UNCORRECTED PROOF MATCOM 2663 1–7 Mathematics and Computers in Simulation xxx (2006) xxx–xxx Shape control of 3D lemniscates  3 Gabriel Arcos a, , Guillermo Montilla b , Jos´ e Ortega c , Marco Paluszny a, ∗ 4 a Laboratorio de Computaci´ on Gr ´ afica y Geometr´ ıa Aplicada, Facultad de Ciencias, Escuela de Matem´ aticas, 5 Universidad Central de Venezuela, Apartado 47809, Los Chaguaramos, Caracas 1041-A, Venezuela 6 b Centro de Procesamiento de Im´ agenes, Facultad de Ingenier´ ıa, Universidad de Carabobo, Valencia, Venezuela 7 c Departamento de Matem ´ aticas, Facultad de Ciencias y Tecnolog´ ıa, Universidad de Carabobo, Valencia, Venezuela 8 9 Abstract 10 A 3D lemniscate is the set of points whose product of squared distances to a given finite family of fixed points is constant. 3D 11 lemniscates are the space analogs of the classical lemniscates in the plane studied in [A. Markushevich, Teor´ ıa de las Funciones 12 Anal´ ıticas, Tomo I, Mir, Mosc´ u, 1970]. They are bounded algebraic surfaces whose degree is twice the number of foci. Within the 13 field of computer aided geometric design (CAGD), 3D lemniscates have been considered in [J.R. Ortega, M. Paluszny, Lemniscatas 14 3D, Revista de Matem´ atica: Teor´ ıa y Aplicaciones 9 (2) (2002) 7–14] only for the case of three foci. This case is simpler than 15 the general case, because most of the parameters that control connectedness and deformation can be computed analytically. We 16 introduce the singularities as shape handles for the control of lemniscate deformation and pay special attention to the case of four 17 foci. 18 © 2006 Published by Elsevier B.V. on behalf of IMACS. 19 Keywords: Shape; Control; 3D lemniscates 20 21 1. Preliminary remarks 22 Given a set {a 1 ,..., a n }⊂ R 3 of fixed points, define the function W : R 3 → R by 23 W (x) = n  i=1 ‖x - a i ‖ 2 . 24 A level set of this function W ρ ={x ∈ R 3 : W (x) = ρ}, will be called a 3D lemniscate with foci a 1 ,..., a n , 25 and radius ρ. 3D lemniscates are implicit bounded algebraic surfaces, whose degree is twice the number of foci 26 (Fig. 1). In general, the lemniscate surface does not have to be connected, although each focus is contained in 27 one connected component. The number of connected components depends on the position of the foci and the 28 radius ρ. 29  This work was partially supported by Grant G97 000651 of Fonacit, Venezuela. ∗ Corresponding author. E-mail addresses: gabrielarcos@cantv.net (G. Arcos), gmontilla@netuno.net.ve (G. Montilla), jortega@uc.edu.ve (J. Ortega), marcopaluszny@gmail.com (M. Paluszny). 1 0378-4754/$32.00 © 2006 Published by Elsevier B.V. on behalf of IMACS. 2 doi:10.1016/j.matcom.2006.06.001