VIII.4 - USING GEOMETRIC CONSTRUCTIONS TO INTERPOLATE ORIENTATION WITH QUATERNIONS

Abstract

This chapter elaborates the use of geometric constructions to interpolate orientation with quaternions. Quaternions have been established as a useful representation for interpolating 3D orientation in computer animation. In keeping with traditional computer animation practices, one would like both interpolating, and approximating splines. This can be derived easily by applying the geometric constructions known for linear splines. A scheme for deriving Bézier control points from a sequence of quaternions is presented. This provides an interpolating spline for quaternions, but the construction is somewhat more complicated than necessary. The most common interpolating spline in use probably is the Catmull–Rom spline. A geometric construction for Catmull–Rom splines is also described. This produces an interpolating spline directly from the control points, without the construction of auxiliary Bézier points. For an approximating spline, a geometric construction for B-splines is well known. Geometric constructions is represented as a triangle with the four control points at the bottom, the result point at the top, and weighting functions for the intermediate results on each arc.