Interpolation of spatial displacements using the Clifford Algebra of E-4

In this paper we show that the Clifford Algebra of four dimensional Euclide an 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 t he components separately. The resulting double quaternion curves are indepe ndent of the coordinate frame in which the key frames are specified Double quaternions represent rotations in E-4 which we use to approximate spatial displacements. The result is a spatial motion interpolation methodology tha t is coordinate frame invariant to a desired degree of accuracy within a bo unded region of three dimensional space. Examples demonstrate the applicati on of this theory to computing distances between spatial displacement, dete rmining the mid-point between two displacements, and generating the spatial motion interpolating a set of key frames.