Abstract The paper describes three models of the real field based on subsets of the integer sequences. The three models are compared to the Harthong–Reeb line. Two of the new models, contrary to the Harthong–Reeb line, provide accurate integer “views” on real numbers at a sequence of growing scales $B^n$ ( $B\ge2$ ).

Geometric Algebra is considered as a very intuitive tool to deal with geometric problems and it appears to be increasingly efficient and useful to deal with computer graphics solutions. For example, the Conformal Geometric Algebra includes circles, spheres, planes and lines as algebraic objects, and intersections between these objects are also algebraic objects. More complex objects such as conics, quadric surfaces can also be expressed and be manipulated using an extension of the conformal Geometric Algebra. However due to high dimension of their representations in Geometric Algebra, implementations of Geometric Algebra that are currently available do not allow efficient realizations of these objects. This paper presents a Geometric Algebra implementation dedicated for both low and high dimensions. The proposed method is a hybrid solution for precomputed code with fast execution and runtime computations with low memory requirement. More specifically, the proposed method combines a precomputed table approach with a recursive method using binary trees. Some rules are defined to select the most appropriate choice, according to the dimension of the algebra and the type of multivectors involved in the product. The resulting implementation is well suited for high dimensional spaces (e.g. algebra of dimension 15) as well as for lower dimensional space. This paper details the integration of this hybrid method as a plug-in into Gaalop, which is a very advanced optimizing code generator. This paper also presents some benchmarks to show the performances of our method, especially in high dimensional spaces.

In this article, we present a rule-based language dedicated to topological operations and based on graph transformations. Generalized maps are described as a particular class of graphs determined by consistency constraints. Hence, topological operations over generalized maps can be specified using graph transformations. The rules we define are provided with syntactic criteria which ensure that graphs computed by applying rules on generalized maps are also generalized maps. We have developed a static analyzer of transformation rules which checks the syntactic criteria in order to ensure the preservation of generalized map consistency constraints. Based on this static analyzer, we have designed a rule-based prototype of a kernel of a topology-based modeler that is generic in dimension. Since adding a new topological operation can be reduced to write a graph transformation rule, we directly obtain an extensible prototype where handled topological objects satisfy built-in consistency. Moreover, first benchmarks show that our prototype is reasonably efficient compared to a reference implementation of 3D generalized maps which use a classical implementation style.

In this paper, we present a code generator transforming a class of CASL specifications into O'Caml programs. This code generator is dedicated to rapid prototyping of CASL specifications especially in the area of geometric modeling where algebraic formalisms have been used since the last decade. A large class of constructive equational specifications is handled by this generator while insuring the correctness of generated O'Caml programs. In particular, CASL specifications with many interpretation models (i.e. incomplete) are automatically supplemented in order to produce a program that implements one of them. Underlying properties, such as termination, completeness and confluence hold when equations satisfy some syntactic criteria given in the paper.