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TECHNICAL PAPERS

# Geometric Interoperability With Epsilon Solidity

[+] Author and Article Information
Jianchang Qi

Spatial Automation Laboratory, Department of Mechanical Engineering, University of Wisconsin–Madison, 1513 University Avenue, Madison, WI 53706qi@students.wisc.edu

Spatial Automation Laboratory, Department of Mechanical Engineering, University of Wisconsin–Madison, 1513 University Avenue, Madison, WI 53706vshapiro@engr.wisc.edu

See Section 3 for additional discussion and examples.

In othor words, every translation can be viewed as a unit process that may or may not introduce additional errors.

As there are inconsistencies between the two papers, we refer to the newer paper (21) for discussion.

Similar translation problems are common whenever tangent surfaces are approximated in the course of translation.

It can be argued that all geometric algorithms sooner or later reduce to a finite number of point membership tests (43).

This is not to say that rule mapping is a trivial issue; for example, the outstanding issue of persistent naming continues to undermine the ability to translate and exchange parametric representations of solids.

For example, a point $x$ is said to be in the $ε$-closure of set $X$, denoted $kε(X)$, if for every $r>ε$, an open ball $B(x,r)$ of radius $r$ centered at $x$ intersects the set $X$. The operations of $ε$-interior$iε$,$ε$-exterior$eε$and$ε$-boundary$∂ε$ are defined similarly.

First articulation of this principle is often attributed to Whitney in the context of manufacturing of fire arms, but it apparently was also employed by others much earlier, for example in the manufacture of clocks at the beginning of the 1700s (45).

J. Comput. Inf. Sci. Eng 6(3), 213-220 (Aug 01, 2005) (8 pages) doi:10.1115/1.2218367 History: Received August 09, 2004; Revised August 01, 2005

## Abstract

Geometric data interoperability is critical in industrial applications where geometric data are transferred (translated) among multiple modeling systems for data sharing and reuse. A big obstacle in data translation lies in that geometric data are usually imprecise and geometric algorithm precisions vary from system to system. In the absence of common formal principles, both industry and academia embraced ad hoc solutions, costing billions of dollars in lost time and productivity. This paper explains how the problem of interoperability, and data translation in particular, may be formulated and studied in terms of a recently developed theory of $ε$-solidity. Furthermore, a systematic classification of problems in data translation shows that in most cases $ε$-solids can be maintained without expensive and arbitrary geometric repairs.

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## Figures

Figure 1

A generic geometric data translation diagram

Figure 2

Even minor changes in geometric primitives during translation may invalidate the model: (a) original model and (b) a failed attempt to repair the translated model

Figure 3

Healing algorithms may drastically change important geometric properties: (a) the original model with smooth blends and (b) an automatically repaired model with sharp corners

Figure 4

Evaluating the same geometric data with different precisions results in inconsistent solids: (a) the original model with small feature in the sending system and (b) the received model with small feature removed after healing

Figure 5

Modeling and representation spaces for ε-solids with limited data accuracy λ and algorithm precision δ

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