Journal
MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING
Volume 22, Issue 7, Pages -Publisher
IOP PUBLISHING LTD
DOI: 10.1088/0965-0393/22/7/075013
Keywords
rotations; equal-area mapping; texture analysis; quaternions; homochoric
Funding
- Air Force Office of Scientific Research, MURI [FA9550-12-1-0458]
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In computational work, data sets must often be represented on the surface of a sphere or inside a ball, requiring uniform grids. We construct a new volume-preserving projection between a cube and the set of unit quaternions. The projection consists of two steps: an equal-volume mapping from the cube to the unit ball, followed by an inverse generalized Lambert projection to either of the two unit quaternion hemispheres. The new projection provides a one-to-one mapping between a grid in the cube and elements of the special orthogonal group SO(3), i.e., 3D rotations. We provide connections to other rotation representation schemes, including the Rodrigues-Frank vector and the homochoric parameterizations, and illustrate the new mapping through example applications relevant to texture analysis.
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