np.sum(n * v, axis=1))[:, np.newaxis] * n

[docs]def dot(v0, v1, out=None, dtype=None):
"""Dot product of two vectors.

The behaviour of this function depends on the format of the input arguments:

* If `v0` and `v1` are 1D, the dot product is returned as a scalar and `out`
is ignored.
* If `v0` and `v1` are 2D, a 1D array of dot products between corresponding
row vectors are returned.
* If either `v0` and `v1` are 1D and 2D, an array of dot products
between each row of the 2D vector and the 1D vector are returned.

Parameters
----------
v0, v1 : array_like
Vector(s) to compute dot products of (e.g. [x, y, z]). `v0` must have
equal or fewer dimensions than `v1`.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Dot product(s) of `v0` and `v1`.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

v0 = np.asarray(v0, dtype=dtype)
v1 = np.asarray(v1, dtype=dtype)

if v0.ndim == v1.ndim == 2 or v0.ndim == 2 and v1.ndim == 1:
toReturn = np.zeros((v0.shape[0],), dtype=dtype) if out is None else out
vr = np.atleast_2d(toReturn)  # make sure we have a 2d view
vr[:] = np.sum(v1 * v0, axis=1)
elif v0.ndim == v1.ndim == 1:
toReturn = np.sum(v1 * v0)
elif v0.ndim == 1 and v1.ndim == 2:
toReturn = np.zeros((v1.shape[0],), dtype=dtype) if out is None else out
vr = np.atleast_2d(toReturn)  # make sure we have a 2d view
vr[:] = np.sum(v1 * v0, axis=1)
else:
raise ValueError("Input arguments have invalid dimensions.")

[docs]def cross(v0, v1, out=None, dtype=None):
"""Cross product of 3D vectors.

The behavior of this function depends on the dimensions of the inputs:

* If `v0` and `v1` are 1D, the cross product is returned as 1D vector.
* If `v0` and `v1` are 2D, a 2D array of cross products between
corresponding row vectors are returned.
* If either `v0` and `v1` are 1D and 2D, an array of cross products
between each row of the 2D vector and the 1D vector are returned.

Parameters
----------
v0, v1 : array_like
Vector(s) in form [x, y, z] or [x, y, z, 1].
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Cross product of `v0` and `v1`.

Notes
-----
* If input vectors are 4D, the last value of cross product vectors is always
set to one.
* If input vectors `v0` and `v1` are Nx3 and `out` is Nx4, the cross product
is computed and the last column of `out` is filled with ones.

Examples
--------

Find the cross product of two vectors::

a = normalize([1, 2, 3])
b = normalize([3, 2, 1])
c = cross(a, b)

If input arguments are 2D, the function returns the cross products of
corresponding rows::

# create two 6x3 arrays with random numbers
shape = (6, 3,)
a = normalize(np.random.uniform(-1.0, 1.0, shape))
b = normalize(np.random.uniform(-1.0, 1.0, shape))
cprod = np.zeros(shape)  # output has the same shape as inputs
cross(a, b, out=cprod)

If a 1D and 2D vector are specified, the cross product of each row of the
2D array and the 1D array is returned as a 2D array::

a = normalize([1, 2, 3])
b = normalize(np.random.uniform(-1.0, 1.0, (6, 3,)))
cprod = np.zeros(a.shape)
cross(a, b, out=cprod)

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

v0 = np.asarray(v0, dtype=dtype)
v1 = np.asarray(v1, dtype=dtype)

if v0.ndim == v1.ndim == 2:  # 2D x 2D
assert v0.shape == v1.shape
toReturn = np.zeros(v0.shape, dtype=dtype) if out is None else out
vr = np.atleast_2d(toReturn)
vr[:, 0] = v0[:, 1] * v1[:, 2] - v0[:, 2] * v1[:, 1]
vr[:, 1] = v0[:, 2] * v1[:, 0] - v0[:, 0] * v1[:, 2]
vr[:, 2] = v0[:, 0] * v1[:, 1] - v0[:, 1] * v1[:, 0]

if vr.shape[1] == 4:
vr[:, 3] = dtype(1.0)

elif v0.ndim == v1.ndim == 1:  # 1D x 1D
assert v0.shape == v1.shape
toReturn = np.zeros(v0.shape, dtype=dtype) if out is None else out
toReturn[0] = v0[1] * v1[2] - v0[2] * v1[1]
toReturn[1] = v0[2] * v1[0] - v0[0] * v1[2]
toReturn[2] = v0[0] * v1[1] - v0[1] * v1[0]

if toReturn.shape[0] == 4:
toReturn[3] = dtype(1.0)

elif v0.ndim == 2 and v1.ndim == 1:  # 2D x 1D
toReturn = np.zeros(v0.shape, dtype=dtype) if out is None else out
vr = np.atleast_2d(toReturn)
vr[:, 0] = v0[:, 1] * v1[2] - v0[:, 2] * v1[1]
vr[:, 1] = v0[:, 2] * v1[0] - v0[:, 0] * v1[2]
vr[:, 2] = v0[:, 0] * v1[1] - v0[:, 1] * v1[0]

if vr.shape[1] == 4:
vr[:, 3] = dtype(1.0)

elif v0.ndim == 1 and v1.ndim == 2:  # 1D x 2D
toReturn = np.zeros(v1.shape, dtype=dtype) if out is None else out
vr = np.atleast_2d(toReturn)
vr[:, 0] = v1[:, 2] * v0[1] - v1[:, 1] * v0[2]
vr[:, 1] = v1[:, 0] * v0[2] - v1[:, 2] * v0[0]
vr[:, 2] = v1[:, 1] * v0[0] - v1[:, 0] * v0[1]

if vr.shape[1] == 4:
vr[:, 3] = dtype(1.0)

else:
raise ValueError("Input arguments have incorrect dimensions.")

[docs]def project(v0, v1, out=None, dtype=None):
"""Project a vector onto another.

Parameters
----------
v0 : array_like
Vector can be Nx2, Nx3, or Nx4. If a 2D array is specified, rows are
treated as separate vectors.
v1 : array_like
Vector to project onto `v0`.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray or float
Projection of vector `v0` on `v1`.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

v0 = np.asarray(v0, dtype=dtype)
v1 = np.asarray(v1, dtype=dtype)

if v0.ndim == v1.ndim == 2 or v0.ndim == 1 and v1.ndim == 2:
toReturn = np.zeros_like(v1, dtype=dtype) if out is None else out
toReturn[:, :] = v1[:, :]
toReturn *= (dot(v0, v1, dtype=dtype) / length(v1))[:, np.newaxis]
elif v0.ndim == v1.ndim == 1:
toReturn = v1 * (dot(v0, v1, dtype=dtype) / np.sum(np.square(v1)))
elif v0.ndim == 2 and v1.ndim == 1:
toReturn = np.zeros_like(v0, dtype=dtype) if out is None else out
toReturn[:, :] = v1[:]
toReturn *= (dot(v0, v1, dtype=dtype) / length(v1))[:, np.newaxis]
else:
raise ValueError("Input arguments have invalid dimensions.")

toReturn += 0.0  # remove negative zeros

[docs]def lerp(v0, v1, t, out=None, dtype=None):
"""Linear interpolation (LERP) between two vectors/coordinates.

Parameters
----------
v0 : array_like
Initial vector/coordinate. Can be 2D where each row is a point.
v1 : array_like
Final vector/coordinate. Must be the same shape as `v0`.
t : float
Interpolation weight factor [0, 1].
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Vector  at `t` with same shape as `v0` and `v1`.

Examples
--------
Find the coordinate of the midpoint between two vectors::

u = [0., 0., 0.]
v = [0., 0., 1.]
midpoint = lerp(u, v, 0.5)  # 0.5 to interpolate half-way between points

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

t = dtype(t)
t0 = dtype(1.0) - t
v0 = np.asarray(v0, dtype=dtype)
v1 = np.asarray(v1, dtype=dtype)

toReturn = np.zeros_like(v0, dtype=dtype) if out is None else out

v0, v1, vr = np.atleast_2d(v0, v1, toReturn)
vr[:, :] = v0 * t0
vr[:, :] += v1 * t

[docs]def distance(v0, v1, out=None, dtype=None):
"""Get the distance between vectors/coordinates.

The behaviour of this function depends on the format of the input arguments:

* If `v0` and `v1` are 1D, the distance is returned as a scalar and `out` is
ignored.
* If `v0` and `v1` are 2D, an array of distances between corresponding row
vectors are returned.
* If either `v0` and `v1` are 1D and 2D, an array of distances
between each row of the 2D vector and the 1D vector are returned.

Parameters
----------
v0, v1 : array_like
Vectors to compute the distance between.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Distance between vectors `v0` and `v1`.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

v0 = np.asarray(v0, dtype=dtype)
v1 = np.asarray(v1, dtype=dtype)

if v0.ndim == v1.ndim == 2 or (v0.ndim == 2 and v1.ndim == 1):
dist = np.zeros((v1.shape[0],), dtype=dtype) if out is None else out
dist[:] = np.sqrt(np.sum(np.square(v1 - v0), axis=1))
elif v0.ndim == v1.ndim == 1:
dist = np.sqrt(np.sum(np.square(v1 - v0)))
elif v0.ndim == 1 and v1.ndim == 2:
dist = np.zeros((v1.shape[0],), dtype=dtype) if out is None else out
dist[:] = np.sqrt(np.sum(np.square(v0 - v1), axis=1))
else:
raise ValueError("Input arguments have invalid dimensions.")

return dist

[docs]def perp(v, n, norm=True, out=None, dtype=None):
"""Project `v` to be a perpendicular axis of `n`.

Parameters
----------
v : array_like
Vector to project [x, y, z], may be Nx3.
n : array_like
Normal vector [x, y, z], may be Nx3.
norm : bool
Normalize the resulting axis. Default is `True`.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Perpendicular axis of `n` from `v`.

Examples
--------
Determine the local `up` (y-axis) of a surface or plane given `normal`::

normal = [0., 0.70710678, 0.70710678]
up = [1., 0., 0.]

yaxis = perp(up, normal)

Do a cross product to get the x-axis perpendicular to both::

xaxis = cross(yaxis, normal)

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

v = np.asarray(v, dtype=dtype)
n = np.asarray(n, dtype=dtype)

toReturn = np.zeros_like(v, dtype=dtype) if out is None else out
v2d, n2d, r2d = np.atleast_2d(v, n, toReturn)

# from GLM `glm/gtx/perpendicular.inl`
r2d[:, :] = v2d - project(v2d, n2d, dtype=dtype)

if norm:
normalize(toReturn, out=toReturn)

toReturn += 0.0  # clear negative zeros

[docs]def bisector(v0, v1, norm=False, out=None, dtype=None):
"""Get the angle bisector.

Computes a vector which bisects the angle between `v0` and `v1`. Input
vectors `v0` and `v1` must be non-zero.

Parameters
----------
v0, v1 : array_like
Vectors to bisect [x, y, z]. Must be non-zero in length and have the
same shape. Inputs can be Nx3 where the bisector for corresponding
rows will be returned.
norm : bool, optional
Normalize the resulting bisector. Default is `False`.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Bisecting vector [x, y, z].

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

v0 = np.asarray(v0, dtype=dtype)
v1 = np.asarray(v1, dtype=dtype)

assert v0.shape == v1.shape

toReturn = np.zeros_like(v0, dtype=dtype) if out is None else out

v02d, v12d, r2d = np.atleast_2d(v0, v1, toReturn)

r2d[:, :] = v02d * length(v12d, dtype=dtype)[:, np.newaxis] + \
v12d * length(v02d, dtype=dtype)[:, np.newaxis]

if norm:
normalize(r2d, out=r2d)

[docs]def angleTo(v, point, degrees=True, out=None, dtype=None):
"""Get the relative angle to a point from a vector.

The behaviour of this function depends on the format of the input arguments:

* If `v0` and `v1` are 1D, the angle is returned as a scalar and `out` is
ignored.
* If `v0` and `v1` are 2D, an array of angles between corresponding row
vectors are returned.
* If either `v0` and `v1` are 1D and 2D, an array of angles
between each row of the 2D vector and the 1D vector are returned.

Parameters
----------
v : array_like
Direction vector [x, y, z].
point : array_like
Point(s) to compute angle to from vector `v`.
degrees : bool, optional
Return the resulting angles in degrees. If `False`, angles will be
returned in radians. Default is `True`.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Distance between vectors `v0` and `v1`.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

v = np.asarray(v, dtype=dtype)
point = np.asarray(point, dtype=dtype)

if v.ndim == point.ndim == 2 or (v.ndim == 2 and point.ndim == 1):
angle = np.zeros((v.shape[0],), dtype=dtype) if out is None else out
u = np.sqrt(length(v, squared=True, dtype=dtype) *
length(point, squared=True, dtype=dtype))
angle[:] = np.arccos(dot(v, point, dtype=dtype) / u)
elif v.ndim == 1 and point.ndim == 2:
angle = np.zeros((point.shape[0],), dtype=dtype) if out is None else out
u = np.sqrt(length(v, squared=True, dtype=dtype) *
length(point, squared=True, dtype=dtype))
angle[:] = np.arccos(dot(v, point, dtype=dtype) / u)
elif v.ndim == point.ndim == 1:
u = np.sqrt(length(v, squared=True, dtype=dtype) *
length(point, squared=True, dtype=dtype))
angle = np.arccos(dot(v, point, dtype=dtype) / u)
else:
raise ValueError("Input arguments have invalid dimensions.")

return np.degrees(angle) if degrees else angle

def sortClockwise(verts):
"""
Sort vertices clockwise from 12 O'Clock (aka vertex (0, 1)).

Parameters
==========
verts : array
Array of vertices to sort
"""
# Blank array of angles
angles = []
# Calculate angle of each vertex
for vert in verts:
# Get angle
ang = angleTo(v=[0, 1], point=vert)
# Flip angle if we're past 6 O'clock
if vert[0] < 0:
ang = 360 - ang
# Append to angles array
angles.append(ang)
# Sort vertices by angles array values
verts = [x for _, x in sorted(zip(angles, verts), key=lambda pair: pair[0])]

return verts

[docs]def surfaceNormal(tri, norm=True, out=None, dtype=None):
"""Compute the surface normal of a given triangle.

Parameters
----------
tri : array_like
Triangle vertices as 2D (3x3) array [p0, p1, p2] where each vertex is a
length 3 array [vx, xy, vz]. The input array can be 3D (Nx3x3) to
specify multiple triangles.
norm : bool, optional
Normalize computed surface normals if ``True``, default is ``True``.
out : ndarray, optional
Optional output array. Must have one fewer dimensions than `tri`. The
shape of the last dimension must be 3.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Surface normal of triangle `tri`.

Examples
--------
Compute the surface normal of a triangle::

vertices = [[-1., 0., 0.], [0., 1., 0.], [1, 0, 0]]
norm = surfaceNormal(vertices)

Find the normals for multiple triangles, and put results in a pre-allocated
array::

vertices = [[[-1., 0., 0.], [0., 1., 0.], [1, 0, 0]],  # 2x3x3
[[1., 0., 0.], [0., 1., 0.], [-1, 0, 0]]]
normals = np.zeros((2, 3))  # normals from two triangles triangles
surfaceNormal(vertices, out=normals)

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

tris = np.asarray(tri, dtype=dtype)
if tris.ndim == 2:
tris = np.expand_dims(tri, axis=0)

if tris.shape[0] == 1:
toReturn = np.zeros((3,), dtype=dtype) if out is None else out
else:
if out is None:
toReturn = np.zeros((tris.shape[0], 3), dtype=dtype)
else:
toReturn = out

# from https://www.khronos.org/opengl/wiki/Calculating_a_Surface_Normal
nr = np.atleast_2d(toReturn)
u = tris[:, 1, :] - tris[:, 0, :]
v = tris[:, 2, :] - tris[:, 1, :]
nr[:, 0] = u[:, 1] * v[:, 2] - u[:, 2] * v[:, 1]
nr[:, 1] = u[:, 2] * v[:, 0] - u[:, 0] * v[:, 2]
nr[:, 2] = u[:, 0] * v[:, 1] - u[:, 1] * v[:, 0]

if norm:
normalize(nr, out=nr)

[docs]def surfaceBitangent(tri, uv, norm=True, out=None, dtype=None):
"""Compute the bitangent vector of a given triangle.

This function can be used to generate bitangent vertex attributes for normal
mapping. After computing bitangents, one may orthogonalize them with vertex
normals using the :func:`orthogonalize` function, or within the fragment
shader. Uses texture coordinates at each triangle vertex to determine the
direction of the vector.

Parameters
----------
tri : array_like
Triangle vertices as 2D (3x3) array [p0, p1, p2] where each vertex is a
length 3 array [vx, xy, vz]. The input array can be 3D (Nx3x3) to
specify multiple triangles.
uv : array_like
Texture coordinates associated with each face vertex as a 2D array (3x2)
where each texture coordinate is length 2 array [u, v]. The input array
can be 3D (Nx3x2) to specify multiple texture coordinates if multiple
triangles are specified.
norm : bool, optional
Normalize computed bitangents if ``True``, default is ``True``.
out : ndarray, optional
Optional output array. Must have one fewer dimensions than `tri`. The
shape of the last dimension must be 3.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Surface bitangent of triangle `tri`.

Examples
--------
Computing the bitangents for two triangles from vertex and texture
coordinates (UVs)::

# array of triangle vertices (2x3x3)
tri = np.asarray([
[(-1.0, 1.0, 0.0), (-1.0, -1.0, 0.0), (1.0, -1.0, 0.0)],   # 1
[(-1.0, 1.0, 0.0), (-1.0, -1.0, 0.0), (1.0, -1.0, 0.0)]])  # 2

# array of triangle texture coordinates (2x3x2)
uv = np.asarray([
[(0.0, 1.0), (0.0, 0.0), (1.0, 0.0)],   # 1
[(0.0, 1.0), (0.0, 0.0), (1.0, 0.0)]])  # 2

bitangents = surfaceBitangent(tri, uv, norm=True)  # bitangets (2x3)

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

tris = np.asarray(tri, dtype=dtype)
if tris.ndim == 2:
tris = np.expand_dims(tri, axis=0)

if tris.shape[0] == 1:
toReturn = np.zeros((3,), dtype=dtype) if out is None else out
else:
if out is None:
toReturn = np.zeros((tris.shape[0], 3), dtype=dtype)
else:
toReturn = out

uvs = np.asarray(uv, dtype=dtype)
if uvs.ndim == 2:
uvs = np.expand_dims(uvs, axis=0)

# based off the implementation from
e1 = tris[:, 1, :] - tris[:, 0, :]
e2 = tris[:, 2, :] - tris[:, 0, :]
d1 = uvs[:, 1, :] - uvs[:, 0, :]
d2 = uvs[:, 2, :] - uvs[:, 0, :]

# compute the bitangent
nr = np.atleast_2d(toReturn)
nr[:, 0] = -d2[:, 0] * e1[:, 0] + d1[:, 0] * e2[:, 0]
nr[:, 1] = -d2[:, 0] * e1[:, 1] + d1[:, 0] * e2[:, 1]
nr[:, 2] = -d2[:, 0] * e1[:, 2] + d1[:, 0] * e2[:, 2]

f = dtype(1.0) / (d1[:, 0] * d2[:, 1] - d2[:, 0] * d1[:, 1])
nr *= f[:, np.newaxis]

if norm:
normalize(toReturn, out=toReturn, dtype=dtype)

[docs]def surfaceTangent(tri, uv, norm=True, out=None, dtype=None):
"""Compute the tangent vector of a given triangle.

This function can be used to generate tangent vertex attributes for normal
mapping. After computing tangents, one may orthogonalize them with vertex
normals using the :func:`orthogonalize` function, or within the fragment
shader. Uses texture coordinates at each triangle vertex to determine the
direction of the vector.

Parameters
----------
tri : array_like
Triangle vertices as 2D (3x3) array [p0, p1, p2] where each vertex is a
length 3 array [vx, xy, vz]. The input array can be 3D (Nx3x3) to
specify multiple triangles.
uv : array_like
Texture coordinates associated with each face vertex as a 2D array (3x2)
where each texture coordinate is length 2 array [u, v]. The input array
can be 3D (Nx3x2) to specify multiple texture coordinates if multiple
triangles are specified. If so `N` must be the same size as the first
dimension of `tri`.
norm : bool, optional
Normalize computed tangents if ``True``, default is ``True``.
out : ndarray, optional
Optional output array. Must have one fewer dimensions than `tri`. The
shape of the last dimension must be 3.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Surface normal of triangle `tri`.

Examples
--------
Compute surface normals, tangents, and bitangents for a list of triangles::

# triangle vertices (2x3x3)
vertices = [[[-1., 0., 0.], [0., 1., 0.], [1, 0, 0]],
[[1., 0., 0.], [0., 1., 0.], [-1, 0, 0]]]

# array of triangle texture coordinates (2x3x2)
uv = np.asarray([
[(0.0, 1.0), (0.0, 0.0), (1.0, 0.0)],   # 1
[(0.0, 1.0), (0.0, 0.0), (1.0, 0.0)]])  # 2

normals = surfaceNormal(vertices)
tangents = surfaceTangent(vertices, uv)
bitangents = cross(normals, tangents)  # or use `surfaceBitangent`

Orthogonalize a surface tangent with a vertex normal vector to get the
vertex tangent and bitangent vectors::

vertexTangent = orthogonalize(faceTangent, vertexNormal)
vertexBitangent = cross(vertexTangent, vertexNormal)

Ensure computed vectors have the same handedness, if not, flip the tangent
vector (important for applications like normal mapping)::

# tangent, bitangent, and normal are 2D
tangent[dot(cross(normal, tangent), bitangent) < 0.0, :] *= -1.0

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

tris = np.asarray(tri, dtype=dtype)
if tris.ndim == 2:
tris = np.expand_dims(tri, axis=0)

if tris.shape[0] == 1:
toReturn = np.zeros((3,), dtype=dtype) if out is None else out
else:
if out is None:
toReturn = np.zeros((tris.shape[0], 3), dtype=dtype)
else:
toReturn = out

uvs = np.asarray(uv, dtype=dtype)
if uvs.ndim == 2:
uvs = np.expand_dims(uvs, axis=0)

# based off the implementation from
e1 = tris[:, 1, :] - tris[:, 0, :]
e2 = tris[:, 2, :] - tris[:, 0, :]
d1 = uvs[:, 1, :] - uvs[:, 0, :]
d2 = uvs[:, 2, :] - uvs[:, 0, :]

# compute the bitangent
nr = np.atleast_2d(toReturn)
nr[:, 0] = d2[:, 1] * e1[:, 0] - d1[:, 1] * e2[:, 0]
nr[:, 1] = d2[:, 1] * e1[:, 1] - d1[:, 1] * e2[:, 1]
nr[:, 2] = d2[:, 1] * e1[:, 2] - d1[:, 1] * e2[:, 2]

f = dtype(1.0) / (d1[:, 0] * d2[:, 1] - d2[:, 0] * d1[:, 1])
nr *= f[:, np.newaxis]

if norm:
normalize(toReturn, out=toReturn, dtype=dtype)

[docs]def vertexNormal(faceNorms, norm=True, out=None, dtype=None):
"""Compute a vertex normal from shared triangles.

This function computes a vertex normal by averaging the surface normals of
the triangles it belongs to. If model has no vertex normals, first use
:func:`surfaceNormal` to compute them, then run :func:`vertexNormal` to
compute vertex normal attributes.

While this function is mainly used to compute vertex normals, it can also
be supplied triangle tangents and bitangents.

Parameters
----------
faceNorms : array_like
An array (Nx3) of surface normals.
norm : bool, optional
Normalize computed normals if ``True``, default is ``True``.
out : ndarray, optional
Optional output array.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Vertex normal.

Examples
--------
Compute a vertex normal from the face normals of the triangles it belongs
to::

normals = [[1., 0., 0.], [0., 1., 0.]]  # adjacent face normals
vertexNorm = vertexNormal(normals)

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

triNorms2d = np.atleast_2d(np.asarray(faceNorms, dtype=dtype))
nFaces = triNorms2d.shape[0]

if out is None:
toReturn = np.zeros((3,), dtype=dtype)
else:
toReturn = out

toReturn[0] = np.sum(triNorms2d[:, 0])
toReturn[1] = np.sum(triNorms2d[:, 1])
toReturn[2] = np.sum(triNorms2d[:, 2])
toReturn /= nFaces

if norm:
normalize(toReturn, out=toReturn, dtype=dtype)

[docs]def fixTangentHandedness(tangents, normals, bitangents, out=None, dtype=None):
"""Ensure the handedness of tangent vectors are all the same.

Often 3D computed tangents may not have the same handedness due to how
texture coordinates are specified. This function takes input surface vectors
are ensures that tangents have the same handedness. Use this function if you
notice that normal mapping shading appears reversed with respect to the
incident light direction. The output array of corrected tangents can be used
inplace of the original.

Parameters
----------
tangents, normals, bitangents : array_like
Input Nx3 arrays of triangle tangents, normals and bitangents. All
arrays must have the same size.
out : ndarray, optional
Optional output array for tangents. If not specified, a new array of
tangents will be allocated.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Array of tangents with handedness corrected.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

tangents = np.asarray(tangents, dtype=dtype)
normals = np.asarray(normals, dtype=dtype)
bitangents = np.asarray(bitangents, dtype=dtype)

toReturn = np.zeros_like(tangents, dtype=dtype) if out is None else out
toReturn[:, :] = tangents
toReturn[dot(cross(normals, tangents, dtype=dtype),
bitangents, dtype=dtype) < 0.0, :] *= -1.0

# ------------------------------------------------------------------------------
# Collision Detection, Interaction and Kinematics
#
[docs]def fitBBox(points, dtype=None):
"""Fit an axis-aligned bounding box around points.

This computes the minimum and maximum extents for a bounding box to
completely enclose `points`. Keep in mind the output in bounds are
axis-aligned and may not optimally fits the points (i.e. fits the points
with the minimum required volume). However, this should work well enough for
applications such as visibility testing (see

Parameters
----------
points : array_like
Nx3 or Nx4 array of points to fit the bounding box to.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Extents (mins, maxs) as a 2x3 array.

--------
computeBBoxCorners : Convert bounding box extents to corners.

"""
dtype = np.float64 if dtype is None else np.dtype(dtype).type

points = np.asarray(points, dtype=dtype)
extents = np.zeros((2, 3), dtype=dtype)

extents[0, :] = (np.min(points[:, 0]),
np.min(points[:, 1]),
np.min(points[:, 2]))
extents[1, :] = (np.max(points[:, 0]),
np.max(points[:, 1]),
np.max(points[:, 2]))

return extents

[docs]def computeBBoxCorners(extents, dtype=None):
"""Get the corners of an axis-aligned bounding box.

Parameters
----------
extents : array_like
2x3 array indicating the minimum and maximum extents of the bounding
box.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
8x4 array of points defining the corners of the bounding box.

Examples
--------
Compute the corner points of a bounding box::

minExtent = [-1, -1, -1]
maxExtent = [1, 1, 1]
corners = computeBBoxCorners([minExtent, maxExtent])

# [[ 1.  1.  1.  1.]
#  [-1.  1.  1.  1.]
#  [ 1. -1.  1.  1.]
#  [-1. -1.  1.  1.]
#  [ 1.  1. -1.  1.]
#  [-1.  1. -1.  1.]
#  [ 1. -1. -1.  1.]
#  [-1. -1. -1.  1.]]

"""
extents = np.asarray(extents, dtype=dtype)

assert extents.shape == (2, 3,)

corners = np.zeros((8, 4), dtype=dtype)
idx = np.arange(0, 8)
corners[:, 0] = np.where(idx[:] & 1, extents[0, 0], extents[1, 0])
corners[:, 1] = np.where(idx[:] & 2, extents[0, 1], extents[1, 1])
corners[:, 2] = np.where(idx[:] & 4, extents[0, 2], extents[1, 2])
corners[:, 3] = 1.0

return corners

[docs]def intersectRayPlane(rayOrig, rayDir, planeOrig, planeNormal, dtype=None):
"""Get the point which a ray intersects a plane.

Parameters
----------
rayOrig : array_like
Origin of the line in space [x, y, z].
rayDir : array_like
Direction vector of the line [x, y, z].
planeOrig : array_like
Origin of the plane to test [x, y, z].
planeNormal : array_like
Normal vector of the plane [x, y, z].
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
tuple or None
Position (`ndarray`) in space which the line intersects the plane and
the distance the intersect occurs from the origin (`float`). `None` is
returned if the line does not intersect the plane at a single point or
at all.

Examples
--------
Find the point in the scene a ray intersects the plane::

# plane information
planeOrigin = [0, 0, 0]
planeNormal = [0, 0, 1]
planeUpAxis = perp([0, 1, 0], planeNormal)

# ray
rayDir = [0, 0, -1]
rayOrigin = [0, 0, 5]

# get the intersect and distance in 3D world space
pnt, dist = intersectRayPlane(rayOrigin, rayDir, planeOrigin, planeNormal)

"""
dtype = np.float64 if dtype is None else np.dtype(dtype).type

# based off the method from GLM
rayOrig = np.asarray(rayOrig, dtype=dtype)
rayDir = np.asarray(rayDir, dtype=dtype)
planeOrig = np.asarray(planeOrig, dtype=dtype)
planeNormal = np.asarray(planeNormal, dtype=dtype)

denom = dot(rayDir, planeNormal, dtype=dtype)
if denom == 0.0:
return None

# distance to collision
dist = dot((planeOrig - rayOrig), planeNormal, dtype=dtype) / denom
intersect = dist * rayDir + rayOrig

return intersect, dist

[docs]def intersectRaySphere(rayOrig, rayDir, sphereOrig=(0., 0., 0.), sphereRadius=1.0,
dtype=None):
"""Calculate the points which a ray/line intersects a sphere (if any).

Get the 3D coordinate of the point which the ray intersects the sphere and
the distance to the point from `orig`. The nearest point is returned if
the line intersects the sphere at multiple locations. All coordinates should
be in world/scene units.

Parameters
----------
rayOrig : array_like
Origin of the ray in space [x, y, z].
rayDir : array_like
Direction vector of the ray [x, y, z], should be normalized.
sphereOrig : array_like
Origin of the sphere to test [x, y, z].
Sphere radius to test in scene units.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
tuple
Coordinate in world space of the intersection and distance in scene
units from `orig`. Returns `None` if there is no intersection.

"""
# based off example from https://antongerdelan.net/opengl/raycasting.html
dtype = np.float64 if dtype is None else np.dtype(dtype).type

rayOrig = np.asarray(rayOrig, dtype=dtype)
rayDir = np.asarray(rayDir, dtype=dtype)
sphereOrig = np.asarray(sphereOrig, dtype=dtype)

d = rayOrig - sphereOrig
b = np.dot(rayDir, d)
c = np.dot(d, d) - np.square(sphereRadius)
b2mc = np.square(b) - c  # determinant

if b2mc < 0.0:  # no roots, ray does not intersect sphere
return None

u = np.sqrt(b2mc)
nearestDist = np.minimum(-b + u, -b - u)
pos = (rayDir * nearestDist) + rayOrig

return pos, nearestDist

[docs]def intersectRayAABB(rayOrig, rayDir, boundsOffset, boundsExtents, dtype=None):
"""Find the point a ray intersects an axis-aligned bounding box (AABB).

Parameters
----------
rayOrig : array_like
Origin of the ray in space [x, y, z].
rayDir : array_like
Direction vector of the ray [x, y, z], should be normalized.
boundsOffset : array_like
Offset of the bounding box in the scene [x, y, z].
boundsExtents : array_like
Minimum and maximum extents of the bounding box.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
tuple
Coordinate in world space of the intersection and distance in scene
units from `rayOrig`. Returns `None` if there is no intersection.

Examples
--------
Get the point on an axis-aligned bounding box that the cursor is over and
place a 3D stimulus there. The eye location is defined by `RigidBodyPose`
object `camera`::

# get the mouse position on-screen
mx, my = mouse.getPos()

# find the point which the ray intersects on the box
result = intersectRayAABB(
camera.pos,
camera.transformNormal(win.coordToRay((mx, my))),
myStim.pos,
myStim.thePose.bounds.extents)

# if the ray intersects, set the position of the cursor object to it
if result is not None:
cursorModel.thePose.pos = result[0]
cursorModel.draw()  # don't draw anything if there is no intersect

Note that if the model is rotated, the bounding box may not be aligned
anymore with the axes. Use `intersectRayOBB` if your model rotates.

"""
# based of the example provided here:
# https://www.scratchapixel.com/lessons/3d-basic-rendering/
# minimal-ray-tracer-rendering-simple-shapes/ray-box-intersection
dtype = np.float64 if dtype is None else np.dtype(dtype).type

rayOrig = np.asarray(rayOrig, dtype=dtype)
rayDir = np.asarray(rayDir, dtype=dtype)
boundsOffset = np.asarray(boundsOffset, dtype=dtype)
extents = np.asarray(boundsExtents, dtype=dtype) + boundsOffset

invDir = 1.0 / rayDir
sign = np.zeros((3,), dtype=int)
sign[invDir < 0.0] = 1

tmin = (extents[sign[0], 0] - rayOrig[0]) * invDir[0]
tmax = (extents[1 - sign[0], 0] - rayOrig[0]) * invDir[0]
tymin = (extents[sign[1], 1] - rayOrig[1]) * invDir[1]
tymax = (extents[1 - sign[1], 1] - rayOrig[1]) * invDir[1]

if tmin > tymax or tymin > tmax:
return None

if tymin > tmin:
tmin = tymin

if tymax < tmax:
tmax = tymax

tzmin = (extents[sign[2], 2] - rayOrig[2]) * invDir[2]
tzmax = (extents[1 - sign[2], 2] - rayOrig[2]) * invDir[2]

if tmin > tzmax or tzmin > tmax:
return None

if tzmin > tmin:
tmin = tzmin

if tzmax < tmax:
tmax = tzmax

if tmin < 0:
if tmax < 0:
return None

return (rayDir * tmin) + rayOrig, tmin

[docs]def intersectRayOBB(rayOrig, rayDir, modelMatrix, boundsExtents, dtype=None):
"""Find the point a ray intersects an oriented bounding box (OBB).

Parameters
----------
rayOrig : array_like
Origin of the ray in space [x, y, z].
rayDir : array_like
Direction vector of the ray [x, y, z], should be normalized.
modelMatrix : array_like
4x4 model matrix of the object and bounding box.
boundsExtents : array_like
Minimum and maximum extents of the bounding box.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
tuple
Coordinate in world space of the intersection and distance in scene
units from `rayOrig`. Returns `None` if there is no intersection.

Examples
--------
Get the point on an oriented bounding box that the cursor is over and place
a 3D stimulus there. The eye location is defined by `RigidBodyPose` object
`camera`::

# get the mouse position on-screen
mx, my = mouse.getPos()

# find the point which the ray intersects on the box
result = intersectRayOBB(
camera.pos,
camera.transformNormal(win.coordToRay((mx, my))),
myStim.thePose.getModelMatrix(),
myStim.thePose.bounds.extents)

# if the ray intersects, set the position of the cursor object to it
if result is not None:
cursorModel.thePose.pos = result[0]
cursorModel.draw()  # don't draw anything if there is no intersect

"""
# based off algorithm:
# https://www.opengl-tutorial.org/miscellaneous/clicking-on-objects/
# picking-with-custom-ray-obb-function/
dtype = np.float64 if dtype is None else np.dtype(dtype).type

rayOrig = np.asarray(rayOrig, dtype=dtype)
rayDir = np.asarray(rayDir, dtype=dtype)
modelMatrix = np.asarray(modelMatrix, dtype=dtype)
boundsOffset = np.asarray(modelMatrix[:3, 3], dtype=dtype)
extents = np.asarray(boundsExtents, dtype=dtype)

tmin = 0.0
tmax = np.finfo(dtype).max
d = boundsOffset - rayOrig

# solve intersects for each pair of planes along each axis
for i in range(3):
axis = modelMatrix[:3, i]
e = np.dot(axis, d)
f = np.dot(rayDir, axis)

if np.fabs(f) > 1e-5:
t1 = (e + extents[0, i]) / f
t2 = (e + extents[1, i]) / f

if t1 > t2:
temp = t1
t1 = t2
t2 = temp

if t2 < tmax:
tmax = t2

if t1 > tmin:
tmin = t1

if tmin > tmax:
return None

else:
# very close to parallel with the face
if -e + extents[0, i] > 0.0 or -e + extents[1, i] < 0.0:
return None

return (rayDir * tmin) + rayOrig, tmin

[docs]def intersectRayTriangle(rayOrig, rayDir, tri, dtype=None):
"""Get the intersection of a ray and triangle(s).

This function can be used to achieve 'pixel-perfect' ray picking/casting on
meshes defined with triangles. However, high-poly meshes may lead to
performance issues.

Parameters
----------
rayOrig : array_like
Origin of the ray in space [x, y, z].
rayDir : array_like
Direction vector of the ray [x, y, z], should be normalized.
tri : array_like
Triangle vertices as 2D (3x3) array [p0, p1, p2] where each vertex is a
length 3 array [vx, xy, vz]. The input array can be 3D (Nx3x3) to
specify multiple triangles.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
tuple
Coordinate in world space of the intersection, distance in scene
units from `rayOrig`, and the barycentric coordinates on the triangle
[x, y]. Returns `None` if there is no intersection.

"""
# based off `intersectRayTriangle` from GLM (https://glm.g-truc.net)
dtype = np.float64 if dtype is None else np.dtype(dtype).type

rayOrig = np.asarray(rayOrig, dtype=dtype)
rayDir = np.asarray(rayDir, dtype=dtype)
triVerts = np.asarray(tri, dtype=dtype)

edge1 = triVerts[1, :] - triVerts[0, :]
edge2 = triVerts[2, :] - triVerts[0, :]

baryPos = np.zeros((2,), dtype=dtype)

p = np.cross(rayDir, edge2)
det = np.dot(edge1, p)

if det > np.finfo(dtype).eps:
dist = rayOrig - triVerts[0, :]

baryPos[0] = np.dot(dist, p)
if baryPos[0] < 0.0 or baryPos[0] > det:
return None

ortho = np.cross(dist, edge1)

baryPos[1] = np.dot(rayDir, ortho)
if baryPos[1] < 0.0 or baryPos[0] + baryPos[1] > det:
return None

elif det < -np.finfo(dtype).eps:
dist = rayOrig - triVerts[0, :]

baryPos[0] = np.dot(dist, p)
if baryPos[0] > 0.0 or baryPos[0] < det:
return None

ortho = np.cross(dist, edge1)

baryPos[1] = np.dot(rayDir, ortho)
if baryPos[1] > 0.0 or baryPos[0] + baryPos[1] < det:
return None
else:
return None

invDet = 1.0 / det
dist = np.dot(edge2, ortho) * invDet
baryPos *= invDet

return (rayDir * dist) + rayOrig, dist, baryPos

[docs]def ortho3Dto2D(p, orig, normal, up, right=None, dtype=None):
"""Get the planar coordinates of an orthogonal projection of a 3D point onto
a 2D plane.

This function gets the nearest point on the plane which a 3D point falls on
the plane.

Parameters
----------
p : array_like
Point to be projected on the plane.
orig : array_like
Origin of the plane to test [x, y, z].
normal : array_like
Normal vector of the plane [x, y, z], must be normalized.
up : array_like
Normalized up (+Y) direction of the plane's coordinate system. Must be
perpendicular to `normal`.
right : array_like, optional
Perpendicular right (+X) axis. If not provided, the axis will be
computed via the cross product between `normal` and `up`.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Coordinates on the plane [X, Y] where the 3D point projects towards
perpendicularly.

Examples
--------
This function can be used with :func:`intersectRayPlane` to find the
location on the plane the ray intersects::

# plane information
planeOrigin = [0, 0, 0]
planeNormal = [0, 0, 1]  # must be normalized
planeUpAxis = perp([0, 1, 0], planeNormal)  # must also be normalized

# ray
rayDir = [0, 0, -1]
rayOrigin = [0, 0, 5]

# get the intersect in 3D world space
pnt = intersectRayPlane(rayOrigin, rayDir, planeOrigin, planeNormal)

# get the 2D coordinates on the plane the intersect occurred
planeX, planeY = ortho3Dto2D(pnt, planeOrigin, planeNormal, planeUpAxis)

"""
dtype = np.float64 if dtype is None else np.dtype(dtype).type

p = np.asarray(p, dtype=dtype)
orig = np.asarray(orig, dtype=dtype)
normal = np.asarray(normal, dtype=dtype)
up = np.asarray(up, dtype=dtype)

toReturn = np.zeros((2,))

offset = p - orig
if right is None:
# derive X axis with cross product
toReturn[0] = dot(offset, cross(normal, up, dtype=dtype), dtype=dtype)
else:
toReturn[0] = dot(offset, np.asarray(right, dtype=dtype), dtype=dtype)

toReturn[1] = dot(offset, up)

[docs]def articulate(boneVecs, boneOris, dtype=None):
"""Articulate an armature.

This function is used for forward kinematics and posing by specifying a list
of 'bones'. A bone has a length and orientation, where sequential bones are
linked end-to-end. Returns the transformed origins of the bones in scene
coordinates and their orientations.

There are many applications for forward kinematics such as posing armatures
and stimuli for display (eg. mocap data). Another application is for getting
the location of the end effector of coordinate measuring hardware, where
encoders measure the joint angles and the length of linking members are
known. This can be used for computing pose from "Sword of Damocles"[1]_ like
hardware or some other haptic input devices which the participant wears (eg.
a glove that measures joint angles in the hand). The computed pose of the
joints can be used to interact with virtual stimuli.

Parameters
----------
boneVecs : array_like
Bone lengths [x, y, z] as an Nx3 array.
boneOris : array_like
Orientation of the bones as quaternions in form [x, y, z, w], relative
to the previous bone.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
tuple
Array of bone origins and orientations. The first origin is root
position which is always at [0, 0, 0]. Use :func:`transform` to
reposition the armature, or create a transformation matrix and use
`applyMatrix` to translate and rotate the whole armature into position.

References
----------
.. [1] Sutherland, I. E. (1968). "A head-mounted three dimensional display".
Proceedings of AFIPS 68, pp. 757-764

Examples
--------
Compute the orientations and origins of segments of an arm::

# bone lengths
boneLengths = [[0., 1., 0.], [0., 1., 0.], [0., 1., 0.]]

# create quaternions for joints
shoulder = mt.quatFromAxisAngle('-y', 45.0)
elbow = mt.quatFromAxisAngle('+z', 45.0)
wrist = mt.quatFromAxisAngle('+z', 45.0)

# articulate the parts of the arm
boxPos, boxOri = mt.articulate(pos, [shoulder, elbow, wrist])

# assign positions and orientations to 3D objects
shoulderModel.thePose.posOri = (boxPos[0, :], boxOri[0, :])
elbowModel.thePose.posOri = (boxPos[1, :], boxOri[1, :])
wristModel.thePose.posOri = (boxPos[2, :], boxOri[2, :])

"""
dtype = np.float64 if dtype is None else np.dtype(dtype).type

boneVecs = np.asarray(boneVecs, dtype=dtype)
boneOris = np.asarray(boneOris, dtype=dtype)

jointOri = accumQuat(boneOris, dtype=dtype)  # get joint orientations
bonesRotated = applyQuat(jointOri, boneVecs, dtype=dtype)  # rotate bones

# accumulate
bonesTranslated = np.asarray(
tuple(itertools.accumulate(bonesRotated[:], lambda a, b: a + b)),
dtype=dtype)
bonesTranslated -= bonesTranslated[0, :]  # offset root length

return bonesTranslated, jointOri

# ------------------------------------------------------------------------------
# Quaternion Operations
#

[docs]def slerp(q0, q1, t, shortest=True, out=None, dtype=None):
"""Spherical linear interpolation (SLERP) between two quaternions.

The behaviour of this function depends on the types of arguments:

* If `q0` and `q1` are both 1-D and `t` is scalar, the interpolation at `t`
is returned.
* If `q0` and `q1` are both 2-D Nx4 arrays and `t` is scalar, an Nx4 array
is returned with each row containing the interpolation at `t` for each
quaternion pair at matching row indices in `q0` and `q1`.

Parameters
----------
q0 : array_like
Initial quaternion in form [x, y, z, w] where w is real and x, y, z
are imaginary components.
q1 : array_like
Final quaternion in form [x, y, z, w] where w is real and x, y, z
are imaginary components.
t : float
Interpolation weight factor within interval 0.0 and 1.0.
shortest : bool, optional
Ensure interpolation occurs along the shortest arc along the 4-D
hypersphere (default is `True`).
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Quaternion [x, y, z, w] at `t`.

Examples
--------
Interpolate between two orientations::

q0 = quatFromAxisAngle(90.0, degrees=True)
q1 = quatFromAxisAngle(-90.0, degrees=True)
# halfway between 90 and -90 is 0.0 or quaternion [0. 0. 0. 1.]
qr = slerp(q0, q1, 0.5)

Example of smooth rotation of an object with fixed angular velocity::

degPerSec = 10.0  # rotate a stimulus at 10 degrees per second

# initial orientation, axis rotates in the Z direction
qr = quatFromAxisAngle([0., 0., -1.], 0.0, degrees=True)
# amount to rotate every second
qv = quatFromAxisAngle([0., 0., -1.], degPerSec, degrees=True)

# ---- within main experiment loop ----
# `frameTime` is the time elapsed in seconds from last `slerp`.
qr = multQuat(qr, slerp((0., 0., 0., 1.), qv, degPerSec * frameTime))
_, angle = quatToAxisAngle(qr)  # discard axis, only need angle

# myStim is a GratingStim or anything with an 'ori' argument which
# accepts angle in degrees
myStim.ori = angle
myStim.draw()

"""
# Implementation based on code found here:
#  https://en.wikipedia.org/wiki/Slerp
#
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

q0 = normalize(q0, dtype=dtype)
q1 = normalize(q1, dtype=dtype)
assert q0.shape == q1.shape

toReturn = np.zeros(q0.shape, dtype=dtype) if out is None else out
toReturn.fill(0.0)
t = dtype(t)
q0, q1, qr = np.atleast_2d(q0, q1, toReturn)

d = np.clip(np.sum(q0 * q1, axis=1), -1.0, 1.0)
if shortest:
d[d < 0.0] *= -1.0
q1[d < 0.0] *= -1.0

theta0 = np.arccos(d)
theta = theta0 * t
sinTheta = np.sin(theta)
s1 = sinTheta / np.sin(theta0)
s0 = np.cos(theta[:, np.newaxis]) - d[:, np.newaxis] * s1[:, np.newaxis]
qr[:, :] = q0 * s0
qr[:, :] += q1 * s1[:, np.newaxis]
qr[:, :] += 0.0

[docs]def quatToAxisAngle(q, degrees=True, dtype=None):
"""Convert a quaternion to `axis` and `angle` representation.

This allows you to use quaternions to set the orientation of stimuli that
have an `ori` property.

Parameters
----------
q : tuple, list or ndarray of float
Quaternion in form [x, y, z, w] where w is real and x, y, z
are imaginary components.
degrees : bool, optional
Indicate `angle` is to be returned in degrees, otherwise `angle` will be
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
tuple
Axis and angle of quaternion in form ([ax, ay, az], angle). If `degrees`
is `True`, the angle returned is in degrees, radians if `False`.

Examples
--------
Using a quaternion to rotate a stimulus a fixed angle each frame::

# initial orientation, axis rotates in the Z direction
qr = quatFromAxisAngle([0., 0., -1.], 0.0, degrees=True)
# rotation per-frame, here it's 0.1 degrees per frame
qf = quatFromAxisAngle([0., 0., -1.], 0.1, degrees=True)

# ---- within main experiment loop ----
# myStim is a GratingStim or anything with an 'ori' argument which
# accepts angle in degrees
qr = multQuat(qr, qf)  # cumulative rotation
_, angle = quatToAxisAngle(qr)  # discard axis, only need angle
myStim.ori = angle
myStim.draw()

"""
dtype = np.float64 if dtype is None else np.dtype(dtype).type
q = normalize(q, dtype=dtype)  # returns ndarray
v = np.sqrt(np.sum(np.square(q[:3])))

if np.count_nonzero(q[:3]):
axis = q[:3] / v
angle = dtype(2.0) * np.arctan2(v, q[3])
else:
axis = np.zeros((3,), dtype=dtype)
axis[0] = 1.
angle = 0.0

axis += 0.0

return axis, np.degrees(angle) if degrees else angle

[docs]def quatFromAxisAngle(axis, angle, degrees=True, dtype=None):
"""Create a quaternion to represent a rotation about `axis` vector by
`angle`.

Parameters
----------
axis : tuple, list, ndarray or str
Axis vector components or axis name. If a vector, input must be length
3 [x, y, z]. A string can be specified for rotations about world axes
(eg. `'+x'`, `'-z'`, `'+y'`, etc.)
angle : float
Rotation angle in radians (or degrees if `degrees` is `True`. Rotations
are right-handed about the specified `axis`.
degrees : bool, optional
Indicate `angle` is in degrees, otherwise `angle` will be treated as
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Quaternion [x, y, z, w].

Examples
--------
Create a quaternion from specified `axis` and `angle`::

axis = [0., 0., -1.]  # rotate about -Z axis
angle = 90.0  # angle in degrees
ori = quatFromAxisAngle(axis, angle, degrees=True)  # using degrees!

"""
dtype = np.float64 if dtype is None else np.dtype(dtype).type
toReturn = np.zeros((4,), dtype=dtype)

if degrees:
else:

try:
axis = VEC_AXES[axis] if isinstance(axis, str) else axis
except KeyError:
raise ValueError(
"Value of `axis` must be either '+X', '-X', '+Y', '-Y', '+Z' or "
"'-Z' or length 3 vector.")

axis = normalize(axis, dtype=dtype)
if np.count_nonzero(axis) == 0:
raise ValueError("Value for `axis` is zero-length.")

toReturn += 0.0  # remove negative zeros

[docs]def quatYawPitchRoll(q, degrees=True, out=None, dtype=None):
"""Get the yaw, pitch, and roll of a quaternion's orientation relative to
the world -Z axis.

You can multiply the quaternion by the inverse of some other one to make the
returned values referenced to a local coordinate system.

Parameters
----------
q : tuple, list or ndarray of float
Quaternion in form [x, y, z, w] where w is real and x, y, z
are imaginary components.
degrees : bool, optional
Indicate angles are to be returned in degrees, otherwise they will be
out : ndarray
Optional output array. Must have same `shape` and `dtype` as what is
expected to be returned by this function of `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Yaw, pitch and roll [yaw, pitch, roll] of quaternion `q`.

"""
# based off code found here:
# https://en.wikipedia.org/wiki/Conversion_between_quaternions_and_Euler_angles
# Yields the same results as PsychXR's LibOVRPose.getYawPitchRoll method.
dtype = np.float64 if dtype is None else np.dtype(dtype).type
q = np.asarray(q, dtype=dtype)

toReturn = np.zeros((3,), dtype=dtype) if out is None else out

sinRcosP = 2.0 * (q[3] * q[0] + q[1] * q[2])
cosRcosP = 1.0 - 2.0 * (q[0] * q[0] + q[1] * q[1])

toReturn[0] = np.arctan2(sinRcosP, cosRcosP)

sinp = 2.0 * (q[3] * q[1] - q[2] * q[0])

if np.fabs(sinp) >= 1.:
toReturn[1] = np.copysign(np.pi / 2., sinp)
else:
toReturn[1] = np.arcsin(sinp)

sinYcosP = 2.0 * (q[3] * q[2] + q[0] * q[1])
cosYcosP = 1.0 - 2.0 * (q[1] * q[1] + q[2] * q[2])

toReturn[2] = np.arctan2(sinYcosP, cosYcosP)

if degrees:
toReturn[:] = np.degrees(toReturn[:])

[docs]def quatMagnitude(q, squared=False, out=None, dtype=None):
"""Get the magnitude of a quaternion.

A quaternion is normalized if its magnitude is 1.

Parameters
----------
q : array_like
Quaternion(s) in form [x, y, z, w] where w is real and x, y, z are
imaginary components.
squared : bool, optional
If ``True`` return the squared magnitude. If you are just checking if a
quaternion is normalized, the squared magnitude will suffice to avoid
the square root operation.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
float or ndarray
Magnitude of quaternion `q`.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

q = np.asarray(q, dtype=dtype)
if q.ndim == 1:
assert q.shape[0] == 4
if squared:
toReturn = np.sum(np.square(q))
else:
toReturn = np.sqrt(np.sum(np.square(q)))
elif q.ndim == 2:
assert q.shape[1] == 4
toReturn = np.zeros((q.shape[0],), dtype=dtype) if out is None else out
if squared:
toReturn[:] = np.sum(np.square(q), axis=1)
else:
toReturn[:] = np.sqrt(np.sum(np.square(q), axis=1))
else:
raise ValueError("Input argument 'q' has incorrect dimensions.")

[docs]def multQuat(q0, q1, out=None, dtype=None):
"""Multiply quaternion `q0` and `q1`.

The orientation of the returned quaternion is the combination of the input
quaternions.

Parameters
----------
q0, q1 : array_like
Quaternions to multiply in form [x, y, z, w] where w is real and x, y, z
are imaginary components. If 2D (Nx4) arrays are specified, quaternions
are multiplied row-wise between each array.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Combined orientations of `q0` amd `q1`.

Notes
-----
* Quaternions are normalized prior to multiplication.

Examples
--------
Combine the orientations of two quaternions::

a = quatFromAxisAngle([0, 0, -1], 45.0, degrees=True)
b = quatFromAxisAngle([0, 0, -1], 90.0, degrees=True)
c = multQuat(a, b)  # rotates 135 degrees about -Z axis

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

q0 = normalize(q0, dtype=dtype)
q1 = normalize(q1, dtype=dtype)
assert q0.shape == q1.shape
toReturn = np.zeros(q0.shape, dtype=dtype) if out is None else out
toReturn.fill(0.0)  # clear array
q0, q1, qr = np.atleast_2d(q0, q1, toReturn)

# multiply quaternions for each row of the operand arrays
qr[:, :3] = np.cross(q0[:, :3], q1[:, :3], axis=1)
qr[:, :3] += q0[:, :3] * np.expand_dims(q1[:, 3], axis=1)
qr[:, :3] += q1[:, :3] * np.expand_dims(q0[:, 3], axis=1)
qr[:, 3] = q0[:, 3]
qr[:, 3] *= q1[:, 3]
qr[:, 3] -= np.sum(np.multiply(q0[:, :3], q1[:, :3]), axis=1)  # dot product
qr += 0.0

[docs]def invertQuat(q, out=None, dtype=None):
"""Get the multiplicative inverse of a quaternion.

This gives a quaternion which rotates in the opposite direction with equal
magnitude. Multiplying a quaternion by its inverse returns an identity
quaternion as both orientations cancel out.

Parameters
----------
q : ndarray, list, or tuple of float
Quaternion to invert in form [x, y, z, w] where w is real and x, y, z
are imaginary components. If `q` is 2D (Nx4), each row is treated as a
separate quaternion and inverted.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Inverse of quaternion `q`.

Examples
--------
Show that multiplying a quaternion by its inverse returns an identity
quaternion where [x=0, y=0, z=0, w=1]::

angle = 90.0
axis = [0., 0., -1.]
q = quatFromAxisAngle(axis, angle, degrees=True)
qinv = invertQuat(q)
qr = multQuat(q, qinv)
qi = np.array([0., 0., 0., 1.])  # identity quaternion
print(np.allclose(qi, qr))   # True

Notes
-----
* Quaternions are normalized prior to inverting.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

q = normalize(q, dtype=dtype)
toReturn = np.zeros(q.shape, dtype=dtype) if out is None else out
qn, qinv = np.atleast_2d(q, toReturn)  # 2d views

# conjugate the quaternion
qinv[:, :3] = -qn[:, :3]
qinv[:, 3] = qn[:, 3]
qinv /= np.sum(np.square(qn), axis=1)[:, np.newaxis]
qinv += 0.0  # remove negative zeros

[docs]def applyQuat(q, points, out=None, dtype=None):
"""Rotate points/coordinates using a quaternion.

This is similar to using `applyMatrix` with a rotation matrix. However, it
is computationally less intensive to use `applyQuat` if one only wishes to
rotate points.

Parameters
----------
q : array_like
Quaternion to invert in form [x, y, z, w] where w is real and x, y, z
are imaginary components.
points : array_like
2D array of vectors or points to transform, where each row is a single
point. Only the x, y, and z components (the first three columns) are
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Transformed points.

Examples
--------
Rotate points using a quaternion::

points = [[1., 0., 0.], [0., -1., 0.]]
quat = quatFromAxisAngle(-90.0, [0., 0., -1.], degrees=True)
pointsRotated = applyQuat(quat, points)
# [[0. 1. 0.]
#  [1. 0. 0.]]

Show that you get the same result as a rotation matrix::

axis = [0., 0., -1.]
angle = -90.0
rotMat = rotationMatrix(axis, angle)[:3, :3]  # rotation sub-matrix only
rotQuat = quatFromAxisAngle(angle, axis, degrees=True)
points = [[1., 0., 0.], [0., -1., 0.]]
isClose = np.allclose(applyMatrix(rotMat, points),  # True
applyQuat(rotQuat, points))

Specifying an array to `q` where each row is a quaternion transforms points
in corresponding rows of `points`::

points = [[1., 0., 0.], [0., -1., 0.]]
quats = [quatFromAxisAngle(-90.0, [0., 0., -1.], degrees=True),
quatFromAxisAngle(45.0, [0., 0., -1.], degrees=True)]
applyQuat(quats, points)

"""
# based on 'quat_mul_vec3' implementation from linmath.h
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

qin = np.asarray(q, dtype=dtype)
points = np.asarray(points, dtype=dtype)

if out is not None:
assert points.shape == out.shape

toReturn = np.zeros(points.shape, dtype=dtype) if out is None else out
pin, pout = np.atleast_2d(points, toReturn)
pout[:, :] = pin[:, :]  # copy values into output array

if qin.ndim == 1:
assert qin.shape[0] == 4
t = cross(qin[:3], pin[:, :3]) * dtype(2.0)
u = cross(qin[:3], t)
t *= qin[3]
pout[:, :3] += t
pout[:, :3] += u
elif qin.ndim == 2:
assert qin.shape[1] == 4 and qin.shape[0] == pin.shape[0]
t = cross(qin[:, :3], pin[:, :3])
t *= dtype(2.0)
u = cross(qin[:, :3], t)
t *= np.expand_dims(qin[:, 3], axis=1)
pout[:, :3] += t
pout[:, :3] += u
else:
raise ValueError("Input arguments have invalid dimensions.")

[docs]def accumQuat(qlist, out=None, dtype=None):
"""Accumulate quaternion rotations.

Chain multiplies an Nx4 array of quaternions, accumulating their rotations.
This function can be used for computing the orientation of joints in an
armature for forward kinematics. The first quaternion is treated as the
'root' and the last is the orientation of the end effector.

Parameters
----------
q : array_like
Nx4 array of quaternions to accumulate, where each row is a quaternion.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified. In this case, the same shape as
`qlist`.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Nx4 array of quaternions.

Examples
--------
Get the orientation of joints in an armature if we know their relative
angles::

shoulder = quatFromAxisAngle('-x', 45.0)  # rotate shoulder down 45 deg
elbow = quatFromAxisAngle('+x', 45.0)  # rotate elbow up 45 deg
wrist = quatFromAxisAngle('-x', 45.0)  # rotate wrist down 45 deg
finger = quatFromAxisAngle('+x', 0.0)  # keep finger in-line with wrist

armRotations = accumQuat([shoulder, elbow, wrist, finger])

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

qlist = np.asarray(qlist, dtype=dtype)
qlist = np.atleast_2d(qlist)

qr = np.zeros_like(qlist, dtype=dtype) if out is None else out
qr[:, :] = tuple(itertools.accumulate(
qlist[:], lambda a, b: multQuat(a, b, dtype=dtype)))

return qr

[docs]def alignTo(v, t, out=None, dtype=None):
"""Compute a quaternion which rotates one vector to align with another.

Parameters
----------
v : array_like
Vector [x, y, z] to rotate. Can be Nx3, but must have the same shape as
`t`.
t : array_like
Target [x, y, z] vector to align to. Can be Nx3, but must have the same
shape as `v`.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Quaternion which rotates `v` to `t`.

Examples
--------
Rotate some vectors to align with other vectors, inputs should be
normalized::

vec = [[1, 0, 0], [0, 1, 0], [1, 0, 0]]
targets = [[0, 1, 0], [0, -1, 0], [-1, 0, 0]]

qr = alignTo(vec, targets)
vecRotated = applyQuat(qr, vec)

numpy.allclose(vecRotated, targets)  # True

Get matrix which orients vertices towards a point::

point = [5, 6, 7]
vec = [0, 0, -1]  # initial facing is -Z (forward in GL)

targetVec = normalize(point - vec)
qr = alignTo(vec, targetVec)  # get rotation to align

M = quatToMatrix(qr)  # 4x4 transformation matrix

"""
# based off Quaternion::align from Quaternion.hpp from OpenMP
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

v = normalize(v, dtype=dtype)
t = normalize(t, dtype=dtype)

if out is None:
if v.ndim == 1:
toReturn = np.zeros((4,), dtype=dtype)
else:
toReturn = np.zeros((v.shape[0], 4), dtype=dtype)
else:
toReturn = out

qr, v2d, t2d = np.atleast_2d(toReturn, v, t)

b = bisector(v2d, t2d, norm=True, dtype=dtype)
cosHalfAngle = dot(v2d, b, dtype=dtype)

nonparallel = cosHalfAngle > 0.0  # rotation is not 180 degrees
qr[nonparallel, :3] = cross(v2d[nonparallel], b[nonparallel], dtype=dtype)
qr[nonparallel, 3] = cosHalfAngle[nonparallel]

if np.all(nonparallel):  # don't bother handling special cases

# deal with cases where the vectors are facing exact opposite directions
ry = np.logical_and(np.abs(v2d[:, 0]) >= np.abs(v2d[:, 1]), ~nonparallel)
rx = np.logical_and(~ry, ~nonparallel)

getLength = lambda x, y: np.sqrt(x * x + y * y)
if not np.all(rx):
invLength = getLength(v2d[ry, 0], v2d[ry, 2])
invLength = np.where(invLength > 0.0, 1.0 / invLength, invLength)  # avoid x / 0
qr[ry, 0] = -v2d[ry, 2] * invLength
qr[ry, 2] = v2d[ry, 0] * invLength

if not np.all(ry):  # skip if all the same edge case
invLength = getLength(v2d[rx, 1], v2d[rx, 2])
invLength = np.where(invLength > 0.0, 1.0 / invLength, invLength)
qr[rx, 1] = v2d[rx, 2] * invLength
qr[rx, 2] = -v2d[rx, 1] * invLength

[docs]def matrixToQuat(m, out=None, dtype=None):
"""Convert a rotation matrix to a quaternion.

Parameters
----------
m : array_like
3x3 rotation matrix (row-major). A 4x4 affine transformation matrix may
be provided, assuming the top-left 3x3 sub-matrix is orthonormal and
is a rotation group.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Rotation quaternion.

Notes
-----
* Depending on the input, returned quaternions may not be exactly the same
as the one used to construct the rotation matrix (i.e. by calling
`quatToMatrix`), typically when a large rotation angle is used. However,
the returned quaternion should result in the same rotation when applied to
points.

Examples
--------
Converting a rotation matrix from the OpenGL matrix stack to a quaternion::

glRotatef(45., -1, 0, 0)

m = np.zeros((4, 4), dtype='float32')  # store the matrix
GL.glGetFloatv(
GL.GL_MODELVIEW_MATRIX,
m.ctypes.data_as(ctypes.POINTER(ctypes.c_float)))

qr = matrixToQuat(m.T)  # must be transposed

Interpolation between two 4x4 transformation matrices::

interpWeight = 0.5

posStart = mStart[:3, 3]
oriStart = matrixToQuat(mStart)

posEnd = mEnd[:3, 3]
oriEnd = matrixToQuat(mEnd)

oriInterp = slerp(qStart, qEnd, interpWeight)
posInterp = lerp(posStart, posEnd, interpWeight)

mInterp = posOriToMatrix(posInterp, oriInterp)

"""
# based off example `Maths - Conversion Matrix to Quaternion` from
# https://www.euclideanspace.com/
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

m = np.asarray(m, dtype=dtype)

if m.shape == (4, 4,) or m.shape == (3, 4,):
m = m[:3, :3]  # keep only rotation group sub-matrix
elif m.shape == (3, 3,):
pass  # fine, nop
else:
raise ValueError("Input matrix `m` must be 3x3 or 4x4.")

toReturn = np.zeros((4,), dtype=dtype) if out is None else out

tr = m[0, 0] + m[1, 1] + m[2, 2]
if tr > 0.0:
s = np.sqrt(tr + 1.0) * 2.0
toReturn[3] = dtype(0.25) * s
toReturn[0] = (m[2, 1] - m[1, 2]) / s
toReturn[1] = (m[0, 2] - m[2, 0]) / s
toReturn[2] = (m[1, 0] - m[0, 1]) / s
elif m[0, 0] > m[1, 1] and m[0, 0] > m[2, 2]:
s = np.sqrt(dtype(1.0) + m[0, 0] - m[1, 1] - m[2, 2]) * dtype(2.0)
toReturn[3] = (m[2, 1] - m[1, 2]) / s
toReturn[0] = dtype(0.25) * s
toReturn[1] = (m[0, 1] + m[1, 0]) / s
toReturn[2] = (m[0, 2] + m[2, 0]) / s
elif m[1, 1] > m[2, 2]:
s = np.sqrt(dtype(1.0) + m[1, 1] - m[0, 0] - m[2, 2]) * dtype(2.0)
toReturn[3] = (m[0, 2] - m[2, 0]) / s
toReturn[0] = (m[0, 1] + m[1, 0]) / s
toReturn[1] = dtype(0.25) * s
toReturn[2] = (m[1, 2] + m[2, 1]) / s
else:
s = np.sqrt(dtype(1.0) + m[2, 2] - m[0, 0] - m[1, 1]) * dtype(2.0)
toReturn[3] = (m[1, 0] - m[0, 1]) / s
toReturn[0] = (m[0, 2] + m[2, 0]) / s
toReturn[1] = (m[1, 2] + m[2, 1]) / s
toReturn[2] = dtype(0.25) * s

# ------------------------------------------------------------------------------
# Matrix Operations
#

[docs]def quatToMatrix(q, out=None, dtype=None):
"""Create a 4x4 rotation matrix from a quaternion.

Parameters
----------
q : tuple, list or ndarray of float
Quaternion to convert in form [x, y, z, w] where w is real and x, y, z
are imaginary components.
out : ndarray or None
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray or None
4x4 rotation matrix in row-major order.

Examples
--------
Convert a quaternion to a rotation matrix::

point = [0., 1., 0., 1.]  # 4-vector form [x, y, z, 1.0]
ori = [0., 0., 0., 1.]
rotMat = quatToMatrix(ori)
# rotate 'point' using matrix multiplication
newPoint = np.matmul(rotMat.T, point)  # returns [-1., 0., 0., 1.]

Rotate all points in an array (each row is a coordinate)::

points = np.asarray([[0., 0., 0., 1.],
[0., 1., 0., 1.],
[1., 1., 0., 1.]])
newPoints = points.dot(rotMat)

Notes
-----
* Quaternions are normalized prior to conversion.

"""
# based off implementations from
# https://github.com/glfw/glfw/blob/master/deps/linmath.h
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
R = np.zeros((4, 4,), dtype=dtype)
else:
dtype = np.dtype(out.dtype).type
R = out
R.fill(0.0)

q = normalize(q, dtype=dtype)
b, c, d, a = q[:]
vsqr = np.square(q)

u = dtype(2.0)
R[0, 0] = vsqr[3] + vsqr[0] - vsqr[1] - vsqr[2]
R[1, 0] = u * (b * c + a * d)
R[2, 0] = u * (b * d - a * c)

R[0, 1] = u * (b * c - a * d)
R[1, 1] = vsqr[3] - vsqr[0] + vsqr[1] - vsqr[2]
R[2, 1] = u * (c * d + a * b)

R[0, 2] = u * (b * d + a * c)
R[1, 2] = u * (c * d - a * b)
R[2, 2] = vsqr[3] - vsqr[0] - vsqr[1] + vsqr[2]

R[3, 3] = dtype(1.0)
R[:, :] += 0.0  # remove negative zeros

return R

[docs]def scaleMatrix(s, out=None, dtype=None):
"""Create a scaling matrix.

The resulting matrix is the same as a generated by a `glScale` call.

Parameters
----------
s : array_like, float or int
Scaling factor(s). If `s` is scalar (float), scaling will be uniform.
Providing a vector of scaling values [sx, sy, sz] will result in an
anisotropic scaling matrix if any of the values differ.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
4x4 scaling matrix in row-major order.

"""
# from glScale
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
S = np.zeros((4, 4,), dtype=dtype)
else:
dtype = np.dtype(out.dtype).type
S = out
S.fill(0.0)

if isinstance(s, (float, int,)):
S[0, 0] = S[1, 1] = S[2, 2] = dtype(s)
else:
S[0, 0] = dtype(s[0])
S[1, 1] = dtype(s[1])
S[2, 2] = dtype(s[2])

S[3, 3] = 1.0

return S

[docs]def rotationMatrix(angle, axis=(0., 0., -1.), out=None, dtype=None):
"""Create a rotation matrix.

The resulting matrix will rotate points about `axis` by `angle`. The
resulting matrix is similar to that produced by a `glRotate` call.

Parameters
----------
angle : float
Rotation angle in degrees.
axis : array_like or str
Axis vector components or axis name. If a vector, input must be length
3. A string can be specified for rotations about world axes (eg. `'+x'`,
`'-z'`, `'+y'`, etc.)
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
4x4 scaling matrix in row-major order. Will be the same array as `out`
if specified, if not, a new array will be allocated.

Notes
-----
* Vector `axis` is normalized before creating the matrix.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
R = np.zeros((4, 4,), dtype=dtype)
else:
dtype = np.dtype(out.dtype).type
R = out
R.fill(0.0)

try:
axis = VEC_AXES[axis] if isinstance(axis, str) else axis
except KeyError:
raise ValueError(
"Value of `axis` must be either '+x', '-x', '+y', '-x', '+z' or "
"'-z' or length 3 vector.")

axis = normalize(axis, dtype=dtype)
if np.count_nonzero(axis) == 0:
raise ValueError("Value for `axis` is zero-length.")

c = np.cos(angle, dtype=dtype)
s = np.sin(angle, dtype=dtype)

xs, ys, zs = axis * s
x2, y2, z2 = np.square(axis)  # type inferred by input
x, y, z = axis
cd = dtype(1.0) - c

R[0, 0] = x2 * cd + c
R[0, 1] = x * y * cd - zs
R[0, 2] = x * z * cd + ys

R[1, 0] = y * x * cd + zs
R[1, 1] = y2 * cd + c
R[1, 2] = y * z * cd - xs

R[2, 0] = x * z * cd - ys
R[2, 1] = y * z * cd + xs
R[2, 2] = z2 * cd + c

R[3, 3] = dtype(1.0)
R[:, :] += 0.0  # remove negative zeros

return R

[docs]def translationMatrix(t, out=None, dtype=None):
"""Create a translation matrix.

The resulting matrix is the same as generated by a `glTranslate` call.

Parameters
----------
t : ndarray, tuple, or list of float
Translation vector [tx, ty, tz].
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
4x4 translation matrix in row-major order. Will be the same array as
`out` if specified, if not, a new array will be allocated.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
T = np.identity(4, dtype=dtype)
else:
dtype = np.dtype(out.dtype).type
T = out
T.fill(0.0)
np.fill_diagonal(T, 1.0)

T[:3, 3] = np.asarray(t, dtype=dtype)

return T

[docs]def invertMatrix(m, out=None, dtype=None):
"""Invert a square matrix.

Parameters
----------
m : array_like
Square matrix to invert. Inputs can be 4x4, 3x3 or 2x2.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Matrix which is the inverse of `m`

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = out.dtype

m = np.asarray(m, dtype=dtype)  # input as array
toReturn = np.empty_like(m, dtype=dtype) if out is None else out
toReturn.fill(0.0)

if m.shape == (4, 4,):
# Special handling of 4x4 matrices, if affine and orthogonal
# (homogeneous), simply transpose the matrix rather than doing a full
# invert.
if isOrthogonal(m[:3, :3]) and isAffine(m):
rg = m[:3, :3]
toReturn[:3, :3] = rg.T
toReturn[:3, 3] = -m[:3, 3].dot(rg)
#toReturn[0, 3] = \
#    -(m[0, 0] * m[0, 3] + m[1, 0] * m[1, 3] + m[2, 0] * m[2, 3])
#toReturn[1, 3] = \
#    -(m[0, 1] * m[0, 3] + m[1, 1] * m[1, 3] + m[2, 1] * m[2, 3])
#toReturn[2, 3] = \
#    -(m[0, 2] * m[0, 3] + m[1, 2] * m[1, 3] + m[2, 2] * m[2, 3])
toReturn[3, 3] = 1.0
else:
toReturn[:, :] = np.linalg.inv(m)
elif m.shape[0] == m.shape[1]:  # square, other than 4x4
toReturn[:, :] = np.linalg.inv(m) if not isOrthogonal(m) else m.T
else:
toReturn[:, :] = np.linalg.inv(m)

[docs]def multMatrix(matrices, reverse=False, out=None, dtype=None):
"""Chain multiplication of two or more matrices.

Multiply a sequence of matrices together, reducing to a single product
matrix. For instance, specifying `matrices` the sequence of matrices (A, B,
C, D) will return the product (((AB)C)D). If `reverse=True`, the product
will be (A(B(CD))).

Alternatively, a 3D array can be specified to `matrices` as a stack, where
an index along axis 0 references a 2D slice storing matrix values. The
product of the matrices along the axis will be returned. This is a bit more
efficient than specifying separate matrices in a sequence, but the
difference is negligible when only a few matrices are being multiplied.

Parameters
----------
matrices : list, tuple or ndarray
Sequence or stack of matrices to multiply. All matrices must have the
same dimensions.
reverse : bool, optional
Multiply matrices right-to-left. This is useful when dealing with
transformation matrices, where the order of operations for transforms
will appear the same as the order the matrices are specified. Default is
'False'. When `True`, this function behaves similarly to
:func:`concatenate`.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Matrix product.

Notes
-----
* You may use `numpy.matmul` when dealing with only two matrices instead of
`multMatrix`.
* If a single matrix is specified, the returned product will have the same
values.

Examples
--------
Chain multiplication of SRT matrices::

translate = translationMatrix((0.035, 0, -0.5))
rotate = rotationMatrix(90.0, (0, 1, 0))
scale = scaleMatrix(2.0)

SRT = multMatrix((translate, rotate, scale))

Same as above, but matrices are in a 3x4x4 array::

matStack = np.array((translate, rotate, scale))

# or ...
# matStack = np.zeros((3, 4, 4))
# matStack[0, :, :] = translate
# matStack[1, :, :] = rotate
# matStack[2, :, :] = scale

SRT = multMatrix(matStack)

Using `reverse=True` allows you to specify transformation matrices in the
order which they will be applied::

SRT = multMatrix(np.array((scale, rotate, translate)), reverse=True)

"""
# convert matrix types
dtype = np.float64 if dtype is None else np.dtype(dtype).type
matrices = np.asarray(matrices, dtype=dtype)  # convert to array

matrices = np.atleast_3d(matrices)
prod = functools.reduce(
np.matmul, matrices[:] if not reverse else matrices[::-1])

if out is not None:
toReturn = out
toReturn[:, :] = prod
else:
toReturn = prod

[docs]def concatenate(matrices, out=None, dtype=None):
"""Concatenate matrix transformations.

Chain multiply matrices describing transform operations into a single matrix
product, that when applied, transforms points and vectors with each
operation in the order they're specified.

Parameters
----------
matrices : list or tuple
List of matrices to concatenate. All matrices must all have the same
size, usually 4x4 or 3x3.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Matrix product.

--------
* multMatrix : Chain multiplication of matrices.

Notes
-----
* This function should only be used for combining transformation matrices.
Use `multMatrix` for general matrix chain multiplication.

Examples
--------
Create an SRT (scale, rotate, and translate) matrix to convert model-space
coordinates to world-space::

S = scaleMatrix([2.0, 2.0, 2.0])  # scale model 2x
R = rotationMatrix(-90., [0., 0., -1])  # rotate -90 about -Z axis
T = translationMatrix([0., 0., -5.])  # translate point 5 units away

# product matrix when applied to points will scale, rotate and transform
# in that order.
SRT = concatenate([S, R, T])

# transform a point in model-space coordinates to world-space
pointModel = np.array([0., 1., 0., 1.])
pointWorld = np.matmul(SRT, pointModel.T)  # point in WCS
# ... or ...
pointWorld = matrixApply(SRT, pointModel)

Create a model-view matrix from a world-space pose represented by an
orientation (quaternion) and position (vector). The resulting matrix will
transform model-space coordinates to eye-space::

# eye pose as quaternion and vector
stimOri = quatFromAxisAngle([0., 0., -1.], -45.0)
stimPos = [0., 1.5, -5.]

# create model matrix
R = quatToMatrix(stimOri)
T = translationMatrix(stimPos)
M = concatenate(R, T)  # model matrix

# create a view matrix, can also be represented as 'pos' and 'ori'
eyePos = [0., 1.5, 0.]
eyeFwd = [0., 0., -1.]
eyeUp = [0., 1., 0.]
V = lookAt(eyePos, eyeFwd, eyeUp)  # from viewtools

# modelview matrix
MV = concatenate([M, V])

You can put the created matrix in the OpenGL matrix stack as shown below.
Note that the matrix must have a 32-bit floating-point data type and needs
to be loaded transposed since OpenGL takes matrices in column-major order::

GL.glMatrixMode(GL.GL_MODELVIEW)

# pyglet
MV = np.asarray(MV, dtype='float32')  # must be 32-bit float!
ptrMV = MV.ctypes.data_as(ctypes.POINTER(ctypes.c_float))

# PyOpenGL
MV = np.asarray(MV, dtype='float32')

Furthermore, you can convert a point from model-space to homogeneous
clip-space by concatenating the projection, view, and model matrices::

# compute projection matrix, functions here are from 'viewtools'
screenWidth = 0.52
screenAspect = w / h
scrDistance = 0.55
frustum = computeFrustum(screenWidth, screenAspect, scrDistance)
P = perspectiveProjectionMatrix(*frustum)

# multiply model-space points by MVP to convert them to clip-space
MVP = concatenate([M, V, P])
pointModel = np.array([0., 1., 0., 1.])
pointClipSpace = np.matmul(MVP, pointModel.T)

"""
return multMatrix(matrices, reverse=True, out=out, dtype=dtype)

[docs]def matrixFromEulerAngles(rx, ry, rz, degrees=True, out=None, dtype=None):
"""Construct a 4x4 rotation matrix from Euler angles.

Rotations are combined by first rotating about the X axis, then Y, and
finally Z.

Parameters
----------
rx, ry, rz : float
Rotation angles (pitch, yaw, and roll).
degrees : bool, optional
Rotation angles are specified in degrees. If `False`, they will be
assumed as radians. Default is `True`.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
4x4 rotation matrix.

Examples
--------
Demonstration of how a combination of axis-angle rotations is equivalent
to a single call of `matrixFromEulerAngles`::

m1 = matrixFromEulerAngles(90., 45., 135.))

# construct rotation matrix from 3 orthogonal rotations
rx = rotationMatrix(90., (1, 0, 0))  # x-axis
ry = rotationMatrix(45., (0, 1, 0))  # y-axis
rz = rotationMatrix(135., (0, 0, 1))  # z-axis
m2 = concatenate([rz, ry, rx])  # note the order

print(numpy.allclose(m1, m2))  # True

Not only does `matrixFromEulerAngles` require less code, it also is
considerably more efficient than constructing and multiplying multiple
matrices.

"""
# from https://www.j3d.org/matrix_faq/matrfaq_latest.html
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
toReturn = np.zeros((4, 4,), dtype=dtype)
else:
dtype = np.dtype(dtype).type
toReturn = out
toReturn.fill(0.0)

angles = np.asarray([rx, ry, rz], dtype=dtype)
if degrees:

a, c, e = np.cos(angles)
b, d, f = np.sin(angles)
bd = b * d

toReturn[0, 0] = c * e
toReturn[0, 1] = -c * f
toReturn[0, 2] = d
toReturn[1, 0] = bd * e + a * f
toReturn[1, 1] = -bd * f + a * e
toReturn[1, 2] = -b * c
toReturn[2, 0] = -ad * e + b * f
toReturn[2, 1] = ad * f + b * e
toReturn[2, 2] = a * c
toReturn[3, 3] = 1.0

[docs]def isOrthogonal(m):
"""Check if a square matrix is orthogonal.

If a matrix is orthogonal, its columns form an orthonormal basis and is
non-singular. An orthogonal matrix is invertible by simply taking the
transpose of the matrix.

Parameters
----------
m : array_like
Square matrix, either 2x2, 3x3 or 4x4.

Returns
-------
bool
`True` if the matrix is orthogonal.

"""
if not isinstance(m, (np.ndarray,)):
m = np.asarray(m)

assert 2 <= m.shape[0] <= 4   # 2x2 to 4x4
assert m.shape[0] == m.shape[1]  # must be square

dtype = np.dtype(m.dtype).type
return np.allclose(np.matmul(m.T, m, dtype=dtype),
np.identity(m.shape[0], dtype))

[docs]def isAffine(m):
"""Check if a 4x4 square matrix describes an affine transformation.

Parameters
----------
m : array_like
4x4 transformation matrix.

Returns
-------
bool
`True` if the matrix is affine.

"""
assert m.shape[0] == m.shape[1] == 4

if not isinstance(m, (np.ndarray,)):
m = np.asarray(m)

dtype = np.dtype(m.dtype).type
eps = np.finfo(dtype).eps

return np.all(m[3, :3] < eps) and (dtype(1.0) - m[3, 3]) < eps

[docs]def applyMatrix(m, points, out=None, dtype=None):
"""Apply a matrix over a 2D array of points.

This function behaves similarly to the following `Numpy` statement::

points[:, :] = points.dot(m.T)

Transformation matrices specified to `m` must have dimensions 4x4, 3x4, 3x3
or 2x2. With the exception of 4x4 matrices, input `points` must have the
same number of columns as the matrix has rows. 4x4 matrices can be used to
transform both Nx4 and Nx3 arrays.

Parameters
----------
m : array_like
Matrix with dimensions 2x2, 3x3, 3x4 or 4x4.
points : array_like
2D array of points/coordinates to transform. Each row should have length
appropriate for the matrix being used.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Transformed coordinates.

Notes
-----
* Input (`points`) and output (`out`) arrays cannot be the same instance for
this function.
* In the case of 4x4 input matrices, this function performs optimizations
based on whether the input matrix is affine, greatly improving performance
when working with Nx3 arrays.

Examples
--------
Construct a matrix and transform a point::

# identity 3x3 matrix for this example
M = [[1.0, 0.0, 0.0],
[0.0, 1.0, 0.0],
[0.0, 0.0, 1.0]]

pnt = [1.0, 0.0, 0.0]

pntNew = applyMatrix(M, pnt)

Construct an SRT matrix (scale, rotate, transform) and transform an array of
points::

S = scaleMatrix([5.0, 5.0, 5.0])  # scale 5x
R = rotationMatrix(180., [0., 0., -1])  # rotate 180 degrees
T = translationMatrix([0., 1.5, -3.])  # translate point up and away
M = concatenate([S, R, T])  # create transform matrix

# points to transform
points = np.array([[0., 1., 0., 1.], [-1., 0., 0., 1.]]) # [x, y, z, w]
newPoints = applyMatrix(M, points)  # apply the transformation

Convert CIE-XYZ colors to sRGB::

sRGBMatrix = [[3.2404542, -1.5371385, -0.4985314],
[-0.969266,  1.8760108,  0.041556 ],
[0.0556434, -0.2040259,  1.0572252]]

colorsRGB = applyMatrix(sRGBMatrix, colorsXYZ)

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(out.dtype).type

m = np.asarray(m, dtype=dtype)
points = np.asarray(points, dtype=dtype)

if out is None:
toReturn = np.zeros_like(points, dtype=dtype)
else:
if id(out) == id(points):
raise ValueError('Output array cannot be same as input.')
toReturn = out

pout, p = np.atleast_2d(toReturn, points)

if m.shape[0] == m.shape[1] == 4:  # 4x4 matrix
if pout.shape[1] == 3:  # Nx3
pout[:, :] = p.dot(m[:3, :3].T)
pout += m[:3, 3]
# find `rcpW` as suggested in OpenXR's xr_linear.h header
# reciprocal of `w` if the matrix is not orthonormal
if not isAffine(m):
rcpW = 1.0 / (m[3, 0] * p[:, 0] +
m[3, 1] * p[:, 1] +
m[3, 2] * p[:, 2] +
m[3, 3])
pout *= rcpW[:, np.newaxis]
elif pout.shape[1] == 4:  # Nx4
pout[:, :] = p.dot(m.T)
else:
raise ValueError(
'Input array dimensions invalid. Should be Nx3 or Nx4 when '
'input matrix is 4x4.')
elif m.shape[0] == 3 and m.shape[1] == 4:  # 3x4 matrix
if pout.shape[1] == 3:  # Nx3
pout[:, :] = p.dot(m[:3, :3].T)
pout += m[:3, 3]
else:
raise ValueError(
'Input array dimensions invalid. Should be Nx3 when input '
'matrix is 3x4.')
elif m.shape[0] == m.shape[1] == 3:  # 3x3 matrix, e.g colors
if pout.shape[1] == 3:  # Nx3
pout[:, :] = p.dot(m.T)
else:
raise ValueError(
'Input array dimensions invalid. Should be Nx3 when '
'input matrix is 3x3.')
elif m.shape[0] == m.shape[1] == pout.shape[1] == 2:  # 2x2 matrix
if pout.shape[1] == 2:  # Nx2
pout[:, :] = p.dot(m.T)
else:
raise ValueError(
'Input array dimensions invalid. Should be Nx2 when '
'input matrix is 2x2.')
else:
raise ValueError(
'Only a square matrix with dimensions 2, 3 or 4 can be used.')

[docs]def posOriToMatrix(pos, ori, out=None, dtype=None):
"""Convert a rigid body pose to a 4x4 transformation matrix.

A pose is represented by a position coordinate `pos` and orientation
quaternion `ori`.

Parameters
----------
pos : ndarray, tuple, or list of float
Position vector [x, y, z].
ori : tuple, list or ndarray of float
Orientation quaternion in form [x, y, z, w] where w is real and x, y, z
are imaginary components.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
4x4 transformation matrix.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
toReturn = np.zeros((4, 4,), dtype=dtype)
else:
dtype = np.dtype(dtype).type
toReturn = out

transMat = translationMatrix(pos, dtype=dtype)
rotMat = quatToMatrix(ori, dtype=dtype)

return np.matmul(transMat, rotMat, out=toReturn)

[docs]def transform(pos, ori, points, out=None, dtype=None):
"""Transform points using a position and orientation. Points are rotated
then translated.

Parameters
----------
pos : array_like
Position vector in form [x, y, z] or [x, y, z, 1].
ori : array_like
Orientation quaternion in form [x, y, z, w] where w is real and x, y, z
are imaginary components.
points : array_like
Point(s) [x, y, z] to transform.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Transformed points.

Examples
--------
Transform points by a position coordinate and orientation quaternion::

# rigid body pose
ori = quatFromAxisAngle([0., 0., -1.], 90.0, degrees=True)
pos = [0., 1.5, -3.]
# points to transform
points = np.array([[0., 1., 0., 1.], [-1., 0., 0., 1.]])  # [x, y, z, 1]
outPoints = np.zeros_like(points)  # output array
transform(pos, ori, points, out=outPoints)  # do the transformation

You can get the same results as the previous example using a matrix by doing
the following::

R = rotationMatrix(90., [0., 0., -1])
T = translationMatrix([0., 1.5, -3.])
M = concatenate([R, T])
applyMatrix(M, points, out=outPoints)

If you are defining transformations with quaternions and coordinates, you
can skip the costly matrix creation process by using `transform`.

Notes
-----
* In performance tests, `applyMatrix` is noticeably faster than `transform`
for very large arrays, however this is only true if you are applying the
same transformation to all points.
* If the input arrays for `points` or `pos` is Nx4, the last column is
ignored.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(dtype).type

pos = np.asarray(pos, dtype=dtype)
ori = np.asarray(ori, dtype=dtype)
points = np.asarray(points, dtype=dtype)

if out is None:
toReturn = np.zeros_like(points, dtype=dtype)
else:
if out.shape != points.shape:
raise ValueError(
"Array 'out' and 'points' do not have matching shapes.")

toReturn = out

pout, points, pos2d = np.atleast_2d(toReturn, points, pos)  # create 2d views

# apply rotation
applyQuat(ori, points, out=pout)

# apply translation
pout[:, 0] += pos2d[:, 0]
pout[:, 1] += pos2d[:, 1]
pout[:, 2] += pos2d[:, 2]

[docs]def scale(sf, points, out=None, dtype=None):
"""Scale points by a factor.

This is useful for converting points between units, and to stretch or
compress points along a given axis. Scaling can be uniform which the same
factor is applied along all axes, or anisotropic along specific axes.

Parameters
----------
sf : array_like or float
Scaling factor. If scalar, all points will be scaled uniformly by that
factor. If a vector, scaling will be anisotropic along an axis.
points : array_like
Point(s) [x, y, z] to scale.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Scaled points.

Examples
--------
Apply uniform scaling to points, here we scale to convert points in
centimeters to meters::

CM_TO_METERS = 1.0 / 100.0
pointsCM = [[1, 2, 3], [4, 5, 6], [-1, 1, 0]]
pointsM = scale(CM_TO_METERS, pointsCM)

Anisotropic scaling along the X and Y axis::

pointsM = scale((SCALE_FACTOR_X, SCALE_FACTOR_Y), pointsCM)

Scale only on the X axis::

pointsM = scale((SCALE_FACTOR_X,), pointsCM)

Apply scaling on the Z axis only::

pointsM = scale((1.0, 1.0, SCALE_FACTOR_Z), pointsCM)

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(dtype).type

points = np.asarray(points, dtype=dtype)
toReturn = np.zeros_like(points, dtype=dtype) if out is None else out
toReturn, points = np.atleast_2d(toReturn, points)  # create 2d views

# uniform scaling
if isinstance(sf, (float, int)):
toReturn[:, :] = points * sf
elif isinstance(sf, (list, tuple, np.ndarray)):  # anisotropic
sf = np.asarray(sf, dtype=dtype)
sfLen = len(sf)
if sfLen <= 3:
toReturn[:, :] = points
toReturn[:, :len(sf)] *= sf
else:
raise ValueError("Scale factor array must have length <= 3.")

[docs]def normalMatrix(modelMatrix, out=None, dtype=None):
"""Get the normal matrix from a model matrix.

Parameters
----------
modelMatrix : array_like
4x4 homogeneous model matrix.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Normal matrix.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(dtype).type

modelMatrix = np.asarray(modelMatrix, dtype=dtype)

toReturn = np.zeros((4, 4), dtype=dtype) if out is None else out
toReturn[:, :] = np.linalg.inv(modelMatrix).T

[docs]def forwardProject(objPos, modelView, proj, viewport=None, out=None, dtype=None):
"""Project a point in a scene to a window coordinate.

This function is similar to `gluProject` and can be used to find the window
coordinate which a point projects to.

Parameters
----------
objPos : array_like
Object coordinates (x, y, z). If an Nx3 array of coordinates is
specified, where each row contains a window coordinate this function
will return an array of projected coordinates with the same size.
modelView : array_like
4x4 combined model and view matrix for returned value to be object
coordinates. Specify only the view matrix for a coordinate in the scene.
proj : array_like
4x4 projection matrix used for rendering.
viewport : array_like
Viewport rectangle for the window [x, y, w, h]. If not specified, the
returned values will be in normalized device coordinates.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Normalized device or viewport coordinates [x, y, z] of the point. The
`z` component is similar to the depth buffer value for the object point.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(dtype).type

toReturn = np.zeros_like(objPos, dtype=dtype) if out is None else out
winCoord, objPos = np.atleast_2d(toReturn, objPos)

# transformation matrix
mvp = np.matmul(proj, modelView)

# must have `w` for this one
if objPos.shape[1] == 3:
temp = np.zeros((objPos.shape[1], 4), dtype=dtype)
temp[:, :3] = objPos
objPos = temp

# transform the points
objNorm = applyMatrix(mvp, objPos, dtype=dtype)

if viewport is not None:
# if we have a viewport, transform it
objNorm[:, :] += 1.0
winCoord[:, 0] = viewport[0] + viewport[2] * objNorm[:, 0]
winCoord[:, 1] = viewport[1] + viewport[3] * objNorm[:, 1]
winCoord[:, 2] = objNorm[:, 2]
winCoord[:, :] /= 2.0
else:
winCoord[:, :] = objNorm

[docs]def reverseProject(winPos, modelView, proj, viewport=None, out=None, dtype=None):
"""Unproject window coordinates into object or scene coordinates.

This function works like `gluUnProject` and can be used to find to an object
or scene coordinate at the point on-screen (mouse coordinate or pixel). The
coordinate can then be used to create a direction vector from the viewer's
eye location. Another use of this function is to convert depth buffer
samples to object or scene coordinates. This is the inverse operation of
:func:`forwardProject`.

Parameters
----------
winPos : array_like
Window coordinates (x, y, z). If `viewport` is not specified, these
should be normalized device coordinates. If an Nx3 array of coordinates
is specified, where each row contains a window coordinate this function
will return an array of unprojected coordinates with the same size.
Usually, you only need to specify the `x` and `y` coordinate, leaving
`z` as zero. However, you can specify `z` if sampling from a depth map
or buffer to convert a depth sample to an actual location.
modelView : array_like
4x4 combined model and view matrix for returned value to be object
coordinates. Specify only the view matrix for a coordinate in the scene.
proj : array_like
4x4 projection matrix used for rendering.
viewport : array_like
Viewport rectangle for the window [x, y, w, h]. Do not specify one if
`winPos` is in already in normalized device coordinates.
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Object or scene coordinates.

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(dtype).type

toReturn = np.zeros_like(winPos, dtype=dtype) if out is None else out
objCoord, winPos = np.atleast_2d(toReturn, winPos)

# get inverse of model and projection matrix
invMVP = np.linalg.inv(np.matmul(proj, modelView))

if viewport is not None:
# if we have a viewport, we need to transform to NDC first
objCoord[:, 0] = ((2 * winPos[:, 0] - viewport[0]) / viewport[2])
objCoord[:, 1] = ((2 * winPos[:, 1] - viewport[1]) / viewport[3])
objCoord[:, 2] = 2 * winPos[:, 2]
objCoord -= 1
objCoord[:, :] = applyMatrix(invMVP, objCoord, dtype=dtype)
else:
# already in NDC, just apply
objCoord[:, :] = applyMatrix(invMVP, winPos, dtype=dtype)

# ------------------------------------------------------------------------------
# Misc. Math Functions
#

[docs]def zeroFix(a, inplace=False, threshold=None):
"""Fix zeros in an array.

This function truncates very small numbers in an array to zero and removes
any negative zeros.

Parameters
----------
a : ndarray
Input array, must be a Numpy array.
inplace : bool
Fix an array inplace. If `True`, the input array will be modified,
otherwise a new array will be returned with same `dtype` and shape with
the fixed values.
threshold : float or None
Threshold for truncation. If `None`, the machine epsilon value for the
input array `dtype` will be used. You can specify a custom threshold as
a float.

Returns
-------
ndarray
Output array with zeros fixed.

"""
toReturn = np.copy(a) if not inplace else a
toReturn += 0.0  # remove negative zeros
threshold = np.finfo(a.dtype).eps if threshold is None else float(threshold)
toReturn[np.abs(toReturn) < threshold] = 0.0  # make zero

[docs]def lensCorrection(xys, coefK=(1.0,), distCenter=(0., 0.), out=None,
dtype=None):
"""Lens correction (or distortion) using the division model with even
polynomial terms.

Calculate new vertex positions or texture coordinates to apply radial
warping, such as 'pincushion' and 'barrel' distortion. This is to compensate
for optical distortion introduced by lenses placed in the optical path of
the viewer and the display (such as in an HMD).

See references[1]_ for implementation details.

Parameters
----------
xys : array_like
Nx2 list of vertex positions or texture coordinates to distort. Works
correctly only if input values range between -1.0 and 1.0.
coefK : array_like or float
Distortion coefficients K_n. Specifying multiple values will add more
polynomial terms to the distortion formula. Positive values will produce
'barrel' distortion, whereas negative will produce 'pincushion'
distortion. In most cases, two or three coefficients are adequate,
depending on the degree of distortion.
distCenter : array_like, optional
X and Y coordinate of the distortion center (eg. (0.2, -0.4)).
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Array of distorted vertices.

Notes
-----
* At this time tangential distortion (i.e. due to a slant in the display)
cannot be corrected for.

References
----------
.. [1] Fitzgibbon, W. (2001). Simultaneous linear estimation of multiple
view geometry and lens distortion. Proceedings of the 2001 IEEE Computer
Society Conference on Computer Vision and Pattern Recognition (CVPR).
IEEE.

Examples
--------
Creating a lens correction mesh with barrel distortion (eg. for HMDs)::

vertices, textureCoords, normals, faces = gltools.createMeshGrid(
subdiv=11, tessMode='center')

# recompute vertex positions
vertices[:, :2] = mt.lensCorrection(vertices[:, :2], coefK=(5., 5.))

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(dtype).type

if isinstance(coefK, (float, int,)):
coefK = (coefK,)

xys = np.asarray(xys, dtype=dtype)
coefK = np.asarray(coefK, dtype=dtype)

d_minus_c = xys - np.asarray(distCenter, dtype=dtype)
r = np.power(length(d_minus_c, dtype=dtype)[:, np.newaxis],
np.arange(len(coefK), dtype=dtype) * 2. + 2.)

toReturn = np.zeros_like(xys, dtype=dtype) if out is None else out

denom = dtype(1.0) + dot(coefK, r, dtype=dtype)
toReturn[:, :] = xys + (d_minus_c / denom[:, np.newaxis])

[docs]def lensCorrectionSpherical(xys, coefK=1.0, aspect=1.0, out=None, dtype=None):
"""Simple lens correction.

Lens correction for a spherical lenses with distortion centered at the
middle of the display. See references[1]_ for implementation details.

Parameters
----------
xys : array_like
Nx2 list of vertex positions or texture coordinates to distort. Assumes
the output will be rendered to normalized device coordinates where
points range from -1.0 to 1.0.
coefK : float
Distortion coefficient. Use positive numbers for pincushion distortion
and negative for barrel distortion.
aspect : float
Aspect ratio of the target window or buffer (width / height).
out : ndarray, optional
Optional output array. Must be same `shape` and `dtype` as the expected
output if `out` was not specified.
dtype : dtype or str, optional
Data type for computations can either be 'float32' or 'float64'. If
`out` is specified, the data type of `out` is used and this argument is
ignored. If `out` is not provided, 'float64' is used by default.

Returns
-------
ndarray
Array of distorted vertices.

References
----------
.. [1] Lens Distortion White Paper, Andersson Technologies LLC,
www.ssontech.com/content/lensalg.html (obtained 07/28/2020)

Examples
--------
Creating a lens correction mesh with barrel distortion (eg. for HMDs)::

vertices, textureCoords, normals, faces = gltools.createMeshGrid(
subdiv=11, tessMode='center')

# recompute vertex positions
vertices[:, :2] = mt.lensCorrection2(vertices[:, :2], coefK=2.0)

"""
if out is None:
dtype = np.float64 if dtype is None else np.dtype(dtype).type
else:
dtype = np.dtype(dtype).type

toReturn = np.empty_like(xys, dtype=dtype) if out is None else out

xys = np.asarray(xys, dtype=dtype)
toReturn[:, 0] = u = xys[:, 0]
toReturn[:, 1] = v = xys[:, 1]
coefKCubed = np.power(coefK, 3, dtype=dtype)

r2 = aspect * aspect * u * u + v * v
r2sqr = np.sqrt(r2, dtype=dtype)
f = 1. + r2 * (coefK + coefKCubed * r2sqr)

toReturn[:, 0] *= f
toReturn[:, 1] *= f

class infrange():
"""
Similar to base Python `range`, but allowing the step to be a float or even
0, useful for specifying ranges for logical comparisons.
"""
def __init__(self, min, max, step=0):
self.min = min
self.max = max
self.step = step

@property
def range(self):
return abs(self.max-self.min)

def __lt__(self, other):
return other > self.max

def __le__(self, other):
return other > self.min

def __gt__(self, other):
return self.min > other

def __ge__(self, other):
return self.max > other

def __contains__(self, item):
if self.step == 0:
return self.min < item < self.max
else:
return item in np.linspace(self.min, self.max, int(self.range/self.step)+1)

def __eq__(self, item):
if isinstance(item, self.__class__):
return all((
self.min == item.min,
self.max == item.max,
self.step == item.step
))
return item in self

return self.__class__(self.min+other, self.max+other, self.step)

def __sub__(self, other):
return self.__class__(self.min - other, self.max - other, self.step)

def __mul__(self, other):
return self.__class__(self.min * other, self.max * other, self.step * other)

def __truedic__(self, other):
return self.__class__(self.min / other, self.max / other, self.step / other)

if __name__ == "__main__":
pass
```