Interpolation of Spatial Displacements Using the Clifford Algebra of E4

[+] Author and Article Information
K. R. Etzel

Naval Sea Systems Command, 2531 Jefferson Davis Hwy., Arlington, VA 22242-5160

J. M. McCarthy

Department of Mechanical and Aerospace Engineering, University of California at Irvine, Irvine, CA 92697

J. Mech. Des 121(1), 39-44 (Mar 01, 1999) (6 pages) doi:10.1115/1.2829427 History: Received November 01, 1996; Revised November 01, 1998; Online December 11, 2007


In this paper we show that the Clifford Algebra of four dimensional Euclidean space yields a set of hypercomplex numbers called “double quaternions.” Interpolation formulas developed to generate Bezier-style quaternion curves are shown to be applicable to double quaternions by simply interpolating the components separately. The resulting double quaternion curves are independent of the coordinate frame in which the key frames are specified. Double quaternions represent rotations in E4 which we use to approximate spatial displacements. The result is a spatial motion interpolation methodology that is coordinate frame invariant to a desired degree of accuracy within a bounded region of three dimensional space. Examples demonstrate the application of this theory to computing distances between spatial displacement, determining the mid-point between two displacements, and generating the spatial motion interpolating a set of key frames.

Copyright © 1999 by The American Society of Mechanical Engineers
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