Orientation interpolation in quaternion space using spherical biarcs

Abstract

We consider the problem of interpolating a smooth curve to a point sequence in the unit quaternion space μ. This problem has application to object orientation interpolation in computer animation, sweep surface generation in solid modeling. Since the unit quaternions form the unit spheres S3 in E4, a simple curve scheme using spherical biarcs is presented to solve this problem. The spherical biarc is a curve on a sphere consisting of two smoothly joining circular arcs. It is shown that for any two given points and two tangents specified at the two points on the unit sphere, S3, there always exist spherical biarcs interpolating these data and these biarcs are easy to construct. This leads to an algorithm for constructing a smooth and locally controllable circular arc spline curve to interpolate a sequence of unit quaternions in μ. We also discuss how to compute in-between quaternions efficiently on the resulting spline curve.