Interpolating Solid Orientations with Circular Blending Quaternion Curves

Abstract

This paper presents a method to smoothly interpolate a sequence of solid orientations using circular blending quaternion curves. We show that, given three solid orientations, there is a circular quaternion curve which interpolates the three orientations. Given four orientations q _i-1, q _i , q _i+1, q _i+2, there are two circular quaternion curves C _i and C_i+1 which interpolate the triples of orientations (q _i-1, q _i , q _i+1) and (q _i , q _i+1, q _i+2) respectively. Thus, both the two.quaternion curves C _i and C_i+1 interpolate the two orientations q _i and q _i+1.

Using a similar blending method to the parabolic blending of Overhauser [15], we generate a quaternion curve Q _i which interpolates the two orientations q _i and q _i+1 while smoothly blending the two circular quaternion curves C _i and C _i+1. The quaternion curve Q _i has the same tangent direction with C _i at q _i and with C _i+1 at q _i+1 respectively. By connecting the quaternion curves Q _i ‘s in a connected sequence, we generate a quaternion path which smoothly interpolates a given sequence of solid orientations.

There are various advantages of this method, in computational complexity as well as in design flexibility, over the previous interpolation methods of solid orientations. Detailed comparisons are made with respect to the Bezier curve of Shoemake [19], the cardinal spline curve of Pletincks [16], and the spherical quadrangle curve of Shoemake [20].

Research supported in part by RIST and KOSEF.