Shape control of 3D lemniscates

A 3D lemniscate is the set of points whose product of squared distance to a given finite family of fixed points is constant. 3D lemniscates are the space analogs of the classical lemniscates in the plane studied in [5]. 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 [3] only in the case of three foci, this case is simpler than the general case because most of the parameters that control connectedness and the deformation may be computed analytically. We introduce the singularities as shapes handles for the control of lemniscate deformation and pay special attention to the case of four foci. 1 Preliminary remarks. Given the set { 1, . . . , n} ⊂ 3 of fixed points define the functionW : 3 → by W ( ) = n ∏ i=1 ‖ − i‖ A level set of this function Wρ = { ∈ 3 : W ( ) = ρ} will be called 3D lemniscate, of foci 1, . . . , n and radius ρ. 3D lemniscates are implicit bounded algebraic surfaces, whose degree is twice the number of foci. In general the ? This work was partially supported by Grant G97 000651 of Fonacit, Venezuela. E-mail addresses: [email protected], [email protected], [email protected] and [email protected]. Preprint submitted to Elsevier Science 27th August 2004 lemniscate surface does not have to be connected, although each focus is contained in one connected component. The number of connected components depends on the position of the foci and the radius ρ. Figure 1. Some interesting 3D lemnisctes. Assume that ρ and 1, . . . , n determine a connected lemniscate then • if we move away one of the foci then the lemniscate stretches in that direction and eventually splits into two connected components • if ρ increases then the lemniscate swells and tends to a spherical shape • if ρ decreases then the surface tightens towards its foci and eventually splits into two or more components Figure 2 shows a sequence of connected lemniscates as one of the foci moves. The surfaces interpolate three given points. See [4]. Figure 2. Focus motion preserving connectivity and the interpolation of three points. If we start with a lemniscate which consists of several components then • as ρ increases the components tend to coalesce • the motion of one of the foci affects the shape of each of the components of the surface As an example consider a focus and C a connected component of the lem-