Formula omitted] spherical Bézier splines

Abstract

The classical de Casteljau algorithm for constructing Bézier curves can be generalised to a sphere of arbitrary dimension by replacing line segments with shortest great circle arcs. The resulting spherical Bézier curves are C ∞ and interpolate the endpoints of their control polygons. In the present paper, we address the problem of piecing these curves together into C 2 splines. For this purpose, we compute the endpoint velocities and accelerations of a spherical Bézier curve of arbitrary degree and use the formulae to define control points that give the curve a desired initial velocity and acceleration. In addition, for uniform splines we establish a simple relationship between the control points of neighbouring curve segments that is necessary and sufficient for C 2 continuity. As illustration, we solve an interpolation problem involving sparse data using both the present method and a normalised polynomial interpolant. The normalised spline exhibits large variations in speed and magnitude of acceleration, whilst the spherical Bézier spline is far better behaved. These considerations are important in applications where velocities and accelerations need to moderated or estimated, notably computer animation and rigid body trajectory planning, where interpolation in the 3-sphere is a fundamental task.