Shape control of 3D lemniscates

A 3D lemniscate is the set of points whose product of squared distances to a given finite family of fixed points is constant. 3D lemniscates are the space analogs of the classical lemniscates in the plane. They are bounded algebraic surfaces whose degree is twice the number of foci. Within the field of computer aided geometric design (CAGD), 3D lemniscates have been considered in [J.R. Ortega, M. Paluszny, Lemniscatas 3D, Revista de Matemática: Teoría y Aplicaciones 9 (2) (2002) 7-14] only for the case of three foci. This case is simpler than the general case, because most of the parameters that control connectedness and deformation can be computed analytically. We introduce the singularities as shape handles for the control of lemniscate deformation and pay special attention to the case of four foci.