Computing the Angularity Tolerance
TL;DR: This paper studies one particular feature of objects, the angularity, and gives an O(n logn) algorithm for this problem, which indicates how well a plane makes a speci ed angle with another plane.
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Abstract: In computational metrology one needs to compute whether an object satisfies specifications of shape within an acceptable tolerance To this end positions on the object are measured, resulting in a collection of points in space From this collection of points one wishes to extract information on flatness, roundness, etc of the object In this paper we study one particular feature of objects, the angularity The angularity indicates how well a plane makes a specified angle with another plane We study the problem in 2-dimensional space (where the planes become lines) and in 3-dimensional space In 2-dimensional space the problem is equivalent to computing the smallest wedge of the given angle that contains all the points We give an O(n2log n) algorithm for this problem In 3-dimensional space we study the more restricted problem where one of the planes is known (a datum plane) In this case the problem is equivalent to asking for the smallest width 3-dimensional strip that contains all the points and makes a given angle with the datum plane We give an O(n log n) algorithm to solve this version We also show that in the case of uncertainty in the measured points, upperbounds and lowerbounds on the width can be computed in similar time bounds
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Citations
Largest and smallest tours and convex hulls for imprecise points
Maarten Löffler,Marc van Kreveld +1 more
- 06 Jul 2006
TL;DR: In this article, the authors studied the problem of computing the smallest and largest possible tours and convex hulls, measured by length and area, for a set of imprecise points, where each point is specified by a region in which the point may lie.
•Journal Article
Largest and smallest tours and convex hulls for imprecise points
Maarten Löffler,Marc van Kreveld +1 more
TL;DR: This work studies the problem of computing the smallest and largest possible tours and convex hulls, measured by length, and in the latter case also by area, and gives polynomial time algorithms for several variants of this problem.
18
On Chebychev fits for pairs of lines and polygons with specified internal angles
Goutam Chatterjee,Bernard Roth +1 more
TL;DR: In this paper, the problem of determining pairs of lines and polygons that best fit a finite set of datapoints is addressed by minimizing the maximum nromal deviation of the data points from the substitute feature.
10
Computing a double-ray center for a planar point set
TL;DR: A double-ray configuration is a configuration in the plane consisting of two rays emanating from one point, and the goal is to find a double-rays configuration that satisfies these criteria.
10
How to cover a point set with a V-shape of minimum width
Boris Aronov,Muriel Dulieu +1 more
TL;DR: This work presents an O(n^2logn) time algorithm to compute, given a set of n points P, a minimum-width balanced V-shape covering P, and describes a PTAS for computing a (1+@e)-approximation of this V- shape in time O((n/@e)logn+(n/ @e^3^/^2)log^2(1/@ e).
References
•Book
Davenport-Schinzel sequences and their geometric applications
Micha Sharir,Pankaj K. Agarwal +1 more
- 01 Jan 1995
TL;DR: A close to linear bound on the maximum length of Davenport--Schinzel sequences enable us to derive sharp bounds on the combinatorial structure underlying various geometric problems, which in turn yields efficient algorithms for these problems.
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Computing the width of a set
TL;DR: It is shown that W(P) can be computed in O(n log n+I) time and O( n) space, where I is the number of antipodal pairs of edges of the convex hull of P, and n is thenumber of vertices.
Efficient randomized algorithms for some geometric optimization problems
Pankaj K. Agarwal,Micha Sharir +1 more
TL;DR: A general technique that yields faster randomized algorithms for solving a number of geometric optimization problems, including computing the width of a point set in 3-space, computing the minimum-width annulus enclosing a set ofn points in the plane, and computing the “biggest stick” inside a simple polygon inThe plane.