Circle and sphere blending with conformal geometric algebra

Abstract

Blending schemes based on circles provide smooth `fair' interpolations between series of points. Here we demonstrate a simple, robust set of algorithms for performing circle blends for a range of cases. An arbitrary level of G-continuity can be achieved by simple alterations to the underlying parameterisation. Our method exploits the computational framework provided by conformal geometric algebra. This employs a five-dimensional representation of points in space, in contrast to the four-dimensional representation typically used in projective geometry. The advantage of the conformal scheme is that straight lines and circles are treated in a single, unified framework. As a further illustration of the power of the conformal framework, the basic idea is extended to the case of sphere blending to interpolate over a surface.