Applications of continuous functions in topological CAD data

TL;DR

This paper explores how continuous functions from topology can be applied to improve the understanding and modeling of topological CAD data, revealing practical relevance for CAD systems.

Contribution

It establishes the formal link between mathematical topology and CAD data models using continuous functions, which has been largely overlooked.

Findings

Continuous functions provide a formal topological framework for CAD data.

Applying topology enhances the understanding of spatial relationships in CAD.

Continuity has practical relevance for CAD system data management.

Abstract

Most CAD or other spatial data models, in particular boundary representation models, are called "topological" and represent spatial data by a structured collection of "topological primitives" like edges, vertices, faces, and volumes. These then represent spatial objects in geo-information- (GIS) or CAD systems or in building information models (BIM). Volume objects may then either be represented by their 2D boundary or by a dedicated 3D-element, the "solid". The latter may share common boundary elements with other solids, just as 2D-polygon topologies in GIS share common boundary edges. Despite the frequent reference to "topology" in publications on spatial modelling the formal link between mathematical topology and these "topological" models is hardly described in the literature. Such link, for example, cannot be established by the often cited nine-intersections model which is too elementary for that purpose. Mathematically, the link between spatial data and the modelled "real world" entities is established by a chain of "continuous functions" - a very important topological notion, yet often overlooked by spatial data modellers. This article investigates how spatial data can actually be considered topological spaces, how continuous functions between them are defined, and how CAD systems can make use of them. Having found examples of applications of continuity in CAD data models it turns out that of continuity has much practical relevance for CAD systems.