Geometric Constraint Solving and Automated Geometric Reasoning (T RACK P ROPOSAL TO SAC 2006, D IJON ,F RANCE ) Xiao-Shan Gao * Academy of Mathematics and Systems Science, Academia Sinica Beijin, China Dominique Michelucci † LE2I, University of Burgundy, Dijon, France Pascal Schreck ‡ LSIIT, Universit´ e Louis Pasteur, Strasbourg,France Abstract We propose to organize at SAC 2006 a track dedicated to the recent trends in the domain of geometric constraint solving (GCS) and automated, or computer aided, deduction in geometry (ADG). Geometric problems are within the heart of many theoretical studies and engineering applications. For instance a large amount of problems from geometric modeling, computer graphics, com- puter vision, computer aided design, and robotics could be reduced to either geometric constraint solving or geometric reasoning. Conversely, a wide range of methods based on very different approaches have been studied for solving geometric constraints and proving geometric theorems. This track will be an interesting opportunity to gather researchers coming from communities concerned by subjects as different as constraint programming, numeric analysis, symbolic computation, CAD, automated reasoning, and computer graphics. CR Categories: G.1.5 [Mathematics of Computing]: Nu- merical Analysis—Roots of Nonlinear Equation; G.4 [Mathe- matics of Computing]: Mathematical Software—; I.2.3 [Com- puting Methodologies]: Artificial Intelligence—Deduction and Theorem Proving; I.3.5 [Computing Methodologies]: Com- puter Graphics—Computational Geometry and Object Modeling; J.6.0 [Computer Application]: Computer-aided Engineering— Computer-aided Design (CAD); K.3.1 [Computing Milieux]: Com- puters and Education—Computer Uses in Education Computer- assisted instruction (CAI); Keywords: geometric constraints solving, geometric model- ing, numerical/symbolic/logic/invariant synergy, computer vision, robotics, geometric theorem proving 1 Rationale The persistent distinction between Automated Geometric Con- straint Solving (GCS) and Automated Deduction in Geometry (ADG) arises mainly from the nature of mathematical tools which are favored by each domain. Roughly speaking, numerical meth- ods are used in GCS while deductive databases, logic or symbolic computation are the preferred approaches in ADG. This distinction seems a little bit restrictive. In fact, it hides a continuum of ap- proaches taking more or less geometry into account. Here are some examples: * xgao@mmrc.iss.ac.cn † Dominique.Michelucci@u-bourgogne.fr ‡ schreck@dpt-info.u-strasbg.fr • The first methods in GCS were mainly based on equational systems resolution using relaxation or Newton-Raphson methods. Recent solvers perform sophisticated constraint sys- tem decompositions based on geometric properties. • Some solvers use intrinsic formulations of constraints in order to throw off any peculiar Cartesian frame. Such approaches are based on geometric concepts as Cayley-Menger determi- nant, tensorial geometry and covariant coordinates, but the equational systems they yield are solved with numeric meth- ods. Note that the solver of CATIA (by Dassault System) which is one of the main geometric modeler on the market uses such an intrinsic formulation of constraints. • Sketchers to architectural designs need to reconstruct 3D ob- jects from 2D sketches (or photographies) which is matter of projective geometry and Grassmann-Cayley algebra. • Deductive databases and logic are required to perform geo- metric constructions in the domain of computer-aided instruc- tion, but “probabilistic provers” can be used to discover some properties If recent trends in GCS seem to point out an increasing influence of geometry in order to have more accurate and sound methods, a lot of work has been done about numerical solvers too. For instance, interval arithmetic has been used to build robust solvers in robotic but also in CAD. Some actual studies are aimed at the exploitation of geometric properties to make such solvers more efficient. In the algebraic case, Bernstein basis and interval arithmetic are used in works by Chenin, Biard, Mourraind and Pavone. In spite of all these studies, solving geometric constraints in the 3D case remains topical. There is not yet a combinatorial criterion of graph rigidity in the 3D case and, consequently, the actual decomposition methods of 3D geometric constraints are not sound. This unsoundness leads to the question of the mis-constrained systems: since modeling with constraints is a kind of programming, there is a need of debuggers, that is, tools which perform a qual- itative study of a constraint system. Recent works propose ways to deal with under or over constrained systems, or to decompose systems in a well constrained part and a mis-constrained one. Related Conference Events: Before all, this track proposal aims to be a meeting between the communities of GCS and ADG. To our knowledge, this ACM event will be the first conference of importance dedicated to this topic. Among the related conference events, the following ones can be noticed :