Epsilon geometry: building robust algorithms from imprecise computations
SCG '89 Proceedings of the fifth annual symposium on Computational geometry
Using tolerances to guarantee valid polyhedral modeling results
SIGGRAPH '90 Proceedings of the 17th annual conference on Computer graphics and interactive techniques
Boundary representation modelling with local tolerances
SMA '95 Proceedings of the third ACM symposium on Solid modeling and applications
Handbook of combinatorics (vol. 2)
A new approach to the surface intersection problem
Computer Aided Geometric Design
Handbook of discrete and computational geometry
Consistent calculations for solids modeling
SCG '85 Proceedings of the first annual symposium on Computational geometry
Representations for Rigid Solids: Theory, Methods, and Systems
ACM Computing Surveys (CSUR)
Analysis of boundary representation model rectification
Proceedings of the sixth ACM symposium on Solid modeling and applications
Topological and geometric properties of interval solid models
Graphical Models
Shape Interrogation for Computer Aided Design and Manufacturing
Shape Interrogation for Computer Aided Design and Manufacturing
Computer Aided Geometric Design
SLEVEs for planar spline curves
Computer Aided Geometric Design
Accuracy and semantics in shape-interrogation applications
Graphical Models - Solid modeling theory and applications
A condition for isotopic approximation
SM '04 Proceedings of the ninth ACM symposium on Solid modeling and applications
Set Membership Classification: A Unified Approach to Geometric Intersection Problems
IEEE Transactions on Computers
ε-Topological formulation of tolerant solid modeling
Computer-Aided Design
On the long-term retention of geometry-centric digital engineering artifacts
Computer-Aided Design
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Algorithms for many geometric queries rely on representations that are comprised of combinatorial (logical, incidence) information, usually in a form of a graph or a cell complex, and geometric data that represents embeddings of the cells in the Euclidean space Ed. Whenever geometric embeddings are imprecise, their incidence relationships may become inconsistent with the associated combinatorial model. Tolerant algorithms strive to compute on such representations despite the inconsistencies, but the meaning and correctness of such computations have been a subject of some controversy.This paper argues that a tolerant algorithm usually assumes that the approximate geometric representation corresponds to a subset of Edthat is homotopy equivalent to the intended exact set. We show that the Nerve Theorem provides systematic means for identifying sufficient conditions for the required homotopy equivalence, and explain how these conditions are used in the context of geometric and solid modeling.