General schemes for unit quaternion curve construction

Abstract

This thesis proposes a new class of unit quaternion curves in the orientation space SO(3), which is a fundamental tool for computer animation involving solid rotations or orientations. We develop a method to transform a curve defined as a weighted sum of basis functions into its unit quaternion analogue. Applying the method to well-known curves including $B\acute{e}zier$, Hermite and B-spline curves, we are able to construct various unit quaternion curves which share nice properties such as continuity and local control property with their original curves. Each of the resulting curves is defined in a closed form, and their expressions are so simple that they are easy to be evaluated and differentiated. We first give a $B\acute{e}zier$ unit quaternion curve with $n$-control points. Then, the cubic $B\acute{e}zier$ quaternion curve is extended to a Hermite unit quaternion curve, which matches given orientations and angular velocities at its ends. Unlike the previous Hermite unit quaternion curves, it is possible to specify arbitrary angular velocities at the curve ends. From experiments, we show that our Hermite unit quaternion curves use less torque than the previous ones. We also give a B-spline unit quaternion curves defined in closed forms and verify that our B-spline unit quaternion curve of order $k$ is $C^{k-2}$-continuous and locally controllable. Since the B-spline unit quaternion curve does not interpolate its control points as the usual B-spline curves in $R^3$, we provide a method to find the control points which makes the B-spline quaternion curve interpolating a given sequence of unit quaternions. The ease of differentiation of our unit quaternion curves makes it possible to do extensive analyses on the curves. We present the formulas for computing high order derivatives of angular velocity of a $B\acute{e}zier$ quaternion curve at the curve ends. By solving the formulas for its control points, we are able to construct a $C^k$-continuous Hermite quat...