Computational Symbolic Geometry

The aim of this work is to present a framework for symbolic computations in Geometry. More precisely, we are interested in problems coming from robotics and vision, therefore we focus on points, linear spaces, spheres, displacements and matrices. The approach chosen consists in dealing with intrinsic properties, in order that we (most of the time) manipulate invariant quantities (independent of the referential frame) and we (as much as possible) avoid using coordinates. The reason for this choice is that computations are done in a more simple, synthetic and natural way than if we used coordinates. For each class of object mentioned before, we give one or more possible formal representations and we describe the relations that exist between the quantities introduced to represent these objects. Here is the general scheme that we follow: if we want to work in a space A, we use a free algebra of polynomials (or a free module) F where the variables represent generators of A and we consider the relations K that exist between these objects: % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm % Wu51MyVXgaruWqVvNCPvMCaebbnrfifHhDYfgasaacH8srps0lbbf9 % q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0RYxir % -Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGacaGa % aeqabaWaaeaaeaaakeaacqaIWaamcqGHsgIRcqWGlbWscqGHsgIRcq % WGgbGrdaWfGaqaaiabgkziUcWcbeqaaiabew9aMbaakiabdgeabjab % gkziUkabicdaWaaa!3E59! $$ 0 \to K \to F\mathop{ \to }\limits^{\phi } A \to 0 $$ The map ∅ associates to a variable of F the object represented by this variable in A and K is the kernel of this map. We call such a map a representation of A. Notice that this representation is not unique. In order to be able to compute modulo K, we give a normal form algorithm which reduces every element of K to 0.