128.59.74.3: /* Theory behind the method */

2013-04-19T17:27:57Z

Theory behind the method

← Older revision		Revision as of 19:27, 19 April 2013
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	Hi, ~~everybody! My~~ name is ~~Maude~~. <br>It is a ~~little about myself~~: ~~I live~~ in ~~France~~, ~~my city~~ of ~~Goussainville.~~ <br>~~It'~~s ~~called often Northern or cultural capital~~ of ~~ILE~~-DE-~~FRANCE~~. ~~I've married 2 years ago~~.<br>I ~~have 2 children - a son~~ (~~Annette~~) and ~~the daughter~~ (~~Edna~~). ~~We all like Racquetball~~.<br><br>~~Also visit my~~ web ~~site~~: [http://~~tinyurl~~.com/~~qjrdtau canada goose homme~~]		A '''Kinetic Triangulation''' data structure is a [[kinetic data structure]] that maintains a [[triangulation (geometry)\|triangulation]] of a set of moving points. Maintaining a kinetic triangulation is important for applications that involve [[motion planning]], such as video games, virtual reality, dynamic simulations and robotics.<ref name="micha"/>

			==Choosing a triangulation scheme==
			The efficiency of a kinetic data structure is defined based on the ratio of the number of internal events to external events, thus good runtime bounds can sometimes be obtained by choosing to use a triangulation scheme that generates a small number of external events.
			For simple [[affine motion]] of the points, the number of discrete changes to the [[convex hull]] is [[big O notation#Family of Bachmann–Landau notations\|estimated by]] <math>\Omega(n^2)</math>,<ref name="convex hull"/> thus the number of changes to any triangulation is also lower bounded by <math>\Omega(n^2)</math>. Finding any triangulation scheme that has a near-quadratic bound on the number of discrete changes is an important open problem.<ref name="micha"/>

			===Delaunay triangulation===
			The [[Delaunay triangulation]] seems like a natural candidate, but a tight analysis of the number of discrete changes that will occur to the Delaunay triangulation (external events) is one of the hardest problems in computational geometry,<ref name="delaunay hardness"/> and the best currently known upper bound is <math>O(n^3)</math>.

			There is a kinetic data structure that [[Kinetic data structure#Performance\|efficiently]] maintains the Delaunay triangulation of a set of moving points,<ref>Gerhard Albers, Leonidas J. Guibas, Joseph S. B. Mitchell, and Thomas Roos. Voronoi diagrams of moving points. Int. J. Comput. Geometry Appl., 8(3):365{380, 1998.</ref> in which the ratio of the total number of events to the number of external events is <math>O(1)</math>.

			===Other triangulations===
			Kaplan et al. developed a [[randomized algorithm\|randomized]] triangulation scheme that experiences an expected number of <math>O(n^2 \beta_{s+2}(n) \log^2 n)</math> external events, where <math>s</math> is the maximum number of times each triple of points can become collinear, <math>\beta_{s+2}(q) = \frac{\lambda_{s+2}(q)}{q}</math>, and <math>\lambda_{s+2}(q)</math> is the maximum length of a [[Davenport-Schinzel sequence]] of order s + 2 on n symbols.<ref name="micha"/>

			===Pseudo-triangulations===
			There is a kinetic data structure (due to Agarwal et al.) which maintains a [[pseudo-triangulation]] in <math>O(n^22^{\sqrt{\log n\log\log n}})</math> events total.<ref>Pankaj K. Agarwal, Julien Basch, Leonidas J. Guibas, John Hershberger, and Li Zhang. Deformable free-space tilings for kinetic collision detection. I. J. Robotic Res., 21(3):179{198, 2002. [http://research.microsoft.com/en-us/um/people/lzha/papers/pset-j.pdf]</ref> All events are [[Kinetic data structure#Certificates Approach\|external]] and require <math>O(\lg n)</math> time to process.

			== References ==
			{{Reflist\|refs=
			<ref name="micha">
			{{cite conference \| url=http://www.math.tau.ac.il/~michas/triank.pdf \| title=A Kinetic Triangulation Scheme for Moving Points in The Plane \| publisher=ACM \| accessdate=May 19, 2012 \| author=Kaplan, Haim; Rubin, Natan; Sharir, Micha \| year=2010 \| month=June \|conference=SCG}}
			</ref>
			<ref name="convex hull">
			{{cite book \| title=Davenport-Schinzel sequences and their geometric applications \| publisher=, Cambridge University Press \| author=Sharir, M,; Agarwal, P.K. \| year=1995 \| location=New York}}
			</ref>
			<ref name="delaunay hardness">
			{{cite web \| url=http://www.cs.smith.edu/~orourke/TOPP/ \| title=The Open Problems Project \| accessdate=May 19, 2012 \| author=Demaine, E.D.; Mitchell, J. S. B. ; O’Rourke, J.}}
			</ref>
			}}
			*Pankaj K. Agarwal, Julien Basch, Mark de Berg, Leonidas J. Guibas, and John Hershberger. Lower bounds for kinetic planar subdivisions. In SCG '99: Proceedings of the fifteenth annual symposium on Computational geometry, pages 247{254, New York, NY, USA, 1999. ACM.[http://research.microsoft.com/en-us/um/people/lzha/papers/pset-j.pdf]

			<!--- Categories --->

			[[Category:Articles created via the Article Wizard]]
			[[Category:Kinetic data structures]]
			[[Category:Triangulation (geometry)]]

en>Darth Vader's Dad: /* References */ Added external link to mentioned web site

2012-05-28T11:00:55Z

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Hi, everybody! My name is Maude. It is a little about myself: I live in France, my city of Goussainville. It's called often Northern or cultural capital of ILE-DE-FRANCE. I've married 2 years ago. I have 2 children - a son (Annette) and the daughter (Edna). We all like Racquetball. Also visit my web site: [http://tinyurl.com/qjrdtau canada goose homme]

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128.59.74.3: /* Theory behind the method */

en>Darth Vader's Dad: /* References */ Added external link to mentioned web site