'''
Routines for working with vectors
These routines can be used with vectors, as well as with matrices containing a vector in each row.
'''
'''
author : Thomas Haslwanter
date : Oct-2020
'''
import numpy as np
# The following construct is required since I want to run the module as a script
# inside the skinematics-directory
import os
import sys
file_dir = os.path.dirname(__file__)
if file_dir not in sys.path:
sys.path.insert(0, file_dir)
import quat
# For deprecation warnings
#import deprecation
import warnings
#warnings.simplefilter('always', DeprecationWarning)
[docs]def normalize(v):
''' Normalization of a given vector (with image)
Parameters
----------
v : array (N,) or (M,N)
input vector
Returns
-------
v_normalized : array (N,) or (M,N)
normalized input vector
.. image:: ../docs/Images/vector_normalize.png
:scale: 33%
Example
-------
>>> skinematics.vector.normalize([3, 0, 0])
array([[ 1., 0., 0.]])
>>> v = [[np.pi, 2, 3], [2, 0, 0]]
>>> skinematics.vector.normalize(v)
array([[ 0.6569322 , 0.41821602, 0.62732404],
[ 1. , 0. , 0. ]])
Notes
-----
.. math::
\\vec{n} = \\frac{\\vec{v}}{|\\vec{v}|}
'''
from numpy.linalg import norm
# Distinguish between a vector and a matrix
if np.array(v).ndim == 1:
vectorFlag = True
else:
vectorFlag = False
# The 'atleast_2d' ensures that the program works on matrices.
# The 'double' avoids trouble 2 lines down, if v is integer.
# And the 'copy' ensures that the input is not modified in the calling program.
v = np.double(np.atleast_2d(v)).copy()
length = norm(v,axis=1)
v[length!=0] = (v[length!=0].T/length[length!=0]).T
if vectorFlag:
v = v.ravel()
return v
[docs]def angle(v1,v2):
'''Angle between two vectors
Parameters
----------
v1 : array (N,) or (M,N)
vector 1
v2 : array (N,) or (M,N)
vector 2
Returns
-------
angle : double or array(M,)
angle between v1 and v2
.. image:: ../docs/Images/vector_angle.png
:scale: 33%
Example
-------
>>> v1 = np.array([[1,2,3],
>>> [4,5,6]])
>>> v2 = np.array([[1,0,0],
>>> [0,1,0]])
>>> skinematics.vector.angle(v1,v2)
array([ 1.30024656, 0.96453036])
Notes
-----
.. math::
\\alpha =arccos(\\frac{\\vec{v_1} \\cdot \\vec{v_2}}{| \\vec{v_1} |
\\cdot | \\vec{v_2}|})
'''
# make sure lists are handled correctly
v1 = np.array(v1)
v2 = np.array(v2)
if v1.ndim < v2.ndim:
v1, v2 = v2, v1
n1 = normalize(v1)
n2 = normalize(v2)
if v2.ndim == 1:
angle = np.arccos(n1.dot(n2))
else:
angle = np.arccos(list(map(np.dot, n1, n2)))
return angle
[docs]def project(v1,v2, projection_type='1D'):
'''Project one vector onto another, or into the plane perpendicular to that vector.
Parameters
----------
v1 : array (N,) or (M,N)
projected vector
v2 : array (N,) or (M,N):
target vector
projection_type : scalar
Has to be one of the following:
- 1D ... projection onto a vector (Default)
- 2D ... projection into the plane perpendicular to that vector
Returns
-------
v_projected : array (N,) or (M,N)
projection of v1 onto v2
.. image:: ../docs/Images/vector_project.png
:scale: 33%
Example
-------
>>> v1 = np.array([[1,2,3],
>>> [4,5,6]])
>>> v2 = np.array([[1,0,0],
>>> [0,1,0]])
>>> skinematics.vector.project(v1,v2)
array([[ 1., 0., 0.],
[ 0., 5., 0.]])
Notes
-----
.. math::
\\vec{n} = \\frac{ \\vec{a} }{| \\vec{a} |}
\\vec{v}_{proj} = \\vec{n} (\\vec{v} \\cdot \\vec{n})
\\mathbf{c}^{image} = \mathbf{R} \cdot \mathbf{c}^{space} + \mathbf{p}_{CS}
*Note* that the orientation of the 2D projection is not uniquely defined.
It is chosen here such that the y-axis points up, and one is "looking down"
rather than "looking up".
'''
v1 = np.atleast_2d(v1)
v2 = np.atleast_2d(v2)
e2 = normalize(v2)
if projection_type == '1D':
if e2.ndim == 1 or e2.shape[0]==1:
return (e2 * list(map(np.dot, v1, e2))).ravel()
else:
return (e2.T * list(map(np.dot, v1, e2))).T
elif projection_type == '2D':
if e2.shape[0] > 1:
raise ValueError('2D projections only implemented for fixed projection-plane!')
x,y,z = e2[0]
projection_matrix = np.array(
[[-y, -x*z, x],
[ x, -y*z, y],
[ 0, x**2+y**2, z]])
if z > 0: # choose a downward-pointing look for the projection
projection_matrix = projection_matrix * np.r_[-1, 1, -1]
projected = v1 @ projection_matrix
projected = projected[:,:2]
if e2.ndim == 1 or e2.shape[0]==1:
return projected.ravel()
else:
return projected
else:
raise ValueError('{0} not allowed as projection_type in vector.project!'.format(projection_type))
[docs]def GramSchmidt(p0,p1,p2):
'''Gram-Schmidt orthogonalization
Parameters
----------
p0 : array (3,) or (M,3)
coordinates of Point 1
p1 : array (3,) or (M,3)
coordinates of Point 2
p2 : array (3,) or (M,3)
coordinates of Point 3
Returns
-------
Rmat : array (9,) or (M,9)
flattened rotation matrix
.. image:: ../docs/Images/GramSchmidt.jpg
:scale: 25%
Example
-------
>>> P0 = np.array([[0, 0, 0], [1,2,3]])
>>> P1 = np.array([[1, 0, 0], [4,1,0]])
>>> P2 = np.array([[1, 1, 0], [9,-1,1]])
>>> GramSchmidt(P0,P1,P2)
array([[ 1. , 0. , 0. , 0. , 1. ,
0. , 0. , 0. , 1. ],
[ 0.6882472 , -0.22941573, -0.6882472 , 0.62872867, -0.28470732,
0.72363112, -0.36196138, -0.93075784, -0.05170877]])
Notes
-----
The flattened rotation matrix corresponds to
.. math::
\\mathbf{R} = [ \\vec{e}_1 \\, \\vec{e}_2 \\, \\vec{e}_3 ]
'''
# If inputs are lists, convert them to arrays:
p0 = np.array(p0)
p1 = np.array(p1)
p2 = np.array(p2)
v1 = np.atleast_2d(p1-p0)
v2 = np.atleast_2d(p2-p0)
ex = normalize(v1)
ey = normalize(v2- project(v2,ex))
ez = np.cross(ex,ey)
return np.hstack((ex,ey,ez))
[docs]def plane_orientation(p0, p1, p2):
'''The vector perpendicular to the plane defined by three points.
Parameters
----------
p0 : array (3,) or (M,3)
coordinates of Point 0
p1 : array (3,) or (M,3)
coordinates of Point 1
p2 : array (3,) or (M,3)
coordinates of Point 2
Returns
-------
n : array (3,) or (M,3)
vector perpendicular to the plane
.. image:: ../docs/Images/vector_plane_orientation.png
:scale: 33%
Example
-------
>>> P0 = np.array([[0, 0, 0], [1,2,3]])
>>> P1 = np.array([[1, 0, 0], [4,1,0]])
>>> P2 = np.array([[1, 1, 0], [9,-1,1]])
>>> plane_orientation(P0,P1,P2)
array([[ 0. , 0. , 1. ],
[-0.36196138, -0.93075784, -0.05170877]])
Notes
-----
.. math::
\\vec{n} = \\frac{ \\vec{a} \\times \\vec{b}} {| \\vec{a} \\times \\vec{b}|}
'''
# If inputs are lists, convert them to arrays:
p0 = np.array(p0)
p1 = np.array(p1)
p2 = np.array(p2)
v01 = p1-p0
v02 = p2-p0
n = np.cross(v01,v02)
return normalize(n)
#@deprecation.deprecated(deprecated_in="1.7", removed_in="1.9",
#current_version=__version__,
#details="Use the ``q_shortest_rotation`` function instead")
[docs]def q_shortest_rotation(v1,v2):
'''Quaternion indicating the shortest rotation from one vector into another.
You can read "qrotate" as either "quaternion rotate" or as "quick
rotate".
Parameters
----------
v1 : ndarray (3,)
first vector
v2 : ndarray (3,)
second vector
Returns
-------
q : ndarray (3,)
quaternion rotating v1 into v2
.. image:: ../docs/Images/vector_q_shortest_rotation.png
:scale: 33%
Example
-------
>>> v1 = np.r_[1,0,0]
>>> v2 = np.r_[1,1,0]
>>> q = qrotate(v1, v2)
>>> print(q)
[ 0. 0. 0.38268343]
'''
# calculate the direction
n = normalize(np.cross(v1,v2))
# make sure vectors are handled correctly
n = np.atleast_2d(n)
# handle 0-quaternions
nanindex = np.isnan(n[:,0])
n[nanindex,:] = 0
# find the angle, and calculate the quaternion
angle12 = angle(v1,v2)
q = (n.T*np.sin(angle12/2.)).T
# if you are working with vectors, only return a vector
if q.shape[0]==1:
q = q.flatten()
return q
[docs]def rotate_vector(vector, q):
'''
Rotates a vector, according to the given quaternions.
Note that a single vector can be rotated into many orientations;
or a row of vectors can all be rotated by a single quaternion.
Parameters
----------
vector : array, shape (3,) or (N,3)
vector(s) to be rotated.
q : array_like, shape ([3,4],) or (N,[3,4])
quaternions or quaternion vectors.
Returns
-------
rotated : array, shape (3,) or (N,3)
rotated vector(s)
.. image:: ../docs/Images/vector_rotate_vector.png
:scale: 33%
Notes
-----
.. math::
q \\circ \\left( {\\vec x \\cdot \\vec I} \\right) \\circ {q^{ - 1}} = \\left( {{\\bf{R}} \\cdot \\vec x} \\right) \\cdot \\vec I
More info under
http://en.wikipedia.org/wiki/Quaternion
Examples
--------
>>> mymat = eye(3)
>>> myVector = r_[1,0,0]
>>> quats = array([[0,0, sin(0.1)],[0, sin(0.2), 0]])
>>> quat.rotate_vector(myVector, quats)
array([[ 0.98006658, 0.19866933, 0. ],
[ 0.92106099, 0. , -0.38941834]])
>>> quat.rotate_vector(mymat, [0, 0, sin(0.1)])
array([[ 0.98006658, 0.19866933, 0. ],
[-0.19866933, 0.98006658, 0. ],
[ 0. , 0. , 1. ]])
'''
vector = np.atleast_2d(vector)
qvector = np.hstack((np.zeros((vector.shape[0],1)), vector))
vRotated = quat.q_mult(q, quat.q_mult(qvector, quat.q_inv(q)))
vRotated = vRotated[:,1:]
if min(vRotated.shape)==1:
vRotated = vRotated.ravel()
return vRotated
[docs]def target2orient(target, orient_type='quat'):
''' Converts a target vector into a corresponding orientation.
Useful for targeting devices, such as eyes, cameras, or missile trackers.
Based on the assumption, that in the reference orientation, the targeting
device points forward.
Parameters
----------
target : array (3,) or (N,3)
Input vector
orient_type : string
Has to be one the following:
- Fick ... Rz * Ry
- nautical ... same as "Fick"
- Helmholtz ... Ry * Rz
- quat ... quaternion
Returns
-------
orientation : array (3,) or (N,3)
Corresponding orientation
For rotation matrices, same sequence as the matrices [deg].
For quaternions, the quaternion vector.
Note that the last column of the sequence angles, and the first column
of the quaterion, will always be zero, because a rotation about
the line-of-sight has no effect.
Example
-------
>>> a = [3,3,0]
>>> b = [5., 0, 5]
>>> skinematics.vector.target2orient(a)
[ 0. 0. 0.38268343]
>>> skinematics.vector.target2orient([a,b])
[[ 0. 0. 0.38268343]
[ 0. -0.38268343 0. ]]
>>> skinematics.vector.target2orient(a, orient_type='nautical')
[ 45. -0. 0.]
'''
if orient_type == 'quat':
orientation = q_shortest_rotation([1,0,0], target)
elif orient_type =='Fick' or orient_type =='nautical':
n = np.atleast_2d(normalize(target))
theta = np.arctan2(n[:,1], n[:,0])
phi = -np.arcsin(n[:,2])
orientation = np.column_stack((theta, phi, np.zeros_like(theta)))
orientation = np.rad2deg(orientation)
elif orient_type == 'Helmholtz':
n = np.atleast_2d(normalize(target))
phi = -np.arctan2(n[:,2], n[:,0])
theta = np.arcsin(n[:,1])
orientation = np.column_stack((phi, theta, np.zeros_like(theta)))
orientation = np.rad2deg(orientation)
else:
raise ValueError('Input parameter {0} not known'.format(orientation))
# For vector input, return a vector:
if orientation.shape[0] == 1:
orientation = orientation.ravel()
return orientation
if __name__=='__main__':
a = [3,3,0]
b = [0, 1, 0]
normalized = normalize(a)
print(normalized)
normalized = normalize(np.cross(a,a))
print(normalized)
