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Some mathematical and representational aspects of solid modeling
1College of Engineering and Applied Science, University of Rochester, Rochester, NY 14627.
IEEE Transactions on Pattern Analysis and Machine Intelligence
|August 27, 2011
Summary
This study presents a rigorous mathematical approach to modeling physical solids for computer representation and geometric algorithms. It formalizes 3-D object properties, enhancing accuracy in applications like automated manufacturing.
Area of Science:
- Computer-aided design and manufacturing (CAD/CAM)
- Computational geometry
- Solid modeling
Background:
- Traditional methods for modeling physical solids in computers often suffer from ambiguities and conceptual complications.
- A need exists for a formal, mathematically rigorous approach to ensure correctness and consistency in geometric algorithms.
Purpose of the Study:
- To present a tripartite approach to mathematical solid modeling, computer representation, and geometric algorithm usage.
- To formalize intuitions about three-dimensional (3-D) objects and operations using mathematical definitions.
- To highlight the advantages of a rigorous, semantics-based approach over ad hoc methods.
Main Methods:
- Developing mathematical definitions to formalize 3-D object properties.
- Utilizing representation-free models and functions to define formal properties of geometric representations.
- Describing common 3-D object representation schemes and their formal/informal properties.
Main Results:
- A formal framework is established for mathematical solid modeling and its computer representation.
- Representation-free models enable the characterization of geometric representations.
- The benefits of mathematical rigor in avoiding ambiguities are demonstrated.
Conclusions:
- A rigorous mathematical approach to solid modeling ensures correctness and validity, crucial for applications like automated manufacturing.
- Formal semantics resolve conceptual complications inherent in ad hoc modeling.
- The presented approach has broad significance beyond discrete goods manufacturing.
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