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.

💡 Research Summary

The paper presents a formal framework for topology‑based geometric modelling that unifies topological structure and geometric/attribute data within a single graph‑based representation. The authors introduce I‑labelled graphs, an extension of partially labelled graphs, where each node can carry a set of labels indexed by a finite set I (e.g., coordinates, curves, colors, material properties). This multi‑label approach allows the simultaneous representation of the embedding (geometric and application‑specific data) together with the pure topological information encoded by arc labels.

The work builds on the well‑known generalised map (G‑map) model for topology. In a G‑map, arcs are labelled with α₀, α₁, … to capture adjacency relations between cells of any dimension (vertices, edges, faces, volumes). The authors embed the G‑map into the I‑labelled graph formalism by treating the pure topological graph as the base (all node labels undefined) and adding the embedding labels on top of it. Consequently, an embedded G‑map must satisfy two families of constraints: (1) topological consistency (pairing, involution, closure of α‑relations) and (2) embedding consistency (each node’s i‑label, when defined, must respect the type and existence required by the corresponding cell).

Transformation rules are defined using the double‑pushout (DPO) approach. A rule r consists of three I‑labelled graphs L ← K → R together with inclusion morphisms. The rule’s left‑hand side L describes a pattern to be matched, K is the preserved kernel, and R is the replacement. The authors impose syntactic conditions on the rule: any label that is undefined (⊥) in L must also be undefined in K and R, ensuring that the rule does not unintentionally create or delete embedding information. A match morphism m : L → G must satisfy the dangling condition (no node removed by the rule is incident to arcs outside the matched subgraph). Under these conditions, the standard DPO construction yields a unique pushout object D and a resulting graph H, guaranteeing that the transformation is well‑defined.

Crucially, the paper proves that the existence and uniqueness results for DPO transformations on partially labelled graphs extend directly to I‑labelled graphs. By projecting an I‑labelled graph onto each index i (producing i‑components) and using the known pushout constructions on those components, the authors assemble a global pushout for the whole multi‑label graph. This ensures that both topological and embedding constraints are preserved automatically during rule application.

To support more expressive operations, the authors introduce rule schemes (or schemata). A scheme contains variables and expressions that compute new label values from the matched context, allowing generic geometric operations such as edge splitting with interpolated vertex coordinates, face extrusion with computed normal vectors, or attribute propagation based on physical models. The scheme is instantiated by evaluating its expressions during rule application, and the same syntactic consistency checks are applied to guarantee that the generated labels respect the embedding constraints.

The paper’s contributions can be summarised as follows:

1. Formal definition of I‑labelled graphs – a categorical model that accommodates multiple independent label families per node.
2. Embedding of G‑maps – a precise mapping of topological structures into the I‑labelled framework, together with explicit consistency constraints.
3. Extension of DPO transformation theory – proof of existence, uniqueness, and preservation of total labelling for multi‑label graphs.
4. Rule schemes with computed labels – a mechanism for parametric geometric transformations while maintaining formal guarantees.
5. Practical relevance – the framework can be applied to CAD, GIS, biomedical modelling, and any domain where complex geometric objects undergo topological and attribute modifications.

Overall, the work provides a rigorous mathematical basis for building transformation‑based modelling tools that automatically enforce both topological correctness and embedding consistency, thereby reducing the risk of errors in complex geometric pipelines and enabling higher‑level, reusable operation specifications.