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Some mathematical and representational aspects of solid modeling.

C M Brown1

  • 1College of Engineering and Applied Science, University of Rochester, Rochester, NY 14627.

IEEE Transactions on Pattern Analysis and Machine Intelligence
|August 27, 2011
PubMed
Summary
This summary is machine-generated.

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.

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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.