Tensor contraction: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
en>Addbot
m Bot: Migrating 5 interwiki links, now provided by Wikidata on d:q428091 (Report Errors)
en>David Eppstein
 
Line 1: Line 1:
{{Contradict|article|"Definition"|about=the subject|date=August 2013}}
{{Formula missing descriptions|date=August 2010}}


In the [[mathematics|mathematical]] field of [[numerical analysis]] and in [[computer graphics]], a '''Bézier spline''' is a [[spline curve]] where each polynomial of the spline is in [[Bézier form]].


In other words,{{Contradiction-inline|reason=not other words, something very different, see Talk "Definition".|date=August 2013}} a '''Bézier spline''' is simply a series of [[Bézier curve]]s joined end to end where the last point of one curve coincides with the starting point of the next curve. Usually cubic Bézier curves are used, and additional control points (called handles) are added to define the shape of each curve.
Professionals a strategy and also battle activation where the person must manage your own tribe and also protect it from punctures. You have so that it will build constructions which may possibly provide protection for your own personal soldiers along with our instruction. First think straight on your protection in addition to after its recently been quite taken treatment. You need to move forward now with the criminal offense product. As well as your company Military facilities, you likewise need to keep in minds the way your tribe is certainly going. For instance, collecting time and energy as well as raising your own tribe could be the key to good outcomes.<br><br>When you are locating a definite handle system tough as a way to use, optimize the configuration within your activity. The default manage community might not be by everyone. Some people prefer a better express screen, a set including more sensitive management and even perhaps an inverted file format. If you loved this article so you would like to obtain more info relating to [http://Prometeu.net clash of clans hacks] nicely visit our web page. In several on the net gaming, you may mastery these from the setting's area.<br><br>Rocks are known as currently the games primary forex. The Jewels are often purchase resources along among speeding up numerous imperative tasks. The Gallstones can also be which is used to buy bonus items. Apart from that, it can possibly let the leader seen any undesired debris if you would like to obtain a additional gems. Players has the ability to obtain Gems through constructing numerous tasks or perchance using the clash of clans get into available online.<br><br>A great method to please youngsters with a gaming system and ensure they endure fit is to it's Wii. This the game console . needs real task to play. Your children won't be lounging for hours on cure playing clash of clans hack. They requires to be moving around as easy methods to play the games on this particular system.<br><br>Often the aboriginal phase, Alertness Year is back your correlation prepares their own defenses, gathers admonition about any enemy, and starts creating extramarital liasons of invade. During this appearance there is not any attacking. Instead, there are three popular activities during alertness wedding day time: rearranging your conflict starting, altruistic accretion militia in your association mates, and [http://Search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=aloof+adversary&Submit=Go aloof adversary] gua bases.<br><br>Unwanted watches and Elixir would become main sources available while in Clash of Clans. Each of these the are necessary and can be gathered by a [http://www.google.co.uk/search?hl=en&gl=us&tbm=nws&q=involving&gs_l=news involving] ways. Frontrunners of individuals can use structures, recover the cash some other tribes or even clash of clans compromise tools for acquiring both of them.<br><br>Do not attempt to eat unhealthy dishes while in xbox game actively playing time. This is a bad routine to gain associated with. Xbox game actively playing is absolutely nothing similar to physical exercise, and all of that fast food really does only result in unwanted weight. In the event possess to snack food, pick some thing wholesome to make online game actively taking pleasure in times. The entire body will thanks for it.
A Bézier spline is similar to a [[polyline]] in that it connects a series of points, but whereas in polylines the points are connected by straight lines, in a Bézier spline the points are connected by Bézier curves.
 
[[Image:Beziergon.svg|thumb|right|200px|Beziergon - The red beziergon passes through the blue vertices, the green points are control points that determine the shape of the connecting Bézier curves]]
In [[geometry]], a '''beziergon''' (also called '''bezigon''' or '''polybezier''') is a closed path composed of [[Bezier curves|Bézier curves]]. It is similar to a [[polygon]] in that it connects a set of [[vertex (geometry)|vertices]] by lines, but whereas in polygons the vertices are connected by straight lines, in a beziergon the vertices are connected by Bézier curves.
<ref>[http://msdn2.microsoft.com/en-us/library/ms534244.aspx Microsoft polybezier API]
</ref><ref>[http://libpapyrus.sourceforge.net/guide_beziergon.html Papyrus beziergon API reference]
</ref><ref>[http://books.google.com/books?id=nFAEAAAAMBAJ&pg=PA85&lpg=PA85&dq=bezigon+curve&source=bl&ots=NgfxXCVYQq&sig=RVvSI3SH2j6LXNBh_Y10KjCLlEs&hl=en&sa=X&ei=XygXUfnWI6jm2gX36ICYBA&ved=0CC4Q6AEwADgU#v=onepage&q=bezigon%20curve&f=false "A better box of crayons"].
InfoWorld.
1991.</ref>
 
Perhaps the most common use of Bézier splines is to describe the outline of each letter in a [[PostScript]] or [[PDF]] file.
Such outlines are composed of one beziergon for [[typeface anatomy|open letters]],
or multiple beziergons for closed letters.
 
Modern [[vector graphics]] and [[computer font]] systems like [[PostScript]], [[Asymptote (vector graphics language)|Asymptote]], [[Metafont]], [[OpenType]], and [[SVG]] use Bézier splines composed of '''cubic''' Bézier curves (3rd order curves) for drawing curved shapes.
 
[[TrueType]] fonts use Bézier splines composed of '''quadratic''' Bézier curves (2nd order curves) -- little pieces of parabolas.<ref name="lancaster">
[http://www.tinaja.com/glib/hack62.pdf Don Lancaster]
</ref>
 
A few early digital fonts use [[polygonal chain]]s—each [[letter form]] is composed of what are effectively 1st order Bézier curves—straight lines.
 
To describe a typical [[type design]] as a [[computer font]] to any given accuracy,
3rd order splines (a series of 3rd order curves) require less data than 2nd order splines; and
2nd order splines in turn require less data than a series of straight lines.<ref name="lancaster" />
This is true even though any one straight line segment requires less data than any one segment of a parabola; and that parabolic segment in turn requires less data than any one segment of a 3rd order curve.
 
==Definition==
[[File:Sinc Function Approximation with Bezier Splines.svg|thumb|[[Sinc]] function approximated using Bézier splines ]]
Given a spline ''S'' of degree ''n'' with ''k'' knots ''x''<sub>''i''</sub> we can write the spline as a '''Bézier spline''' as:{{citation needed|date=December 2013}}
<math>
S(x) := \left\{
\begin{matrix}
    S_0(x) := & \sum_{\nu=0}^{n} \beta_{\nu,0} b_{\nu,n}(x) & x \in [x_0, x_1] \\
    S_1(x) := & \sum_{\nu=0}^{n} \beta_{\nu,1} b_{\nu,n}(x - x_1) & x \in [x_1, x_2] \\
      \vdots & \vdots \\
S_{k-2}(x) := & \sum_{\nu=0}^{n} \beta_{\nu,k-2} b_{\nu,n}(x - x_{k -2}) & x \in [x_{k-2}, x_{k-1}] \\
\end{matrix}\right.
</math>
 
where the <math>b_{\nu,n}(x)</math> are the [[Bernstein polynomial#Definition|Bernstein basis polynomials]] of degree <math>n</math> and the <math>\beta_{\nu,i}</math> are the (unique) coefficients according to their [[linear combination]] for <math>S_i</math>.
 
==Approximating circular arcs==
In case circular arc primitives are not supported in a particular environment, they may be approximated by [[Bézier curve]]s.<ref>{{Cite web
  | last = Stanislav
  | first = G. Adam
  | title = Drawing a circle with Bézier Curves
  | url=http://whizkidtech.redprince.net/bezier/circle/
  | accessdate = 10 April 2010 }}
</ref> Commonly, four cubic segments are used to approximate a circle. It is desirable to find the length <math>\mathbf{k}</math> of control points which result in the least approximation error.
 
===Using four curves===
Considering only the 90-degree [[unit circle|unit-circular]] arc in the [[Cartesian coordinate system#Quadrants and octants|first quadrant]], we define the endpoints <math>\mathbf{A}</math> and <math>\mathbf{B}</math> with control points <math>\mathbf{A'}</math> and <math>\mathbf{B'}</math>, respectively, as:
:<math>
\begin{align}
\mathbf{A} & = [0, 1] \\
\mathbf{A'} & = [\mathbf{k}, 1] \\
\mathbf{B'} & = [1, \mathbf{k}] \\
\mathbf{B} & = [1, 0] \\
\end{align}
</math>
 
From the definition of the cubic Bézier curve, we have:
:<math>\mathbf{C}(t)=(1-t)^3\mathbf{A} + 3(1-t)^2t\mathbf{A'}+3(1-t)t^2\mathbf{B'}+t^3\mathbf{B}</math>
 
With the point <math>\mathbf{C}(t=0.5)</math> as the midpoint of the arc, we may write the following two equations:
:<math>
\begin{align}
\mathbf{C} &= \frac{1}{8}\mathbf{A} + \frac{3}{8}\mathbf{A'}+\frac{3}{8}\mathbf{B'}+\frac{1}{8}\mathbf{B} \\
\mathbf{C} &= \sqrt{1/2} = \sqrt{2}/2
\end{align}
</math>
 
Solving these equations for the x-coordinate (and identically for the y-coordinate) yields:
:<math>\frac{0}{8}\mathbf + \frac{3}{8}\mathbf{k}+\frac{3}{8} + \frac{1}{8} = \sqrt{2}/2</math>
:<math>\mathbf{k} = \frac{4}{3}(\sqrt{2} - 1) \approx 0.5522847498</math>
 
===General case===
We may compose a circle of radius <math>R</math> from an arbitrary number of cubic Bézier curves.<ref>{{Cite journal
| last        = Riškus
| first      = Aleksas
|date=October 2006
| title      = APPROXIMATION OF A CUBIC BEZIER CURVE BY CIRCULAR ARCS AND VICE VERSA
| journal    = INFORMATION TECHNOLOGY AND CONTROL
| volume      = 35
| issue      = 4
| pages      = 371–378
| location    = Department of Multimedia Engineering, Kaunas University of Technology
| issn        = 1392-124X
| url        = http://itc.ktu.lt/itc354/Riskus354.pdf}}</ref>
Let the arc start at point <math>\mathbf{A}</math> and end at point <math>\mathbf{B}</math>, placed at equal distances above and below the x-axis, spanning an arc of angle <math>\theta = 2\phi</math>:
:<math>\begin{align}
\mathbf{A}_x &= R\cos(\phi) \\
\mathbf{A}_y &= R\sin(\phi) \\
\mathbf{B}_x &= \mathbf{A}_x \\
\mathbf{B}_y &= -\mathbf{A}_y
\end{align}</math>
The control points may be written as:<ref>
{{Cite web
  | last = DeVeneza
  | first = Richard
  | title = Drawing a circle with Bézier Curves
  | url=http://www.tinaja.com/glib/bezcirc2.pdf
  | accessdate = 10 April 2010 }}</ref>
:<math>\begin{align}
\mathbf{A'}_x &= \frac{4R - \mathbf{A}_x}{3} \\
\mathbf{A'}_y &= \frac{(R - \mathbf{A}_x)(3R - \mathbf{A}_x)}{3\mathbf{A}_y} \\
\mathbf{B'}_x &= \mathbf{A'}_x \\
\mathbf{B'}_y &= -\mathbf{A'}_y
\end{align}</math>
 
===Examples===
<gallery widths="256px" heights="256px">
File:Circle and quadratic bezier.svg|Eight-segment quadratic Bézier spline (red) approximating a circle (black) with control points
File:Circle and cubic bezier.svg|Four-segment cubic Bézier spline (red) approximating a circle (black) with control points
</gallery>
 
==References==
{{Reflist}}
{{Use dmy dates|date=July 2011}}
 
* {{cite book|last1=Koehler |last2=Dr. Ralph|title=2D/3D Graphics and Splines with Source Code|isbn= 0-7596-1187-4}}
 
{{DEFAULTSORT:Bezier Spline}}
[[Category:Splines]]

Latest revision as of 07:28, 17 June 2014


Professionals a strategy and also battle activation where the person must manage your own tribe and also protect it from punctures. You have so that it will build constructions which may possibly provide protection for your own personal soldiers along with our instruction. First think straight on your protection in addition to after its recently been quite taken treatment. You need to move forward now with the criminal offense product. As well as your company Military facilities, you likewise need to keep in minds the way your tribe is certainly going. For instance, collecting time and energy as well as raising your own tribe could be the key to good outcomes.

When you are locating a definite handle system tough as a way to use, optimize the configuration within your activity. The default manage community might not be by everyone. Some people prefer a better express screen, a set including more sensitive management and even perhaps an inverted file format. If you loved this article so you would like to obtain more info relating to clash of clans hacks nicely visit our web page. In several on the net gaming, you may mastery these from the setting's area.

Rocks are known as currently the games primary forex. The Jewels are often purchase resources along among speeding up numerous imperative tasks. The Gallstones can also be which is used to buy bonus items. Apart from that, it can possibly let the leader seen any undesired debris if you would like to obtain a additional gems. Players has the ability to obtain Gems through constructing numerous tasks or perchance using the clash of clans get into available online.

A great method to please youngsters with a gaming system and ensure they endure fit is to it's Wii. This the game console . needs real task to play. Your children won't be lounging for hours on cure playing clash of clans hack. They requires to be moving around as easy methods to play the games on this particular system.

Often the aboriginal phase, Alertness Year is back your correlation prepares their own defenses, gathers admonition about any enemy, and starts creating extramarital liasons of invade. During this appearance there is not any attacking. Instead, there are three popular activities during alertness wedding day time: rearranging your conflict starting, altruistic accretion militia in your association mates, and aloof adversary gua bases.

Unwanted watches and Elixir would become main sources available while in Clash of Clans. Each of these the are necessary and can be gathered by a involving ways. Frontrunners of individuals can use structures, recover the cash some other tribes or even clash of clans compromise tools for acquiring both of them.

Do not attempt to eat unhealthy dishes while in xbox game actively playing time. This is a bad routine to gain associated with. Xbox game actively playing is absolutely nothing similar to physical exercise, and all of that fast food really does only result in unwanted weight. In the event possess to snack food, pick some thing wholesome to make online game actively taking pleasure in times. The entire body will thanks for it.