Maximizing the overlap of two planar convex sets under rigid motions

Hee Kap Ahn; Otfried Cheong; Chong Dae Park; Chan Su Shin; Antoine Vigneron

doi:10.1016/j.comgeo.2006.01.005

Maximizing the overlap of two planar convex sets under rigid motions

Hee Kap Ahn, Otfried Cheong^*, Chong Dae Park, Chan Su Shin, Antoine Vigneron

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

21 Scopus citations

Abstract

Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any >0, we compute a rigid motion such that the area of overlap is at least 1- times the maximum possible overlap. Our algorithm uses O(1/) extreme point and line intersection queries on P and Q, plus O((1/ ²)log(1/)) running time. If only translations are allowed, the extra running time reduces to O((1/)log(1/)). If P and Q are convex polygons with n vertices in total that are given in an array or balanced tree, the total running time is O((1/)logn+(1/²)log(1/)) for rigid motions and O((1/)logn+(1/)log(1/)) for translations.

Original language	English (US)
Pages (from-to)	3-15
Number of pages	13
Journal	Computational Geometry: Theory and Applications
Volume	37
Issue number	1 SPEC. ISS.
DOIs	https://doi.org/10.1016/j.comgeo.2006.01.005
State	Published - May 2007
Externally published	Yes

Bibliographical note

Funding Information:
✩ Work by the first three authors was supported by the Brain Korea 21 Project, The School of Information Technology, KAIST, in 2005. Work by the fourth author was supported by the Hankuk University of Foreign Studies Research Fund of 2005. Work by the fifth author was supported by the National University of Singapore under grant R-252-000-166-112. * Corresponding author. E-mail addresses: [email protected] (H.-K. Ahn), [email protected] (O. Cheong), [email protected] (C.-D. Park), [email protected] (C.-S. Shin), [email protected] (A. Vigneron).

Keywords

Approximation algorithm
Convex shape
Geometric pattern matching
Sublinear algorithm

ASJC Scopus subject areas

Computer Science Applications
Geometry and Topology
Control and Optimization
Computational Theory and Mathematics
Computational Mathematics

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10.1016/j.comgeo.2006.01.005

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title = "Maximizing the overlap of two planar convex sets under rigid motions",

abstract = "Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any >0, we compute a rigid motion such that the area of overlap is at least 1- times the maximum possible overlap. Our algorithm uses O(1/) extreme point and line intersection queries on P and Q, plus O((1/ 2)log(1/)) running time. If only translations are allowed, the extra running time reduces to O((1/)log(1/)). If P and Q are convex polygons with n vertices in total that are given in an array or balanced tree, the total running time is O((1/)logn+(1/2)log(1/)) for rigid motions and O((1/)logn+(1/)log(1/)) for translations.",

keywords = "Approximation algorithm, Convex shape, Geometric pattern matching, Sublinear algorithm",

author = "Ahn, {Hee Kap} and Otfried Cheong and Park, {Chong Dae} and Shin, {Chan Su} and Antoine Vigneron",

note = "Funding Information: ✩ Work by the first three authors was supported by the Brain Korea 21 Project, The School of Information Technology, KAIST, in 2005. Work by the fourth author was supported by the Hankuk University of Foreign Studies Research Fund of 2005. Work by the fifth author was supported by the National University of Singapore under grant R-252-000-166-112. * Corresponding author. E-mail addresses: [email protected] (H.-K. Ahn), [email protected] (O. Cheong), [email protected] (C.-D. Park), [email protected] (C.-S. Shin), [email protected] (A. Vigneron).",

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N1 - Funding Information: ✩ Work by the first three authors was supported by the Brain Korea 21 Project, The School of Information Technology, KAIST, in 2005. Work by the fourth author was supported by the Hankuk University of Foreign Studies Research Fund of 2005. Work by the fifth author was supported by the National University of Singapore under grant R-252-000-166-112. * Corresponding author. E-mail addresses: [email protected] (H.-K. Ahn), [email protected] (O. Cheong), [email protected] (C.-D. Park), [email protected] (C.-S. Shin), [email protected] (A. Vigneron).

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AB - Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any >0, we compute a rigid motion such that the area of overlap is at least 1- times the maximum possible overlap. Our algorithm uses O(1/) extreme point and line intersection queries on P and Q, plus O((1/ 2)log(1/)) running time. If only translations are allowed, the extra running time reduces to O((1/)log(1/)). If P and Q are convex polygons with n vertices in total that are given in an array or balanced tree, the total running time is O((1/)logn+(1/2)log(1/)) for rigid motions and O((1/)logn+(1/)log(1/)) for translations.

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Maximizing the overlap of two planar convex sets under rigid motions

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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