# Source code for clifford.tools

```
"""
.. currentmodule:: clifford.tools
========================================
tools (:mod:`clifford.tools`)
========================================
Algorithms and tools of various kinds.
Tools for specific ga's
---------------------------------
.. autosummary::
:toctree: generated/
g3
g3c
Determining Rotors From Frame Pairs or Orthogonal Matrices
-----------------------------------------------------------
Given two frames that are related by a orthogonal transform, we seek a rotor
which enacts the transform. Details of the mathematics and psuedo-code used the
create the algorithms below can be found at Allan Cortzen's website.
http://ctz.dk/geometric-algebra/frames-to-versor-algorithm/
There are also some helper functions which can be used to translate matrices
into GA frames, so an orthogonal (or complex unitary ) matrix can be directly
translated into a Verser.
.. autosummary::
:toctree: generated/
orthoFrames2Verser
orthoMat2Verser
mat2Frame
"""
from __future__ import absolute_import, division
from __future__ import print_function, unicode_literals
from functools import reduce
from math import sqrt
from numpy import eye, array, sign, zeros, sin,arccos
import itertools
from .. import Cl, gp, Frame
from .. import eps as global_eps
from warnings import warn
def omoh(A, B):
'''
Determines homogenzation scaling for two Frames related by a Rotor
This is used as part of the frames2Versor algorithm, when the
frames are given in CGA. It is requried because the model assumes,
`B = R*A*~R`
but if data is given in the original space, only
`lambda*B' == homo(B)`
is observable. We need to determine lambda before the Cartan-based
algorithm can be used. The name of this function is inverses of
`homo`, which is the method used to homogenize
Parameters
--------------
A : list of vectors, or clifford.Frame
the set of vectors before the transform
B : list of vectors, or clifford.Frame
the set of vectors after the transform, and homogenzation.
ie B=(B/B|einf)
Returns
---------
out : list of floats
weights on `B`, which produce inhomogenous versions of `B`. If
you multiply the input `B` by `lam`, it will fulfill `B = R*A*~R`
Examples
----------
lam = ohom(A,B):
B_ohom = Frame([B[k]*lam[k] for k in range(len(B)])
'''
if len(A) != len(B) or len(A) < 3:
raise ValueError('input must be >=3 long and len(a)==len(b)')
idx = range(len(A))
lam = zeros(len(A))
for i in idx:
j, k = [p for p in idx if p != i][:2]
lam[i] = \
float((A[i] * A[j])(0) * (A[i] * A[k])(0) * (B[j] * B[k])(0)) / \
float((B[i] * B[j])(0) * (B[i] * B[k])(0) * (A[j] * A[k])(0))
lam[i] = sqrt(float(lam[i]))
return lam
[docs]def mat2Frame(A, layout=None, is_complex=None):
'''
Translates a (possibly complex) matrix into a real vector frame
The rows and columns are interpreted as follows
* M,N = shape(A)
* M = dimension of space
* N = number of vectors
If A is complex M and N are doubled.
Parameters
------------
A : ndarray
MxN matrix representing vectors
'''
# TODO: could simplify this by just implementing the real case and then
# recursively calling this for A.real, and A.imag, then combine results
# M = dimension of space
# N = number of vectors
M, N = A.shape
if is_complex is None:
if A.dtype == 'complex':
is_complex = True
else:
is_complex = False
if is_complex:
N = N * 2
M = M * 2
if layout is None:
layout, blades = Cl(M)
e_ = layout.basis_vectors
e_ = [e_['e%i' % k] for k in range(layout.firstIdx, layout.firstIdx + M)]
a = [0 ^ e_[0]] * N
if not is_complex:
for n in range(N):
for m in range(M):
a[n] = (a[n]) + ((A[m, n]) ^ e_[m])
else:
for n in range(N // 2):
n_ = 2 * n
for m in range(M // 2):
m_ = 2 * m
a[n_] = (a[n_]) + ((A[m, n].real) ^ e_[m_]) \
+ ((A[m, n].imag) ^ e_[m_ + 1])
a[n_ + 1] = (a[n_ + 1]) + ((-A[m, n].imag) ^ e_[m_]) \
+ ((A[m, n].real) ^ e_[m_ + 1])
return a, layout
def frame2Mat(B, A=None, is_complex=None):
if is_complex is not None:
raise NotImplementedError()
if A is None:
# assume we have orthonormal initial frame
A = B[0].layout.basis_vectors_lst
# you need float() due to bug in clifford
M = [float(b | a) for b in B for a in A]
M = array(M).reshape(len(B), len(B))
def orthoFrames2Verser_dist(A, B, eps=None):
'''
Determines verser for two frames related by an orthogonal transform
The frames themselves do not have to be othorgonal.
Based on [1,2]. This works in Euclidean spaces and, under special
circumstances in other signatures. see [1] for limitaions/details
[1] http://ctz.dk/geometric-algebra/frames-to-versor-algorithm/
[2] Reconstructing Rotations and Rigid Body Motions from Exact Point
Correspondences Through Reflections, Daniel Fontijne and Leo Dorst
'''
# TODO: should we test to see if A and B are related by rotation?
# TODO: implement reflect/rotate based on distance (as in[1])
# keep copy of original frames
A = A[:]
B = B[:]
if len(A) != len(B):
raise ValueError('len(A)!=len(B)')
if eps is None:
eps = global_eps()
# store each reflector in a list
r_list = []
# find the vector pair with the largest distance
dist = [abs((a - b) ** 2) for a, b in zip(A, B)]
k = dist.index(max(dist))
while dist[k] >= eps:
r = (A[k] - B[k]) / abs(A[k] - B[k]) # determine reflector
r_list.append(r) # append to our list
A = A[1:] # remove current vector pair
B = B[1:]
if len(A) == 0:
break
# reflect remaining vectors
for j in range(len(A)):
A[j] = -r * A[j] * r
# find the next pair based on current distance
dist = [abs((a - b) ** 2) for a, b in zip(A, B)]
k = dist.index(max(dist))
# print(str(len(r_list)) + ' reflections found')
R = reduce(gp, r_list[::-1])
return R, r_list
[docs]def orthoFrames2Verser(B, A=None, delta=1e-3, eps=None, det=None,
remove_scaling=False):
'''
Determines verser for two frames related by an orthogonal transform
Based on [1,2]. This works in Euclidean spaces and, under special
circumstances in other signatures. see [1] for limitaions/details
Parameters
-----------
B : list of vectors, or clifford.Frame
the set of vectors after the transform, and homogenzation.
ie B=(B/B|einf)
A : list of vectors, or clifford.Frame
the set of vectors before the transform. If `None` we assume A is
the basis given B.layout.basis_vectors
delta : float
Tolerance for reflection/rotation determination. If the normalized
distance between A[i] and B[i] is larger than delta, we use
reflection, otherwise use rotation.
eps: float
Tolerance on spinor determination. if pseudoscalar of A differs
in magnitude from pseudoscalar of B by eps, then we have spinor.
If `None`, use the `clifford.eps()` global eps.
det : [+1,-1,None]
The sign of the determinant of the versor, if known. If it is
known a-priori that the versor is a rotation vs a reflection, this
fact might be needed to correctly append an additional reflection
which leaves transformed points invariant. See 4.6.3 [2].
remove_scaling : Bool
Remove the effects of homogenzation from frame B. This is needed
if you are working in CGA, but the input data is given in the
original space. See `omoh` method for more. See 4.6.2 of [2]
Returns
---------
R : clifford.Multivector
the Versor.
rs : list of clifford.Multivectors
ordered list of found reflectors/rotors.
References
------------
[1] http://ctz.dk/geometric-algebra/frames-to-versor-algorithm/
[2] Reconstructing Rotations and Rigid Body Motions from Exact Point
Correspondences Through Reflections, Daniel Fontijne and Leo Dorst
'''
# Checking and Setup
if A is None:
# assume we have orthonormal initial frame
bv = B[0].layout.basis_vectors
A = [bv[k] for k in sorted(bv.keys())]
# make copy of original frames, so we can rotate A
A = A[:]
B = B[:]
if len(A) != len(B):
raise ValueError('len(A)!=len(B)')
if eps is None:
eps = global_eps()
# Determine if we have a spinor
spinor = False
# store peudoscalar of frame B, in case known det (see end)
try:
B = Frame(B)
B_En = B.En
except:
pass
N = len(A)
# Determine and remove scaling factors caused by homogenization
if remove_scaling == True:
lam = omoh(A, B)
B = Frame([B[k] * lam[k] for k in range(N)])
try:
# compute ratio of volumes for each frame. take Nth root
A = Frame(A[:])
B = Frame(B[:])
alpha = abs(B.En / A.En) ** (1. / N)
if abs(alpha - 1) > eps:
spinor = True
# we have a spinor, remove the scaling (add it back in at the end)
B = [b / alpha for b in B]
except:
# probably A and B are not pure vector correspondence
# whatever, it might still work
pass
# Find the Verser
# store each reflector/rotor in a list, make full verser at the
# end of the loop
r_list = []
for k in range(N):
a, b = A[0], B[0]
r = a - b # determine reflector
if abs(b ** 2) > eps:
d = abs(r ** 2) / abs(b ** 2) # conditional rotation tolerance
else:
# probably b is a null vector, make our best guess for tol!
d = abs(r ** 2)
if d >= delta:
# reflection part
r_list.append(r)
A = A[1:] # remove current vector pair
B = B[1:]
for j in range(len(A)):
A[j] = -r * A[j] * r.inv()
else:
# rotation part
# if k==N: # see paper for explaination
# break
R = b * (a + b)
if abs(R) > eps: # abs(R) can be <eps in null space
r_list.append(R) # append to our list
A = A[1:] # remove current vector pair
B = B[1:]
for j in range(len(A)):
A[j] = R * A[j] * R.inv()
R = reduce(gp, r_list[::-1])
# if det is known a priori check to see if it's correct, if not add
# an extra reflection which leaves all points in B invarianct
if det is not None:
I = R.pseudoScalar
our_det = (R * I * ~R * I.inv())(0)
if sign(float(our_det)) != det:
R = B_En.dual() * R
if abs(R) < eps:
warn('abs(R)<eps. likely to be inaccurate')
R = R / abs(R)
if spinor:
R = R * sqrt(alpha)
return R, r_list
[docs]def orthoMat2Verser(A, eps=None, layout=None, is_complex=None):
'''
Translates an orthogonal (or unitary) matrix to a Verser
`A` is interpreted as the frame produced by transforming a
orthonormal frame by an orthogonal transform. Given this relation,
this function will find the verser which enacts this transform.
Parameters
------------
'''
B, layout = mat2Frame(A, layout=layout, is_complex=is_complex)
N = len(B)
# if (A.dot(A.conj().T) -eye(N/2)).max()>eps:
# warn('A doesnt appear to be a rotation. ')
A, layout = mat2Frame(eye(N), layout=layout, is_complex=False)
return orthoFrames2Verser(A=A, B=B, eps=eps)
def rotor_decomp(V, x):
'''
Rotor decomposition of rotor V
Given a rotor V, and a vector x, this will decompose V into a
series of two rotations, U and H, where U leaves x
invariant and H contains x.
Limited to 4D for now
Parameters
---------------
V : clifford.MultiVector
rotor
x : clifford.MultiVector
vector
Returns
-------
H : clifford.Multivector
rotor which contains x
U : clifford.Multivector
rotor which leaves x invariant
References
----------------
[1] : Space Time Algebra, D. Hestenes. AppendixB, Theroem 4
'''
H2 = V * x * ~V * x.inv() # inv needed to handle signatures
H = (1 + H2) / sqrt(abs(float(2 * (1 + H2(0)))))
U = H * x * V * x.inv()
return H, U
def sinc(x):
return sin(x)/x
def log_rotor(V):
'''
Logarithm of a simple rotor
'''
if (V(2)**2).grades() !=[0]:
print(V)
#raise ValueError('Bivector is not a Blade.')
if abs(V(2))<global_eps():
return log(float(V(0)))
# numpy's trig correctly chooses hyperbolic or not with Complex args
theta = arccos(complex(V(0)))
return V(2)/sinc(theta).real
```