Rule-based transformations for geometric modelling

The context of this paper is the use of formal methods for topology-based geometric modelling. Topology-based geometric modelling deals with objects of various dimensions and shapes. Usually, objects are defined by a graph-based topological data structure and by an embedding that associates each topological element (vertex, edge, face, etc.) with relevant data as their geometric shape (position, curve, surface, etc.) or application dedicated data (e.g. molecule concentration level in a biological context). We propose to define topology-based geometric objects as labelled graphs. The arc labelling defines the topological structure of the object whose topological consistency is then ensured by labelling constraints. Nodes have as many labels as there are different data kinds in the embedding. Labelling constraints ensure then that the embedding is consistent with the topological structure. Thus, topology-based geometric objects constitute a particular subclass of a category of labelled graphs in which nodes have multiple labels. We previously introduced a formal approach of topological modelling based on graph transformation rules. Topological operations, that only modify the topological structure of objects, can be defined such that the topological consistency of constructed objects is ensured with syntactic conditions on rules. In this paper, we follow the same approach in order to deal with geometric operations, that can modify both the topological structure and the embedding. Thus, we define syntactic conditions on rules to ensure the consistency of the embedding during transformations.