3D Rotations using Rodrigues Rotation Formula

Posted on May 31, 2019 by machinelearning1

The problem we are addressing here is the rotation of a general 3D vector v about a given axis of rotation denoted by \hat{n} by \theta radians.

The Rodrigues Rotation Formula is as follows:

v^{'} = (1-\cos(\theta))(v.\hat{n})\hat{n} + \cos(\theta) v + \sin(\theta)(\hat{n}\times v)

Example: for a sanity check we can consider \theta=0 that will result in

v^{'} = (1-\cos(0))(v.\hat{n})\hat{n} + \cos(0) v + \sin(0)(\hat{n}\times v) = v

Example:

Rotate v=[2,0,0] about \hat{n}=[0,0,1] by \theta = \pi/2.

import numpy as np
v = np.array([2.,0.,0.])
n = np.array([0.,0.,1.])
theta = np.pi/2
vp = (1-np.cos(theta))*(np.dot(v,n))*n + np.cos(theta)*v + np.sin(theta)*(np.cross(n,v))
>>> vp
array([1.2246468e-16, 2.0000000e+00, 0.0000000e+00])

 

 

 

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