CODEWITHSUNDEEP
arraysnumpyvectororthogonalcross-product
Physicist
Physicist

Reputation: 3048

generating two orthogonal vectors that are orthogonal to a particular direction

What is the simplest and most efficient ways in numpy to generate two orthonormal vectors a and b such that the cross product of the two vectors equals another unit vector k, which is already known?

I know there are infinitely many such pairs, and it doesn't matter to me which pairs I get as long as the conditions axb=k and a.b=0 are satisfied.

Upvotes: 18

Views: 30278

Answers (4)

Mark
Mark

Reputation: 87

Another solution in n-dimensional space might be:

import numpy as np
vector = np.array([1,2,3,4])
vector = vector[np.newaxis,:]
orthogonal_vertices = np.linalg.svd(vector)[-1]

Upvotes: 0

behzad.nouri
behzad.nouri

Reputation: 77951

The Gram-Schmidt procedure will do exactly this. For example:

>>> k  # an arbitrary unit vector k is not array. k is must be numpy class. np.array
np.array([ 0.59500984,  0.09655469, -0.79789754])

To obtain the 1st one:

>>> x = np.random.randn(3)  # take a random vector
>>> x -= x.dot(k) * k       # make it orthogonal to k
>>> x /= np.linalg.norm(x)  # normalize it

To obtain the 2nd one:

>>> y = np.cross(k, x)      # cross product with k

and to verify:

>>> np.linalg.norm(x), np.linalg.norm(y)
(1.0, 1.0)
>>> np.cross(x, y)          # same as k
array([ 0.59500984,  0.09655469, -0.79789754])
>>> np.dot(x, y)            # and they are orthogonal
-1.3877787807814457e-17
>>> np.dot(x, k)
-1.1102230246251565e-16
>>> np.dot(y, k)
0.0

Upvotes: 22

René Wirnata
René Wirnata

Reputation: 151

Sorry, I can't put it as a comment because of a lack of reputation.

Regarding @behzad.nouri's answer, note that if k is not a unit vector the code will not give an orthogonal vector anymore!

The correct and general way to do so is to subtract the longitudinal part of the random vector. The general formula for this is here

So you simply have to replace this in the original code:

>>> x -= x.dot(k) * k / np.linalg.norm(k)**2

Upvotes: 15

william liang
william liang

Reputation: 11

Assume the vector that supports the orthogonal basis is u.

b1 = np.cross(u, [1, 0, 0])   # [1, 0, 0] can be replaced by other vectors, just get a vector orthogonal to u
b2 = np.cross(u, b1)
b1, b2 = b1 / np.linalg.norm(b1), b2 / np.linalg.norm(b2)

A shorter answer if you like.

Get a transformation matrix

B = np.array([b1, b2])
TransB = np.dot(B.T, B)
u2b = TransB.dot(u) # should be like [0, 0, 0]

Upvotes: 1

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