Lemniscates 3D: a CAGD primitive?

Abstract

A 3D lemniscate is an implicitly given surface which generalizes the well-known Bernoulli lemniscates curves and the Cassini ovals in 2D. It is characterized by placing a finite number of points in space (the foci) and choosing a constant (radius), its algebraic degree is twice the number of foci and it is always contained in the union of certain spheres centered at the foci. The distribution of the foci gives a rough idea of the 3D shapes that could be modeled with any of the connected components of the lemniscate. The position of the foci can be used to stretch and to produce knoblike features. Given a set of foci, for a small radius the lemniscate consists of a number of spherelike surfaces centered at the foci which do not touch each other. As the radius increases the disconnected pieces coalesce producing interesting surfaces. In order to make 3D lemniscates a potentially useful primitive for CAGD it is necessary to control the coalescing/splitting of the connected components of the lemniscate while we move the foci and change the radius, simultaneously. In this paper we offer tools towards this control. We look closely at the case of four noncoplanar foci.