Smooth interpolation of orientations with angular velocity constraints using quaternions

In this paper we present methods to smoothly interpolate orientations, given N rotational keyframes of an object along a trajectory. The methods allow the user to impose constraints on the rotational path, such as the angular velocity at the endpoints of the trajectory. We convert the rotations to quaternions, and then spline in that non-Euclidean space. Analogous to the mathematical foundations of flat-space spline curves, we minimize the net "tangential acceleration" of the quaternion path. We replace the flat-space quantities with curved-space quantities, and numerically solve the resulting equation with finite difference and optimization methods. 1 Introduction The problem of using spline curves to smoothly interpolate mathematical quantities in flat Euclidean spaces is a well-studied problem in computer graphics [bartels et al 87], [kochanek&bartels 84]. Many quantities important to computer graphics, however, such as rotations, lie in non-Euclidean spaces. In 1985, a method to...