Finding the upper envelope of n line segments in O(n log n) time
Information Processing Letters
Using tolerances to guarantee valid polyhedral modeling results
SIGGRAPH '90 Proceedings of the 17th annual conference on Computer graphics and interactive techniques
Davenport-Schinzel sequences and their geometric applications
Davenport-Schinzel sequences and their geometric applications
Handbook of computational geometry
Handbook of computational geometry
Probabilistic range queries in moving objects databases with uncertainty
Proceedings of the 3rd ACM international workshop on Data engineering for wireless and mobile access
Efficient indexing methods for probabilistic threshold queries over uncertain data
VLDB '04 Proceedings of the Thirtieth international conference on Very large data bases - Volume 30
Tolerance envelopes of planar mechanical parts with parametric tolerances
Computer-Aided Design
Point distance and orthogonal range problems with dependent geometric uncertainties
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling
Uncertain geometry with dependencies
Proceedings of the 14th ACM Symposium on Solid and Physical Modeling
A method for planning safe trajectories in image-guided keyhole neurosurgery
MICCAI'10 Proceedings of the 13th international conference on Medical image computing and computer-assisted intervention: Part III
Geometric computation and optimization on tolerance dimensioning
Computer-Aided Design
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Classical computational geometry algorithms handle geometric constructs whose shapes and locations are exact. However, many real-world applications require the modeling of objects with geometric uncertainties. Existing geometric uncertainty models cannot handle dependencies among objects. This results in the overestimation of errors. We have developed the Linear Parametric Geometric Uncertainty Model, a general, computationally efficient, worst-case, linear approximation of geometric uncertainty that supports dependencies among uncertainties. In this paper, we present the properties of the uncertainty zones of a line and circle, defined using this model, and describe efficient algorithms to compute them. We show that the line's envelope has linear space complexity and is computed in low polynomial time. The circle's envelope has quadratic complexity and is also computed in low polynomial time.