A C²-Continuous Unit Quaternion Interpolatory Spline Curve

In order to be able to quickly generate smooth unit quaternion interpolatory spline curves and realtimely control 3D object keyframe animations, this paper presents a quartic-polynomial-based unit quaternion interpolatory spline curve, and proves its interpolatory property and C² continuity. First, proper quartic polynomial blending functions are chosen to generate a C²-continuous spline curve in Euclidean space which can interpolate a given sequence of data points. Then, via the product of several exponential functions where the exponents are the cumulative forms of the blending functions, and the bases are constant unit quaternions representing the given keyframe orientations, the spline curve in Euclidean space is extended to one in unit quaternion space S³. The proposed curve can automatically pass through a given sequence of keyframe orientations accurately, and avoid the iterative process in solving the quaternionic nonlinear system of equations when obtaining the spline control points from given data points, as used in classic B-spline unit quaternion interpolatory curves, and improve the computational efficiency. Experimental results demonstrated the effectiveness of the proposed scheme in the application of 3D keyframe animations.