Fast Random Rotation Matrices

• Main Author: James Arvo
• Other Authors: The Pennsylvania State University CiteSeerX Archives
• Format: Text
• Language: English
• Published: 1992
• Subjects: North Pole
• Online Access: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.1357; http://www.cs.caltech.edu/~arvo/papers/RotationMat.ps

line zp and containing the origin. Such a reflection is given by the Householder matrix H = I \Gamma 2vv T (2) where v is a unit vector parallel to zp (See, for instance, [Golub 85]). To turn this into a rotation we need only apply one more reflection (making the determinant positive). A convenient reflection for this purpose is reflection through the origin; that is, scaling by \Gamma1. Thus, the final rotation matrix can be expressed as the product M = \GammaH R (3) where R is the simple rotation in equation 1. The rotation matrix M will be uniformly distributed within SO(3), the set of all rotations in 3-space, if H takes the north pole to every point on the sphere with equal probability density. This will hold if the image of z under the random r