Shape Modeling: New Theories and Algorithms – SHAN

Today, 3D geometric models play a central role in all industrial processes (design and engineering, numerical simulations, marketing, project review, pilot training, NC simulations, machining, ergonomic studies, maintenance operations). However, constructing their underlying mathematical representations is difficult and often done manually. In CAD systems, a kind of surfaces, B-spline surfaces and especially NURBS (Non Uniform Rational B Splines) are usually adopted for Shape Modeling. NURBS are parametric surfaces. With NURBS, the designer can create curves and surfaces with control points or weights very easily, but there are many computational problems with NURBS, because they do not have a mathematical close form. Moreover, up to now, almost all geometric modeling tool kits are based on traditional mathematics. They ignore the fact that computers can only represent a finite set of real numbers and simply use the formula (a-?<b) and (b<a+?) to compare whether two real numbers a and b are equal to each other or not. In this ANR/NSF project, we propose to fully study the fundamental aspects of new mathematical representation of curves and surfaces, and explore alternative research avenues by introducing new optimization techniques. We will apply these techniques for solving some fundamental problems such as projecting points and curves on surfaces, fitting, skinning and stitching. However, first of all, we propose a new approach to construct a new system of continuity theories, called floating-point continuity, to bridge the gap between the traditional mathematics and modern computer science. Based on the floating-point continuities and epsilon-geometry continuities concepts introduced in our recent publications, we will propose several key geometric modeling operators based in this new approach. Moreover, based on our academic and industrial experience, all the novel approaches will be tested on CAD and Engineering applications and compared with classical methods. Our team INRIA-Tsinghua, has proven his efficiency since 2005: more than twenty joint scientific publications in the best international journals (Computer Aided Design-Elsevier, Computer Aided Geometry Design-Elsevier) as well as technological transfers with European companies. In this Project we add both ANR and NSF granted Ph.D and Post Doctors working together and we will associate French and Chinese scientists who have a strong expertise in the domain.