Minimum area circumscribing Polygons

Aggarwal, Alok; Chang, Jyun-Sheng; Yap, Chee

doi:10.1007/bf01898354

Cited by 43 publications

(36 citation statements)

References 8 publications

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“…Aggarwal et al [2] have shown that the minimum-area circumscribing k-gon can be found in O(n 2 log n log k) time. They made use of an interleaving lemma similar to the ones described above, and also used a similar dynamic programming phase consisting of steps in which the maximum value in each row of a matrix must be found.…”

Section: Finding the Extremal Polygons Of A Convex Polygonmentioning

confidence: 99%

Geometric applications of a matrix-searching algorithm

Aggarwal¹,

Klawe

Moran³

et al. 1987

Algorithmica

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354

171

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Let A be a matrix with real entries and let j(i) be the index of the leftmost column containing the maximum value in row i ofA. A is said to be monotone if il > i2 implies thatj(il) >-j(i2). A is totally monotone if all of its submatrices are monotone. We show that finding the maximum entry in each row of an arbitrary n x m monotone matrix requires O(m log n) time, whereas if the matrix is totally monotone the time is O(m) when m-> n and is O(m(l+log(n/m))) when m < n.The problem of finding the maximum value within each row of a totally monotone matrix arises in several geometric algorithms such as the all-farthest-neighbors problem for the vertices of a convex polygon. Previously only the property of monotonicity, not total monotonicity, had been used within these algorithms. We use the | bound on finding the maxima of wide totally monotone matrices to speed up these algorithms by a factor of log n.

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Section: Finding the Extremal Polygons Of A Convex Polygonmentioning

confidence: 99%

Geometric applications of a matrix-searching algorithm

Aggarwal¹,

Klawe

Moran³

et al. 1987

Algorithmica

Self Cite

354

171

View full text Add to dashboard Cite

show abstract

“…Such a proof would most likely be nontrivial. Since we also did not have an implementation of the algorithm described in [Aggarwal85] available to us, we could not do any empirical comparisons between our approximations and the true minimum areas. However, from manual inspection of our results, our algorithm always produced results that are within our expectation of being good approximations of the smallest possible quadrilaterals.…”

Section: Discussionmentioning

confidence: 99%

“…Aggarwal et al presented a general technique to compute the smallest convex k-sided polygon to enclose a given convex n-sided polygon [Aggarwal85]. Their method runs in O(n 2 log n log k) time, however, and can be difficult to implement.…”

Section: Computing Tight Bounding Quadrilateralmentioning

confidence: 99%

Computing a View Frustum to Maximize an Object’s Image Area

Low¹,

Ilie²

2005

Graphics Tools---the JGT Editors' Choice

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This paper presents a method to compute a view frustum for a 3D object viewed from a given viewpoint, such that the object is completely enclosed in the frustum and the object's image area is also near-maximal in the given 2D rectangular viewing region. This optimization can be used to improve the resolution of shadow maps and texture maps for projective texture mapping. Instead of doing the optimization in 3D space to find a good view frustum, our method uses a 2D approach. The basic idea of our approach is as follows. First, from the given viewpoint, a conveniently-computed view frustum is used to project the 3D vertices of the object to their corresponding 2D image points. A tight 2D bounding quadrilateral is then computed to enclose these 2D image points. Next, considering the projective warp between the bounding quadrilateral and the rectangular viewing region, our method applies a technique of camera calibration to compute a new view frustum that generates an image that covers the viewing region as much as possible.

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“…were dealt with in [6] and [1], with an improvement in [2]. Another approach, pursued in [12] and [13], provides simple, linear time algorithms that guarantee a best-order, worst-case (symmetric metric) error (instead of best possible error for the specific polygon).…”

Section: Introductionmentioning

confidence: 98%

Hausdorff approximation of convex polygons

Lopez

Reisner

2005

Computational Geometry

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We develop algorithms for the approximation of convex polygons with n vertices by convex polygons with fewer (k) vertices. The approximating polygons either contain or are contained in the approximated ones. The distance function between convex bodies which we use to measure the quality of the approximation is the Hausdorff metric. We consider two types of problems: min -#, where the goal is to minimize the number of vertices of the output polygon, for a given distance ε, and min -ε, where the goal is to minimize the error, for a given maximum number of vertices. For min -# problems, our algorithms are guaranteed to be within one vertex of the optimal, and run in O(n log n) and O(n) time, for inner and outer approximations, respectively. For min -ε problems, the error achieved is within an arbitrary factor α > 1 from the best possible one, and our inner and outer approximation algorithms run in O(f (α, P ) · n log n) and O(f (α, P ) · n) time, respectively. Where the factor f (α, P ) has reciprocal logarithmic growth as α decreases to 1, this factor depends on the shape of the approximated polygon P .  (M.A. Lopez), reisner@math.haifa.ac.il (S. Reisner).

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Minimum area circumscribing Polygons

Abstract: We show that the smallest k-gon circumscribing a convex n-gon can be computed in O (n z log n log k) time.

Cited by 43 publications

References 8 publications

Geometric applications of a matrix-searching algorithm

Geometric applications of a matrix-searching algorithm

Computing a View Frustum to Maximize an Object’s Image Area

Hausdorff approximation of convex polygons

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