Direct Linear Interpolation of Geometric Objects in Conformal Geometric Algebra

Abstract

Typically we do not add objects in conformal geometric algebra (CGA), rather we apply operations that preserve grade, usually via rotors, such as rotation, translation, dilation, or via reflection and inversion. However, here we show that direct linear interpolation of conformal geometric objects can be both intuitive and of practical use. We present a method that generates useful interpolations of point pairs, lines, circles, planes and spheres and describe algorithms and proofs of interest for computer vision applications that use this direct averaging of geometric objects.