Surface Deformation by Normalized Operators

In order to get an optimal use of all the interesting features for free-form mod-eling provided by sophisticated geometric models (Bezier, B-Spline, NURBS), we propose a deformation technique based on the concept of normalized operators. Such an operator can be viewed as a black-box which encloses | and hides to the user | the high number and the complexity of the parameters involved during a deformation. The normalization property of an operator means that it does not modify the structure of the geometric model on which it is applied. Therefore, sequential applications of such operators can be used in order to create a progressive deformation of the object. 1 Motivations One of the goals of geometric modeling is to enable the creation of objects which are as complex as objects that exist in the real world. A classical principle to reach such a goal is to use a two-pass modeling process, the rst pass is intended to give the object a coarse shape and the second pass consists in applying several deformations which enable ne control of the geometrical description. Several models have been proposed to express the geometry of an object, and the most recent of them allow free-form modeling and are well adapted to this two-pass modeling process. One example are the famous spline parametric surfaces, obtained by a tensorial product of two spline curves. Usually such a surface is deened by a set of so-called control points that act on the shape of the surface. Therefore, moving these control points enables local or global deformations of the surface. Moreover, according to the kind of spline surfaces (piecewise, uniform, non uniform, rational, etc), there are several other parameters that provide additional control of the shape, and which can be used during the deformation process. This particular point will be detailled more precisely in Section 2 for non uniform rational B-spline surfaces that involve a great number of parameters in their deenition. Interactive displacement of control points deening a spline surface can be viewed as a low-level deformation technique and can eeectively provide free-form modeling. Unfortunately, getting the exact desired shape at the end of the process, is usually a painfull operation. Indeed, the number of control points that have to be moved can rapidly become unmanageable as the complexity of the object increases. Therefore, the need of high-level deformation techniques has arised quite soonly for CAD applications.