Fast Random Rotation Matrices
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...
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ftciteseerx:oai:CiteSeerX.psu:10.1.1.53.1357 2023-05-15T17:39:47+02:00 Fast Random Rotation Matrices James Arvo The Pennsylvania State University CiteSeerX Archives 1992 application/postscript http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.1357 http://www.cs.caltech.edu/~arvo/papers/RotationMat.ps en eng http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.1357 http://www.cs.caltech.edu/~arvo/papers/RotationMat.ps Metadata may be used without restrictions as long as the oai identifier remains attached to it. http://www.cs.caltech.edu/~arvo/papers/RotationMat.ps text 1992 ftciteseerx 2016-01-08T10:31:48Z 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 Text North Pole Unknown North Pole |
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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 |
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The Pennsylvania State University CiteSeerX Archives |
format |
Text |
author |
James Arvo |
spellingShingle |
James Arvo Fast Random Rotation Matrices |
author_facet |
James Arvo |
author_sort |
James Arvo |
title |
Fast Random Rotation Matrices |
title_short |
Fast Random Rotation Matrices |
title_full |
Fast Random Rotation Matrices |
title_fullStr |
Fast Random Rotation Matrices |
title_full_unstemmed |
Fast Random Rotation Matrices |
title_sort |
fast random rotation matrices |
publishDate |
1992 |
url |
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.1357 http://www.cs.caltech.edu/~arvo/papers/RotationMat.ps |
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North Pole |
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http://www.cs.caltech.edu/~arvo/papers/RotationMat.ps |
op_relation |
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.1357 http://www.cs.caltech.edu/~arvo/papers/RotationMat.ps |
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Metadata may be used without restrictions as long as the oai identifier remains attached to it. |
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