Declarative Mesh Subdivision Using Topological Rewriting in MGS

Abstract

Abstract. Mesh subdivision algorithms are usually specified informally using graphical schemes defining local mesh refinements. These algorithms are then implemented efficiently in an imperative framework. The implementation is cumbersome and implies some tricky indices management. Smith et al. (2004) asks the question of the declarative programming of such algorithms in an index-free way. In this paper, we positively answer this question by presenting a rewriting framework where mesh refinements are described by simple rules. This framework is based on a notion of topological chain rewriting. Topological chains generalize the notion of labeled graph to higher dimensional objects. This framework has been implemented in the domain specific language MGS. The same generic approach has been used to implement Loop as well as Butterfly, Catmull-Clark and Kobbelt subdivision schemes.

Building Topological Spaces for Musical Objects

“…It acts as a container and the values as the elements of the data structure. Transformations of topological collections are defined by rewriting rules [19] specifying replacement of sub-collections that can be recursively performed to build new spaces.…”

“…There exist several ways to control the application of a set of rules on a collection but these details are not necessary for the comprehension of the work presented here. A formal specification of topological rewriting is given in [19]. We sketch here only the specification of patterns.…”

The Modeling and the Simulation of the Fluid Machines of Synthetic Biology

“…A dedicated definition has been developed in [19]. However, thanks to the term structure of a topological collection, transformations can be defined in the framework of set rewriting, following an approach similar to that taken in [31] for hyper-graphs: using the additive representation of topological collections, topological rewriting can be simply defined as an adapted version of conditional first-order associative-commutative term rewriting, see [36] for the details.…”

Using a graph grammar system in the finite element method

“…This was done using a quasi-context sensitive graph grammar. Spicher et al (2010) modeled mesh refinements by a simple rewriting framework, based on topological chain rewriting. Both approaches allow us only to model uniform refinements, because non-uniform mesh transformations are context dependent and cannot be modelled by the quasi-context sensitive graph grammar.…”