Semiglobal deformation and correction of free- form surfaces using a mechanical alternative J.C. Leon, P. Veron Laboratoire Sols, Solides, Structures UMR-CNRS 5521, BP 53, 38041 Grenoble Cedex 9, France E-mail: Philippe.veron@img.fr Jean-caude.leon@img.fr Free-form surfaces deformation is difficult since more than one control point must be moved to achieve satisfying results. Simul- taneous movement of control vertices pro- vides semiglobal deformation. The method uses an analogy between the control poly- hedron of a surface and the mechanical equilibrium of a bar network. Changes of mechanical parameters produce real-time shape modifications. A toolkit of basic deformation functions achieve surface in- flation, tweaking and shrinking. Each equilibrium position of the network is the solution of a linear system of equations. The surface can be deformed on a local, semiglobal or global basis. Higher-level interactive functions suppress undesired bumps or wrinkles and provide real-time surface adjustment and precise control of curvature distribution. Key words: Free-form surfaces — computer- aided design — mechanical model — semi- global deformation — surface correction 1Correspondence to: J.C. Leon 1 Introduction Geometric models like Be´ zier, B-Spline or NURBS (Bartels et al. 1987; Farin 1988; Le´on 1991) are deeply involved in free-form surface description used for computer-aided design and other design purposes. All these methods can build models with the help of software. Often, geometric models require modifications during or after their construction to meet aesthetic and/or fucntional criteria. Usually, the modification of a model is a long and tedious task carried out through basic functions. These functions use the well-known geometric properties of the model surface patches and their associated control polyhedron vertices. The core of the patch deformation process is the displace- ment of polyhedron vertices. Hence, the orienta- tion, the amplitude and the direction of each vertex displacement and the number of vertices moved are the unknowns of the problem. Com- monly, all these parameters must be set by the user. The user can control the displacement of only one vertex or, eventually, a continuous row of vertices when their movement depends linearly upon the displacement of a master vertex. With either of the previous situations, such a deforma- tion process is highly iterative, time consuming and rather tedious. Several alternatives have been proposed to de- form free forms. Geometrical or mechanical tech- niques summarise the directions of current work in that area. Work originated by Barr (1984) and Sederberg and Parry (1986) and further developed by Coquillart (1990) and Coquillart and Jance`ne (1991) requires the creation of an auxiliary, Be´zier — or B-Spline — based geometry in which the model is immersed. However, deformations in that auxiliary geometry are carried out with stan- dard Be´zier — or B-Spline — type deformation tools that still lead to dozens of control para- meters in the sense previously defined. A precise control of the deformation process can be difficult to achieve since the auxiliary geometry is kept as simple as possible. Further reduction of control parameters, as well as improvements of the under- standing of the behaviour of the method, has been proposed (Hsu et al. 1992). The reduction of the number of control parameters, as seen by the user, is one of the aspects of the present work. Some mechanical approaches (Terzoploulos et al. 1987), and particularly the work of Celnicker and Gossard (1991) employ finite element models The Visual Computer (1997) 13 : 109—126 ( Springer-Verlag 1997 109