Geometric Modeling: Consistency Preservation Using Two-Layered Variable Substitutions

In the context of topology-based geometric modeling, operations transform objects regarding both their topological structure (i.e. cell subdivision: vertex, edge, face, etc.) and their embeddings (i.e. relevant data: vertex positions, face colors, volume densities, etc.). Graph transformations with variables allow us to generically handle those operations. We use two types of variables: orbit variables to abstract topological cells and node variables to abstract embedding data. We show how these variables can be simultaneously used, and we provide syntactic conditions on rules to ensure that they preserve object consistency. This rule-based approach is the cornerstone of Jerboa, a tool that allows a fast and safe prototyping of geometric modelers.