Downforce - Revision history

en>AnomieBOT: Dating maintenance tags: {{CN}}

2014-06-11T19:20:33Z

Dating maintenance tags: {{CN}}

← Older revision		Revision as of 21:20, 11 June 2014
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	~~In the [[mathematics\|mathematical]] subfield of [[numerical analysis]] the '''de Boor's algorithm'''~~ is a ~~fast and [[numerically stable]] [[algorithm]] for evaluating [[spline curve]]s in [[B-spline]] form~~. ~~It is~~ a ~~generalization of~~ the ~~[[de Casteljau's algorithm]] for [[Bézier curve]]s~~. ~~The algorithm was devised by [[Carl R. de Boor]]. Simplified, potentially faster variants of~~ the de Boor algorithm have been created but they suffer from comparatively lower stability.<ref>{{cite journal \|last=Lee \|first=E. T. Y. \|date=December 1982 \|title=A Simplified B-Spline Computation Routine \|journal=Computing \|volume=29 \|issue~~=4 \|pages=365–371 \|publisher=Springer-Verlag\|doi=10.1007/BF02246763}}</ref><ref>{{cite journal \| author = Lee~~, ~~E. T. Y. \| journal = Computing \| issue = 3 \| pages = 229–238 \| publisher = Springer-Verlag \| doi=10.1007/BF02240069\|title = Comments on some B-spline algorithms \| volume = 36 \| year = 1986}}</ref>~~		A lagging computer is absolutely annoying plus can be very a headache. Almost every individual that utilizes a computer faces this problem some time or the different. If the computer furthermore suffers within the same issue, there are it difficult to continue working as normal. In such a condition, the thought, "what could I do to create my PC run quicker?" is recurring and infuriating. There's a answer, but!<br><br>Registry is not moreover important to fast computer boot up, and important to the total performance of a computer. If you have a registry error, we might face blue screen, freezing or even crash. It's mandatory to regularly clean up the invalid, lost, junk registry keys to keep the computer healthy plus running swiftly.<br><br>When you compare registry products you want a swiftly acting registry cleaning. It's no good spending hours and the PC waiting for the registry cleaning to complete its task. You wish the cleaner to complete its task inside minutes.<br><br>Paid registry cleaners on the alternative hand, I have found, are usually cheap. They offer standard, free updates or at least inexpensive changes. This follows because the software manufacturer must confirm their product is most effective in staying before its competitors.<br><br>Another usual cause of PC slow down is a corrupt registry. The registry is a important component of computers running on Windows platform. When this gets corrupted a PC will slowdown, or worse, not begin at all. Fixing the registry is simple with the utilize of a system plus [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities 2014].<br><br>Windows relies heavily on this database, storing everything from a latest emails to your Internet favorites in there. Because it's so crucial, a computer is constantly adding plus updating the files inside it. This is okay, nevertheless it could make a computer run slow, when the computer accidentally breaks its important registry files. This is a extremely popular issue, and really makes a computer run slower every day. What arises is that since a computer is frequently using 100's of registry files at when, it sometimes gets confused plus create some of them unreadable. This then makes your computer run slow, considering Windows takes longer to read the files it needs.<br><br>To accelerate your computer, we just should be able to get rid of all these junk files, permitting the computer to find exactly what it wants, when it wants. Luckily, there's a tool which allows you to do this conveniently and immediately. It's a tool called a 'registry cleaner'.<br><br>If you like to have a computer with quick running speed, you'd better install a good registry cleaner to clean the useless files for we. As long as we take care of the computer, it may keep in good condition.

	~~== Introduction ==~~

	~~The general setting is~~ as ~~follows~~. ~~We would like~~ to ~~construct a curve whose shape~~ is ~~described by~~ a ~~sequence of ''p'' points~~ <~~math~~>~~\mathbf{d}_0, \mathbf{d}_1, \dots, \mathbf{d}_{p-1}~~<~~/math~~>, ~~which plays~~ the ~~role~~ of a ~~''control polygon''~~. ~~The curve can be described as~~ a ~~function <math> \mathbf{s}(x)</math> of one parameter ''x''. To pass through the sequence of points, the curve must satisfy <math>\mathbf{s}(u_0)=\mathbf{d}_0~~, ~~\dots,~~
	~~\mathbf{s}(u_{p-1})=\mathbf{d}_{p-1}</math>. But this is not quite the case: in general~~ we ~~are satisfied that the curve "approximates" the control polygon. We assume that ''u<sub>0</sub>~~, .~~.., u<sub>p-1</sub>~~'~~' are given~~ to ~~us along with <math>\mathbf{d}_0~~, ~~\mathbf{d}_1~~, ~~\dots, \mathbf{d}_{p-1}</math>.~~

	~~One approach~~ to ~~solve this problem is by [[spline (mathematics)\|spline]]s~~. ~~A spline is a curve that is a piecewise ''n~~<~~sup~~>th<~~/sup~~>~~'' degree polynomial~~. ~~This means that, on any interval~~ '~~'<nowiki>[</nowiki>u<sub>i</sub>, u<sub>i+1</sub>)'',~~ the ~~curve must be equal~~ to ~~a polynomial of degree at most ''n''~~. ~~It may be equal~~ to ~~different polynomials on different intervals~~. ~~The polynomials must be ''synchronized'': when the polynomials from intervals ''~~<~~nowiki~~>[<~~/nowiki>u<sub>i-1</sub~~>, ~~u<sub>i</sub>)'' and ''<nowiki>[</nowiki>u<sub>i</sub>~~, ~~u<sub>i+1</sub>)'' meet~~ at the ~~point ''u<sub>i</sub>'', they~~ must ~~have the same value at this point and~~ their ~~derivatives must be equal (to ensure that the curve~~ is ~~smooth)~~.

	~~De Boor's algorithm is an algorithm which, given ''u~~<~~sub~~>0<~~/sub~~>~~, .~~.., ~~u<sub>p-1</sub>'' and <math>\mathbf{d}_0~~, ~~\mathbf{d}_1, \dots, \mathbf{d}_{p-1}</math>, finds the value of spline curve <math>\mathbf{s}(x)</math>~~ at ~~a point ''x''. It uses [[Big O notation\|O]](n<sup>2</sup>) operations~~. ~~Notice that~~ the ~~running time of~~ the ~~algorithm depends only on degree ''n'' and not on the number of points ''p''.~~

	~~== Outline~~ of ~~the algorithm==~~
	~~Suppose we want to evaluate the spline curve for~~ a ~~parameter value <math> x \in~~ [~~u_{\ell},u_{\ell+1}] </math>.~~
	~~We can express the curve as~~

	~~:<math> \mathbf{s}(x) = \sum_{i=0}^{p-1} \mathbf{d}_i N_i^n(x) , </math>~~
	~~where<ref>~~http://~~www~~.~~cs.mtu.edu/~shene/COURSES~~/~~cs3621/NOTES/spline/B~~-~~spline/bspline~~-~~basis~~.~~html~~<~~/ref~~>
	<~~math~~>~~N_i^n(x)=\frac{x-u_i}{u_{i+n}-u_i}N_i^{n-1}(x) + \frac{u_{i+n+1}-x}{u_{i+n+1}-u_{i+1}}N_{i+1}^{n-1}(x)~~ ,~~</math>~~
	~~and~~
	~~<math>N_i^0(x)=\left\{\begin{matrix} 1, & \mbox{if }x \~~in ~~[u_{i},u_{i+1}) \\ 0, & \mbox{otherwise } \end{matrix}\right~~.~~</math>~~


	~~Due to the spline locality property~~,
	~~:<math> \mathbf{s}(x) = \sum_{i=\ell-n}^{\ell} \mathbf{d}_i N_i^n(x) </math>~~
	So the ~~value <math>\mathbf{s}(x)</math>~~ is ~~determined by the control points <math> \mathbf{d}_{\ell-n}~~,~~\mathbf{d}_{\ell-n+1}~~,~~\dots~~,~~\mathbf{d}_{\ell} </math>; the other control points <math>\mathbf{d}_i</math> have no influence~~. ~~De Boor~~'s ~~algorithm~~, ~~described in the next section~~, ~~is a procedure which efficiently calculates~~ the ~~expression for~~ <~~math~~> ~~\mathbf{s}(x)~~ <~~/math~~>.

	~~== The algorithm ==~~

	~~Suppose <math> x \in [u_{\ell}~~,~~u_{\ell+1}) </math>~~ and ~~<math> \mathbf{d}_i^{[0]} = \mathbf{d}_i </math> for <math>i = \ell-n, \dots, \ell</math>~~.
	~~Now calculate~~
	:<~~math~~> ~~\mathbf{d}_i^{[k]} = (1-\alpha_{k,i}) \mathbf{d}_{i-1}^{[k-1]} + \alpha_{k,i} \mathbf{d}_i^{[k-1]}; \qquad k=1,\dots,n; \quad i=\ell-n+k,\dots,\ell~~ <~~/math~~>
	with
	~~:<math> \alpha_{k~~,~~i} = \frac{x-u_i}{u_{i+n+1-k}-u_i}. </math>~~
	~~Then <math> \mathbf{s}(x) = \mathbf{~~d~~}_{\ell}^{[n]} </math>~~.

	~~== See also ==~~
	*[[De Casteljau's algorithm]]
	*[[Bézier curve]]
	*[[NURBS]]

	~~== External links ==~~
	*[http://www.cs.~~mtu.edu/~shene/COURSES/cs3621/NOTES/spline/de-Boor.html De Boor's Algorithm]~~
	*[http://www.css.taylor.edu/~btoll/s99/424/res/ucdavis/CAGDNotes/Deboor-Cox-Calculation/Deboor-Cox-Calculation.html The DeBoor-Cox Calculation]

	~~== References ==~~
	~~<references/>~~

	~~[[Category:Numerical analysis]]~~
	~~[[Category:Splines]]~~
	~~[[Category:Interpolation]]~~

2001:630:D0:ED07:C586:5E92:1DBB:87FF: /* Fundamental principles */

2014-01-18T16:43:44Z

Fundamental principles

← Older revision		Revision as of 18:43, 18 January 2014
Line 1:		Line 1:
			In the [[mathematics\|mathematical]] subfield of [[numerical analysis]] the '''de Boor's algorithm''' is a fast and [[numerically stable]] [[algorithm]] for evaluating [[spline curve]]s in [[B-spline]] form. It is a generalization of the [[de Casteljau's algorithm]] for [[Bézier curve]]s. The algorithm was devised by [[Carl R. de Boor]]. Simplified, potentially faster variants of the de Boor algorithm have been created but they suffer from comparatively lower stability.<ref>{{cite journal \|last=Lee \|first=E. T. Y. \|date=December 1982 \|title=A Simplified B-Spline Computation Routine \|journal=Computing \|volume=29 \|issue=4 \|pages=365–371 \|publisher=Springer-Verlag\|doi=10.1007/BF02246763}}</ref><ref>{{cite journal \| author = Lee, E. T. Y. \| journal = Computing \| issue = 3 \| pages = 229–238 \| publisher = Springer-Verlag \| doi=10.1007/BF02240069\|title = Comments on some B-spline algorithms \| volume = 36 \| year = 1986}}</ref>

			== Introduction ==

	~~Track down educational titles~~. ~~It isn~~'~~t generally plainly showcased listed of primary blockbusters when it comes to game stores or electronic portions~~, ~~however are in~~ the ~~vicinity~~ of. ~~Speak to other moms and mothers and fathers or question employees with respect to specific suggestions,~~ as ~~details really exist that information by helping cover his or her~~'~~s learning languages~~, ~~learning concepts and practicing mathematics.~~<br><br>~~Obtain a gaming program for him or her~~. ~~Similar to required assignments time,~~ this ~~video recordings game program will aide you to manage a child~~'~~s tradition. When the times have been set~~, ~~stick to any schedule~~. ~~Do Not necessarily quite back as a result of whining or selling~~. ~~The schedule is only reliable if you just keep going~~.<br><br>~~If you happen~~ to ~~getting a online game for your little one~~, ~~look for one and enables numerous customers to perform with each other. Video gaming can thought of as a solitary action. Nevertheless~~, ~~it is important if you want to motivate your youngster starting to be social~~, ~~and multi~~-~~player clash of clans hack~~ is ~~capable of performing~~ that. ~~They give sisters and brothers on top of~~ that ~~buddies to all~~ on ~~take a moment as laugh and compete against each other.~~<br><br>~~Explore~~ the ~~evaluations and see the trailers before buying~~ a ~~relevant video game~~. ~~Cause it~~ to ~~one thing you are thinking about before you get it~~. ~~These video games aren~~'~~t low~~-~~cost~~, and ~~also you doesn~~'~~t just get nearly as much cash whenever you industry inside a employed game title which you have simply just utilized several times.~~<br><br>~~Your dog~~'s ~~important to agenda your apple~~ is ~~consistently attach from association war complications~~ . ~~because association wars are usually fought inside a improved breadth absolutely -- them war zone~~. ~~When it comes to the war region~~, ~~you adapt~~ and ~~advance confrontation bases instead of approved villages; therefore~~, ~~your towns resources~~, ~~trophies~~, and ~~absorber are never~~ in ~~danger~~.<br><br>~~Make sure that your [~~http://~~browse~~.~~deviantart~~.~~com~~/?q=~~corporation~~+~~build corporation build]~~ and ~~buy a handful new laboratory so you can~~ to ~~research improved barbarians~~. ~~Eventually~~, in ~~predicament you take part involving game~~ for ~~most months~~, ~~you might finally select the nirvana of five~~-~~star barbarians~~.<br><br>~~Basically~~, ~~it would alone acquiesce all of us to make sure you tune 2 volume concerns~~. ~~If you liked this posting and you would like to acquire extra facts concerning clash of clans cheats;~~ [http://~~circuspartypanama~~.~~com resources], kindly check out the site~~. ~~If you appetite for you to songs added than in which probably~~ - ~~as Supercell finely acquainted t had currently all~~-~~important~~ - you phrases assorted beeline segments. Theoretically they could circuit alike added bulk equipment. If they capital to help allegation added in or beneath for a two day skip, they are able to calmly familiarize 1 appended segment.		The general setting is as follows. We would like to construct a curve whose shape is described by a sequence of ''p'' points <math>\mathbf{d}_0, \mathbf{d}_1, \dots, \mathbf{d}_{p-1}</math>, which plays the role of a ''control polygon''. The curve can be described as a function <math> \mathbf{s}(x)</math> of one parameter ''x''. To pass through the sequence of points, the curve must satisfy <math>\mathbf{s}(u_0)=\mathbf{d}_0, \dots,
			\mathbf{s}(u_{p-1})=\mathbf{d}_{p-1}</math>. But this is not quite the case: in general we are satisfied that the curve "approximates" the control polygon. We assume that ''u<sub>0</sub>, ..., u<sub>p-1</sub>'' are given to us along with <math>\mathbf{d}_0, \mathbf{d}_1, \dots, \mathbf{d}_{p-1}</math>.

			One approach to solve this problem is by [[spline (mathematics)\|spline]]s. A spline is a curve that is a piecewise ''n<sup>th</sup>'' degree polynomial. This means that, on any interval ''<nowiki>[</nowiki>u<sub>i</sub>, u<sub>i+1</sub>)'', the curve must be equal to a polynomial of degree at most ''n''. It may be equal to different polynomials on different intervals. The polynomials must be ''synchronized'': when the polynomials from intervals ''<nowiki>[</nowiki>u<sub>i-1</sub>, u<sub>i</sub>)'' and ''<nowiki>[</nowiki>u<sub>i</sub>, u<sub>i+1</sub>)'' meet at the point ''u<sub>i</sub>'', they must have the same value at this point and their derivatives must be equal (to ensure that the curve is smooth).

			De Boor's algorithm is an algorithm which, given ''u<sub>0</sub>, ..., u<sub>p-1</sub>'' and <math>\mathbf{d}_0, \mathbf{d}_1, \dots, \mathbf{d}_{p-1}</math>, finds the value of spline curve <math>\mathbf{s}(x)</math> at a point ''x''. It uses [[Big O notation\|O]](n<sup>2</sup>) operations. Notice that the running time of the algorithm depends only on degree ''n'' and not on the number of points ''p''.

			== Outline of the algorithm==
			Suppose we want to evaluate the spline curve for a parameter value <math> x \in [u_{\ell},u_{\ell+1}] </math>.
			We can express the curve as

			:<math> \mathbf{s}(x) = \sum_{i=0}^{p-1} \mathbf{d}_i N_i^n(x) , </math>
			where<ref>http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/B-spline/bspline-basis.html</ref>
			<math>N_i^n(x)=\frac{x-u_i}{u_{i+n}-u_i}N_i^{n-1}(x) + \frac{u_{i+n+1}-x}{u_{i+n+1}-u_{i+1}}N_{i+1}^{n-1}(x) ,</math>
			and
			<math>N_i^0(x)=\left\{\begin{matrix} 1, & \mbox{if }x \in [u_{i},u_{i+1}) \\ 0, & \mbox{otherwise } \end{matrix}\right.</math>


			Due to the spline locality property,
			:<math> \mathbf{s}(x) = \sum_{i=\ell-n}^{\ell} \mathbf{d}_i N_i^n(x) </math>
			So the value <math>\mathbf{s}(x)</math> is determined by the control points <math> \mathbf{d}_{\ell-n},\mathbf{d}_{\ell-n+1},\dots,\mathbf{d}_{\ell} </math>; the other control points <math>\mathbf{d}_i</math> have no influence. De Boor's algorithm, described in the next section, is a procedure which efficiently calculates the expression for <math> \mathbf{s}(x) </math>.

			== The algorithm ==

			Suppose <math> x \in [u_{\ell},u_{\ell+1}) </math> and <math> \mathbf{d}_i^{[0]} = \mathbf{d}_i </math> for <math>i = \ell-n, \dots, \ell</math>.
			Now calculate
			:<math> \mathbf{d}_i^{[k]} = (1-\alpha_{k,i}) \mathbf{d}_{i-1}^{[k-1]} + \alpha_{k,i} \mathbf{d}_i^{[k-1]}; \qquad k=1,\dots,n; \quad i=\ell-n+k,\dots,\ell </math>
			with
			:<math> \alpha_{k,i} = \frac{x-u_i}{u_{i+n+1-k}-u_i}. </math>
			Then <math> \mathbf{s}(x) = \mathbf{d}_{\ell}^{[n]} </math>.

			== See also ==
			*[[De Casteljau's algorithm]]
			*[[Bézier curve]]
			*[[NURBS]]

			== External links ==
			*[http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/de-Boor.html De Boor's Algorithm]
			*[http://www.css.taylor.edu/~btoll/s99/424/res/ucdavis/CAGDNotes/Deboor-Cox-Calculation/Deboor-Cox-Calculation.html The DeBoor-Cox Calculation]

			== References ==
			<references/>

			[[Category:Numerical analysis]]
			[[Category:Splines]]
			[[Category:Interpolation]]

en>Consolator: /* Book */

2012-08-23T23:38:01Z

Book

New page

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