Abstract Labeled graphs are particularly well adapted to represent objects in the context of topology-based geometric modeling. Thus, graph transformation theory is used to implement modeling operations and check their consistency. This article defines a class of graph transformation rules dedicated to embedding computations. Objects are here defined as a particular subclass of labeled graphs in which arc labels encode their topological structure (i.e., cell subdivision: vertex, edge, face) and node labels encode their embedding (i.e., relevant data: vertex positions, face colors, volume density). Object consistency is defined by labeling constraints which must be preserved by modeling operations that modify topology and/or embedding. Dedicated graph transformation variables allow us to access the existing embedding from the underlying topological structure (e.g., collecting all the points of a face) in order to compute the new embedding using user-provided functions (e.g., compute the barycenter of several points). To ensure the safety of the defined operations, we provide syntactic conditions on rules that preserve the object consistency constraints.

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.