Rule-based transformations for geometric modelling

Abstract

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

Topological consistency preservation with graph transformation schemes

“…At that point, the possibility to deal with domain-related information (called embeddings) was the main bottleneck of the approach. The introduction of I-labeled graph in [33] solved this issue and defined DPO rewriting for graphs where nodes are labeled by a family of labels. Jerboa [17] offers a graphical rule editor along with a syntax checker that guarantees the consistency of G-map rules (both for the topological aspect and the omission of geometric aspect).…”

“…In topology-based geometric modeling, these data are referred to as embedding. In [33], embeddings were defined as a family of node labels with appropriate consistency constraints and conditions. Since rules can modify both the topology and the embeddings of an object, conditions to preserve the embedding consistency in meta-rules were studied in [34].…”

Preserving consistency in geometric modeling with graph transformations

“…Similarly, in Figure 10(b), nodes a, b, c, d, e, and f that belong to the same face are labeled with the same yellow color. This property is captured by an embedding constraint [7]: Definition 6 (Embedded graph and embedded generalized map). Let π : ⟨o⟩ → τ be an embedding operation with ⟨o⟩ an orbit type and τ a data type.…”

“…To extend their usability, we showed that they offer a safer design of topological operations to produce a topological-based geometric modeler [8], which was more accessible than traditional ad hoc implementations that are fastidious to code and vulnerable to consistency bugs. In [7], we defined a new graph category that allows nodes to have multiple labels in order to represent the multiple embedding types simultaneously (e.g., a face being labeled by both its color and its porosity). We proved the existence of graph transformations in this category and introduced a new type of dedicated variables and operators to express embedding transformations in rules.…”

“…In this article, we propose a generic graph transformation approach that allows the implementation of any application domain's modeling operations. More precisely, this paper addresses the embedding aspect of modeling operations considering a representation of embedded generalized maps as labeled graphs introduced in [7] and in which node labels and associated labeling constraints encode the object embedding. Our first contribution is to give conditions on the relabeling graph transformations of [28] to guarantee the preservation of the embedding consistency.…”