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| In mathematics, the '''Fréchet distance''' is a measure of similarity between [[curve]]s that takes into account the location and ordering of the points along the curves. It is named after [[Maurice Fréchet]].
| | If an existing Word - Press code is found vulnerable, Word - Press will immediately issue an update for that. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. * A community forum for debate of the product together with some other customers in the comments spot. They found out all the possible information about bringing up your baby and save money at the same time. You can customize the appearance with PSD to Word - Press conversion ''. <br><br>You just download ready made templates to a separate directory and then choose a favorite one in the admin panel. While direct advertising is limited to few spots in your site and tied to fixed monthly payment by the advertisers, affiliate marketing can give you unlimited income as long as you can convert your traffic to sales. A Wordpress plugin is a software that you can install into your Wordpress site. Now, I want to anxiety that not every single query will be answered. For a Wordpress website, you don't need a powerful web hosting account to host your site. <br><br>The least difficult and very best way to do this is by acquiring a Word - Press site. By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply. If Gandhi was empowered with a blogging system, every event in his life would have been minutely documented so that it could be recounted to the future generations. The animation can be quite subtle these as snow falling gently or some twinkling start in the track record which are essentially not distracting but as an alternative gives some viewing enjoyment for the visitor of the internet site. If you have any questions on starting a Word - Press food blog or any blog for that matter, please post them below and I will try to answer them. <br><br>If you cherished this informative article in addition to you would want to be given more info about [http://www.tough-luxe.com/hello-world/ wordpress backup] i implore you to stop by our website. Digg Digg Social Sharing - This plugin that is accountable for the floating social icon located at the left aspect corner of just about every submit. Quttera - Quttera describes itself as a 'Saa - S [Software as a Service] web-malware monitoring and alerting solution for websites of any size and complexity. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. IVF ,fertility,infertility expert,surrogacy specialist in India at Rotundaivf. If your blog employs the permalink function, This gives your SEO efforts a boost, and your visitors will know firsthand what's in the post when seeing the URL. <br><br>Internet is not only the source for information, it is also one of the source for passive income. Sanjeev Chuadhary is an expert writer who shares his knowledge about web development through their published articles and other resource. This allows updates to be sent anyone who wants them via an RSS reader or directly to their email. In addition, Word - Press design integration is also possible. Likewise, professional publishers with a multi author and editor setup often find that Word - Press lack basic user and role management capabilities. |
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| ==Intuitive definition==
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| The Fréchet distance between two curves is the minimum length of a leash required to connect a dog and its owner, constrained on two separate paths, as they walk without backtracking along their respective curves from one endpoint to the other. The definition is symmetric with respect to the two curves. Imagine a dog walking along one curve and the dog's owner walking along the other curve, connected by a leash. Both walk continuously along their respective curve from the prescribed start point to the prescribed end point of the curve. Both may vary their speed, and even stop, at arbitrary positions and for arbitrarily long. However, neither can backtrack. The Fréchet distance between the two curves is the length of the shortest leash (not the shortest leash that is sufficient for all walks, but the shortest leash of all the leashes) that is sufficient for traversing both curves in this manner.
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| ==Formal definition==
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| Let <math>S</math> be a [[metric space]]. A [[curve]] <math>A</math> in <math>S</math> is a [[continuous function (topology)|continuous]] [[map (mathematics)|map]] from the [[unit interval]] into <math>S</math>, i.e., <math>A : [0,1] \rightarrow S</math>. A [[parametric equation|reparameterization]] <math>\alpha</math> of <math>[0,1]</math> is a continuous, [[monotonic function|non-decreasing]], [[surjection]] <math>\alpha: [0,1] \rightarrow [0,1]</math>.
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| Let <math>A</math> and <math>B</math> be two given curves in <math>S</math>. Then, the '''Fréchet distance''' between <math>A</math> and <math>B</math> is defined as the [[infimum]] over all reparameterizations <math>\alpha</math> and <math>\beta</math> of <math>[0,1]</math> of the maximum over all <math>t \in [0,1]</math> of the distance in <math>S</math> between <math>A(\alpha(t))</math> and <math>B(\beta(t))</math>. In mathematical notation, the Fréchet distance <math>F(A,B)</math> is
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| <math>F(A,B) = \inf_{\alpha, \beta}\,\,\max_{t \in [0,1]} \,\, \Bigg \{d \Big ( A(\alpha(t)), \, B(\beta(t)) \Big ) \Bigg \}</math>
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| where <math>d</math> is the [[metric (mathematics)|distance function]] of <math>S</math>.
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| Informally, we can think of the parameter <math>t</math> as "time". Then, <math>A(\alpha(t))</math> is the position of the dog and <math>B(\beta(t))</math> is the position of the dog's owner at time <math>t</math> (or vice versa). The length of the leash between them at time <math>t</math> is the distance between <math>A(\alpha(t))</math> and <math>B(\beta(t))</math>. Taking the infimum over all possible reparametrizations of <math>[0,1]</math> corresponds to choosing the walk along the given paths where the maximum leash length is minimized. The restriction that <math>\alpha</math> and <math>\beta</math> be non-decreasing means that neither the dog nor its owner can backtrack.
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| The Fréchet metric takes into account the flow of the two curves because the pairs of points whose distance contributes to the Fréchet distance sweep continuously along their respective curves. This makes the Fréchet distance a better measure of similarity for curves than alternatives, such as the [[Hausdorff distance]], for arbitrary point sets. It is possible for two curves to have small Hausdorff distance but large Fréchet distance.
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| The Fréchet distance and its variants find application in several problems, from [[morphing]]<ref name="efrat02new">{{citation
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| | last1 = Efrat | first1 = Alon
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| | last2 = Guibas | first2 = Leonidas J. | author2-link = Leonidas J. Guibas
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| | last3 = Har-Peled | first3 = Sariel
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| | last4 = Mitchell | first4 = Joseph S. B. | author4-link = Joseph S. B. Mitchell
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| | last5 = Murali | first5 = T. M.
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| | doi = 10.1007/s00454-002-2886-1
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| | issue = 4
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| | journal = [[Discrete and Computational Geometry]]
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| | pages = 535–569
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| | title = New similarity measures between polylines with applications to morphing and polygon sweeping
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| | url = http://valis.cs.uiuc.edu/~sariel/research/papers/01/morph/morph.pdf
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| | volume = 28
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| | year = 2002}}.</ref> and [[handwriting recognition]]<ref name=sri07frechet>{{citation
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| | last1 = Sriraghavendra | first1 = R.
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| | last2 = Karthik | first2 = K.
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| | last3 = Bhattacharyya | first3 = Chiranjib
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| | contribution = Fréchet distance based approach for searching online handwritten documents
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| | doi = 10.1109/ICDAR.2007.121
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| | pages = 461–465
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| | title = Proc. 9th International Conference on Document Analysis and Recognition (ICDAR '07)
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| | url = http://eprints.iisc.ernet.in/26359/
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| | year = 2007}}.</ref> to [[structural alignment|protein structure alignment]].<ref name="minghui08protein">{{citation
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| | last1 = Minghui | first1 = Jiang
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| | last2 = Ying | first2 = Xu
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| | last3 = Binhai | first3 = Zhu
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| | doi = 10.1142/S0219720008003278
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| | issue = 1
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| | journal = Journal of Bioinformatics and Computational Biology
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| | pages = 51–64
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| | title = Protein structure-structure alignment with discrete Fréchet distance
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| | volume = 6
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| | year = 2008
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| | pmid = 18324745}}.</ref> Alt and Godau<ref name="alt95frechet">{{citation
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| | last1 = Alt | first1 = Helmut
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| | last2 = Godau | first2 = Michael
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| | doi = 10.1142/S0218195995000064
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| | issue = 1–2
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| | journal = International Journal of Computational Geometry and Applications
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| | pages = 75–91
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| | title = Computing the Fréchet distance between two polygonal curves
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| | url = http://www.cs.uu.nl/people/marc/asci/ag-cfdbt-95.pdf
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| | volume = 5
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| | year = 1995}}.</ref> were the first to describe a polynomial-time algorithm to compute the Fréchet distance between two [[polygonal chain|polygonal curves]] in [[Euclidean space]]. The running time of their algorithm is <math>O(mn \cdot \log(mn))</math> for two polygonal curves with ''m'' and ''n'' segments.
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| == The free-space diagram ==
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| [[Image:free-space-diagram.png|thumb|alt=example of a free-space diagram|Free-space diagram of for the red and the blue curve. In contrast to the definition in the text, which uses the parameter interval [0,1] for both curves, the curves are parameterized by arc length in this example.]]
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| An important tool for calculating the Fréchet distance of two curves is the free-space diagram, which was introduced by Alt and Godau.<ref name="alt95frechet"/>
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| The free-space diagram between two curves for a given distance threshold ε is a two-dimensional region in the parameter space that consist of all point pairs on the two curves at distance at most ε:
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| <math>D_\varepsilon(A,B) := \{\,(\alpha,\beta)\in[0,1]^2\mid d( A(\alpha), B(\beta))\le\varepsilon \,\}
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| </math>
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| The Fréchet distance <math>F(A,B)</math> is at most ε if and only if the free-space diagram <math>D_\varepsilon(A,B)</math> contains a path which from the lower left corner to the upper right corner which is monotone both in the horizontal and in the vertical direction.
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| ==Variants==
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| The '''weak Fréchet distance''' is a variant of the classical Fréchet distance without the requirement that the endpoints move monotonically along their respective curves — the dog and its owner are allowed to backtrack to keep the leash between them short. Alt and Godau<ref name="alt95frechet"/> describe a simpler algorithm to compute the weak Fréchet distance between polygonal curves.
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| The '''discrete Fréchet distance''', also called the '''coupling distance''', is an approximation of the Fréchet metric for polygonal curves, defined by Eiter and Mannila.<ref name="eiter94discrete">{{citation
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| | last1 = Eiter | first1 = Thomas
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| | last2 = Mannila | first2 = Heikki | author2-link = Heikki Mannila
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| | publisher = Christian Doppler Laboratory for Expert Systems, TU Vienna, Austria
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| | series = Tech. Report CD-TR 94/64
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| | title = Computing discrete Fréchet distance
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| | url = http://www.kr.tuwien.ac.at/staff/eiter/et-archive/cdtr9464.pdf
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| | year = 1994}}.</ref> The discrete Fréchet distance considers only positions of the leash where its endpoints are located at vertices of the two polygonal curves and never in the interior of an edge. This special structure allows the discrete Fréchet distance to be computed in polynomial time by an easy dynamic programming algorithm.
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| When the two curves are embedded in a metric space other than Euclidean space, such as a [[monotone polygon#3D|polyhedral terrain]] or some Euclidean space with obstacles, the distance between two points on the curves is most naturally defined as the length of the [[shortest path]] between them. The leash is required to be a [[geodesic]] joining its endpoints. The resulting metric between curves is called the '''geodesic Fréchet distance'''.<ref name="efrat02new" /><ref name="cook08geodesic">{{citation
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| | last1 = Cook | first1 = Atlas F., IV
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| | last2 = Wenk | first2 = Carola
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| | publisher = University of Texas at San Antonio
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| | series = Tech. Report CS-TR-2008-0010
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| | title = Geodesic Fréchet distance with polygonal obstacles
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| | url = http://www.win.tue.nl/~acook/papers/CS-TR-2008-010.pdf
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| | year = 2008}}.</ref><ref name="maheshwari05frechet">{{citation
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| | last1 = Maheshwari | first1 = Anil
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| | last2 = Yi | first2 = Jiehua
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| | contribution = On computing Fréchet distance of two paths on a convex polyhedron
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| | pages = 41–44
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| | title = Proc. 21st European Workshop on Computational Geometry
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| | url = http://www.eurocg.org/www.win.tue.nl/EWCG2005/Proceedings/11.pdf
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| | year = 2005}}.</ref> Cook and Wenk<ref name="cook08geodesic"/> describe a polynomial-time algorithm to compute the geodesic Fréchet distance between two polygonal curves in a [[simple polygon]].
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| If we further require that the leash must move continuously in the ambient metric space, then we obtain the notion of the '''homotopic Fréchet distance'''<ref name="chambers09homotopic">{{citation | |
| | last1 = Chambers | first1 = Erin Wolf
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| | last2 = Colin de Verdière | first2 = Éric
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| | last3 = Erickson | first3 = Jeff
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| | last4 = Lazard | first4 = Sylvain
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| | last5 = Lazarus | first5 = Francis
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| | last6 = Thite | first6 = Shripad
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| | doi = 10.1016/j.comgeo.2009.02.008
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| | journal = Computational Geometry: Theory and Applications
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| | title = Homotopic Fréchet distance between curves, or Walking your dog in the woods in polynomial time
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| | url = http://www.ist.caltech.edu/~sthite/pubs/frechet-cgta.pdf
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| | year = 2009
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| | volume = 43
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| | issue = 3
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| | pages = 295–311}}.</ref> between two curves. The leash cannot switch discontinuously from one position to another — in particular, the leash cannot jump over obstacles, and can sweep over a mountain on a terrain only if it is long enough. The motion of the leash describes a [[homotopy]] between the two curves. Chambers ''et al.''<ref name="chambers09homotopic"/> describe a polynomial-time algorithm to compute the homotopic Fréchet distance between polygonal curves in the Euclidean plane with obstacles.
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| ==Examples==
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| The Fréchet distance between two concentric circles of radius <math>r_1</math> and <math>r_2</math> respectively is <math>|r_1 - r_2|.</math>
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| The longest leash is required when the owner stands still and the dog travels to the opposite side of the circle (<math>r_1 + r_2</math>), and the shortest leash when both owner and dog walk at a constant speed around the circle (<math>|r_1 - r_2|</math>).
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| *{{citation
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| | last1 = de Berg| first1 = Mark
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| | contribution = Analyzing Trajectories of Moving Objects
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| | pages = 11–75
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| | title = Computational Geometry, Two Selected Topics
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| | url = http://www.win.tue.nl/~mdberg/Onderwijs/AdvAlg2011/Course%20Notes/AA-geom-lecture4.pdf}}.
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| *{{citation
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| | last1 = Aronov | first1 = Boris | author1-link = Boris Aronov
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| | last2 = Har-Peled | first2 = Sariel
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| | last3 = Knauer | first3 = Christian
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| | last4 = Wang | first4 = Yusu
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| | last5 = Wenk | first5 = Carola
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| | contribution = Fréchet distance for curves, revisited
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| | doi = 10.1007/11841036_8
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| | pages = 52–63
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| | publisher = Springer-Verlag
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| | series = Lecture Notes in Computer Science
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| | title = Proc. 14th European Symposium on Algorithms
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| | url = http://www.cs.utsa.edu/~carola/ESA06A_wang.pdf
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| | volume = 4168
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| | year = 2006}}.
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| *{{citation
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| | last1 = Alt | first1 = Helmut
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| | last2 = Buchin | first2 = Maike
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| | doi = 10.1007/s00454-009-9152-8
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| | journal = [[Discrete and Computational Geometry]]
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| | pages = 78–99
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| | title = Can we compute the similarity between surfaces?
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| | volume = 43
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| | year = 2010
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| | arxiv = cs.CG/0703011}}.
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| {{DEFAULTSORT:Frechet Distance}}
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| [[Category:Metric geometry]]
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| [[Category:Topology]]
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| [[Category:Geometric algorithms]]
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