doi:10.1016/j.cad.2005.10.010 
Copyright © 2005 Elsevier Ltd All rights reserved.
ε-Topological formulation of tolerant solid modeling
Spatial Automation Laboratory, Department of Mechanical Engineering, University of Wisconsin-Madison, 1513 University Avenue, Madison, Wisconsin 53706, USA
Received 31 August 2005;
accepted 31 October 2005.
Available online 8 February 2006.
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Abstract
Classical theory of solid modeling relies on the notion of regular sets and presupposes exactness in both geometric data and algorithms. In contrast, modeling, exchange and translation of geometric models in engineering applications usually involve data approximations and algorithms with different numerical precisions. We argue that an appropriate formulation of these geometric modeling problems require finite size neighborhoods, leading to the notion of ε-topological operations. These operations are then used to formulate the definitions of ε-regularity and ε-solid that extend and subsume the corresponding classical concepts as exact special cases. Furthermore, the proposed theory suggests how the classical solid modeling paradigm should be extended in order to deal with the outstanding problems in geometric robustness, validation, and data translation. In particular, it explains why the current methods for validating boundary representaetions are not always sufficient and demonstrates that widely adapted geometric repairs are often unnecessary for maintaining solidity in the presence of numerical inaccuracies.
Keywords: Solid modeling; ε-Topological operations; ε-Regularity; ε-Solidity; Geometric robustness; Geometric data translation; Tolerant modeling
Fig. 1. Classical topological operations and the corresponding ε-topological operations. (a) Classical topological operations are defined using infinitesimal ε=0 neighborhoods. (b) ε-topological operations are defined using finite size neighborhoods. (c) Both ε-interior and ε-exterior decrease, and ε-boundary grows as ball radius ε increases.
Fig. 2. A theory of ε-solidity must tolerate imperfections of size less than ε near the theoretical boundary of a solid: (a) dangling edges or isolated points; (b) small cracks and voids; (c) misaligned or redundant vertices and edges.
Fig. 3. ε-Regular sets and intervals tolerate imperfections of size less than ε near the set boundary: (a) growing i0(X) to cover dangling edges and isolated points; (b) shrinking k0(X) to cover small cracks and voids; (c) an interval [X_, X+] covers the imperfection by iε(X+) and kε(X_).
Fig. 4. A set interval becomes ε-regular if the ε is big enough: (a) set interval [X_, X+]; (b) X+
kε(X_); (c) iε(X+)X_.
Fig. 5. Any set contained in an ε-regular interval is ε-regular with the same or smaller value of ε: (a) set instance X with dangling pieces and inner cracks, contained in an ε-regular set interval; (b) Xkεi0(X); (c) iεk0(X)X.
Fig. 6. Comparison of the theoretically exact PMC on CSG representation (top row) and PMCε recognizing limited precision of PMCδ against individual primitives (bottom row). (a) Two 2D closed regular sets A and B, A=k0i0(A), B=k0i0(B). (b) Standard intersection of A and B, A∩B allows dangling ‘face’ of any size. (c) Classical regularized intersection of A and B gives k0i0(A∩B). (d) Testing interior and boundary sets of A and B by finite precision δ-PMC. (e) Neighborhood of point p has to be larger than δ to include points from iδ(A) and iδ(B) for regularization. (f) The ε-regularization removes points which are ε far from iδ(A)∩iδ(B).
Fig. 7. Modeling and representation spaces for ε-solids with limited data accuracy λ and algorithm precision δ.
Fig. 8. Solid validation procedure based on boundary representation is not sufficient under indicated tolerances: (a) vertex merging algorithm indicates that the representation is a single point, and hence is invalid; (b) non-empty iδ and bounded kδ indicate a valid boundary representation of a solid.
Fig. 9. Repair of boundary representation is not necessary to assure validity under tolerances: (a) common errors in boundary representation indicate invalid object; (b) a suitable choice of PMCδ procedure can classify points of the interior and exterior in the presence of errors.
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