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| [[Image:Superellipse star.svg|thumb|200px|right|The superellipse with ''n'' = {{fraction|1|2}}, ''a'' = ''b'' = 1]]
| | == overwhelmed win over the soul of the house == |
| [[Image:Superellipse rounded diamond.svg|thumb|200px|right|The superellipse with ''n'' = {{fraction|3|2}}, ''a'' = ''b'' = 1]]
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| [[Image:Superellipse chamfered square.svg|thumb|200px|right|[[Squircle]], the superellipse with ''n'' = 4, ''a'' = ''b'' = 1]]
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| A '''superellipse''', also known as a '''Lamé curve''' after [[Gabriel Lamé]] is a geometric figure defined in the [[Cartesian coordinate system]] as the set of all points (''x'', ''y'') with
| | Kind of near-blind reverence, like pious believers like.<br><br>when cumulative prestige to some level, but also in the hearts quietly sublimation, and no doubt, these alliance members fear for Xiao Yan, also in this [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ 腕時計 スタンダード] sublimation, the more [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ電波ソーラー腕時計レディース] fanatical, in this case, 'medicine' old, who are somewhat surprised, but it did not stop, people always need to have a faith, because this belief, they will have a high degree of cohesion, and Tianfu alliance among the ability to be this belief, apparently with only Xiao Yan<br><br>league carnival, which lasted two weeks later, just gradually subsided, but although unusual member of the excitement [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ 電波時計 腕時計] a little crazy, but league executives, but it is still maintained a [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 腕時計 メンズ] calm, overwhelmed [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ ソーラー 腕時計] win over the soul of the house, for the prestigious Union does has immeasurable benefits, but since the top of this living place, it will inevitably be many slings and arrows, the past trend of strong soul of the house, in |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://commonnet.no-ip.biz/mediawiki/index.php/User:Peklzzgihfsu#Chapter_XIII_Small_Square http://commonnet.no-ip.biz/mediawiki/index.php/User:Peklzzgihfsu#Chapter_XIII_Small_Square]</li> |
| | |
| | <li>[http://www.jnjn.net/plus/feedback.php?aid=781 http://www.jnjn.net/plus/feedback.php?aid=781]</li> |
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| | <li>[http://www.sqlcourse.com/cgi-bin/interpreter.cgi http://www.sqlcourse.com/cgi-bin/interpreter.cgi]</li> |
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| | </ul> |
|
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|
| : <math>\left|\frac{x}{a}\right|^n\! + \left|\frac{y}{b}\right|^n\! = 1</math>
| | == 'Come on == |
|
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| where ''n'', ''a'' and ''b'' are positive numbers.
| | Tsing Yi with the girls in the juvenile behind.<br><br>'Come on, Ling old, say good-bye.'<br><br>Xiao Yan raised his head, looked before the mountain, he could feel the breath of a large number of forceful, where fast cohesion, and, in the [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html カシオ腕時計 g-shock] other two directions Mountains, also came bursts of spatial fluctuations, it [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ ソーラー 腕時計] seems Yan Lei II strong family is dispatched<br><br>voice down, Xiao Yan is no longer hesitated, holding the hand Kaoru children, stature flash, is swept [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ スタンダード 腕時計] into the sky.<br><br>'Xiao Yan Master, I wish triumphant [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 腕時計 メンズ casio] return!'<br><br>looked back duo Xiao Yan Ling shadow is smile, muttered to himself.<br><br>when Xiao Yan horizon when the two appear in the first mountain, where the air is already shadows scattered, many of the ancient tribe of the strong launch into the sky, forming a neat formation in the sky, the vast surging between breath, like a remote one, that like power and influence, even into the territory of the soul of Emperor [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html casio 腕時計 g-shock] Xiao Yan, are |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://www.ftkx17.com/plus/feedback.php?aid=320 http://www.ftkx17.com/plus/feedback.php?aid=320]</li> |
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| | </ul> |
|
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| This formula defines a [[closed curve]] contained in the [[rectangle]] −''a'' ≤ ''x'' ≤ +''a'' and −''b'' ≤ ''y'' ≤ +''b''. The parameters ''a'' and ''b'' are called the ''semi-diameters'' of the curve.
| | == and under this acid itch == |
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| When ''n'' is between 0 and 1, the superellipse looks like a four-armed star with [[wikt:concave|concave]] (inwards-curved) sides. For ''n'' = 1/2, in particular, each of the four arcs is a [[Bézier curve#Quadratic Bézier curves|Quadratic Bézier curve]] defined by the two axes; as a result, each arc is a segment of a [[parabola]].
| | Gu fine blood into the body, Xiao Yan is covered with the skin immediately harnessed into a bright Jinmang, [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html casio 時計] the vast energy in the body as [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 腕時計 チタン] the storm swept in general open, had exhausted the meridians, almost in an instant, that is once again being gurgle vindictive, quickly filled.<br><br>Moreover, in this energy crisis sweeping the room, Xiao Yan also feel the body's bones, muscles and even the meridians, etc., are among faint heard a strange acid itch, and under this acid itch, as if bones, muscles, etc., are slowly becoming more and more tough and [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html 時計 カシオ] strong<br><br>'This is the essence and blood cologne energy [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ gショック 腕時計] inherent in it, really amazing.'<br><br>feel the change in the body, Xiao Yan could not help but come to mind is some amazing promise of 'color', this ability, almost all can, and some higher-order Dan 'medicine' compared to this really worthy [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html カシオ腕時計 g-shock] cologne Void Top-like existence of Warcraft community, just fine blood force, is that it has |
| | | 相关的主题文章： |
| When ''n'' is 1 the curve is a [[rhombus]] with corners (±''a'', 0) and (0, ±''b''). When ''n'' is between 1 and 2, it looks like a rhombus with those same corners but with [[convex set|convex]] (outwards-curved) sides. The [[curvature]] increases without [[Limit of a sequence|limit]] as one approaches the corners.
| | <ul> |
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| When ''n'' is 2, the curve is an ordinary [[ellipse]] (in particular, a [[circle]] if ''a'' = ''b''). When ''n'' is greater than 2, it looks superficially like a [[rectangle]] with [[chamfer]]ed (rounded) corners. The curvature is zero at the points (±''a'', 0) and (0, ±''b'').
| | <li>[http://www.sedationresource.com/cgi-bin/commerce.cgi http://www.sedationresource.com/cgi-bin/commerce.cgi]</li> |
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| If ''n'' < 2, the figure is also called a '''hypoellipse'''; if ''n'' > 2, a '''hyperellipse'''.
| | <li>[http://www.gziot.org.cn/plus/feedback.php?aid=1271 http://www.gziot.org.cn/plus/feedback.php?aid=1271]</li> |
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| When ''n'' ≥ 1 and ''a'' = ''b'', the superellipse is the boundary of a [[ball (mathematics)|ball]] of '''R'''<sup>2</sup> in the [[norm (mathematics)#p-norm|''n''-norm]].
| | <li>[http://www.jonarne.net/cgi-bin/gbook.cgi http://www.jonarne.net/cgi-bin/gbook.cgi]</li> |
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| The extreme points of the superellipse are (±''a'', 0) and (0, ±''b''), and its four "corners" are (±''sa, ±sb''), where <math>s=2^{-1/n}</math> (sometimes called the "superness" <ref>Donald Knuth: ''The METAFONTbook'', p. 126</ref>).
| | </ul> |
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| ==Mathematical properties==
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| When ''n'' is a nonzero [[rational number]] ''p'' / ''q'' (in lowest terms), then the superellipse is a [[algebraic curve|plane algebraic curve]]. For positive ''n'' the order is ''pq''; for negative ''n'' the order is 2''pq''. In particular, when ''a'' and ''b'' are both one and ''n'' is an even integer, then it is a [[Fermat curve]] of degree ''n''. In that case it is nonsingular, but in general it will be [[Singular point of an algebraic variety|singular]]. If the numerator is not even, then the curve is pasted together from portions of the same algebraic curve in different orientations.
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| [[Image:Lame anima.gif|thumb|200px|right|Animation]]
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| The curve is given by the [[parametric equation]]s
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| :<math>\left.
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| \begin{align}
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| x\left(\theta\right) &= \plusmn a\cos^{\frac{2}{n}} \theta \\
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| y\left(\theta\right) &= \plusmn b\sin^{\frac{2}{n}} \theta
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| \end{align} \right\} \qquad 0 \le \theta < \frac{\pi}{2} </math>
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| or
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| :<math>
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| \begin{align}
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| x\left(\theta\right) &= {|\cos \theta|}^{\frac{2}{n}} \cdot a \sgn(\cos \theta) \\
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| y\left(\theta\right) &= {|\sin \theta|}^{\frac{2}{n}} \cdot b \sgn(\sin \theta).
| |
| \end{align}
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| </math>
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| The area inside the superellipse can be expressed in terms of the [[gamma function]], Γ(''x''), as
| |
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| :<math> \mathrm{Area} = 4 a b \frac{\left(\Gamma \left(1+\tfrac{1}{n}\right)\right)^2}{\Gamma \left(1+\tfrac{2}{n}\right)} . </math>
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| The [[pedal curve]] is relatively straightforward to compute. Specifically, the pedal of
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| :<math>\left(\frac{x}{a}\right)^n\! + \left(\frac{y}{b}\right)^n\! = 1</math>
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| is given in [[polar coordinate]]s by<ref>{{cite book | author=J. Edwards | title=Differential Calculus
| |
| | publisher= MacMillan and Co.| location=London | pages=164| year=1892
| |
| |url=http://books.google.com/books?id=unltAAAAMAAJ&pg=PA164#v=onepage&q&f=false}}</ref>
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| : <math>(a \cos \theta)^{\tfrac{n}{n-1}}+(b \sin \theta)^{\tfrac{n}{n-1}}=r^{\tfrac{n}{n-1}}.</math>{{clear}}
| |
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| ==Generalizations==
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| {{Expand section|date=June 2008}}
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| [[Image:Superellipse generalization.png|thumb|right|Example of the generalized superellipse with ''m'' ≠ ''n''.]]
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| The superellipse is further generalized as:
| |
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| :<math>\left|\frac{x}{a}\right|^m + \left|\frac{y}{b}\right|^n = 1; \qquad m, n > 0.</math>
| |
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| or
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| :<math>
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| \begin{align}
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| x\left(\theta\right) &= {|\cos \theta|}^{\frac{2}{m}} \cdot a \sgn(\cos \theta) \\
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| y\left(\theta\right) &= {|\sin \theta|}^{\frac{2}{n}} \cdot b \sgn(\sin \theta).
| |
| \end{align}
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| </math>
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| (Note that <math>\theta</math> is NOT a physical angle of the figure, but just a parameter)
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| {{clear}}
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| == History ==
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| The general Cartesian notation of the form comes from the French mathematician [[Gabriel Lamé]] (1795–1870) who generalized the equation for the ellipse.
| |
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| [[File:MeliorSuperEllipse.svg|thumb|176px|The outer outlines of the letters 'o' and 'O' in Zapf's Melior typeface are described by superellipses with ''n'' = {{nowrap|log(1/2) / log (7/9)}} ≈ 2.758]]
| |
| [[Hermann Zapf]]'s [[typeface]] [[Melior (typeface)|Melior]], published in 1952, uses superellipses for letters such as ''o''. <!--Many web sites say Zapf actually drew the shapes of Melior by hand without knowing the mathematical concept of the superellipse, and only later did Piet Hein point out to Zapf that his curves were extremely similar to the mathematical construct, but these web sites do not cite any primary source of this account.--> Thirty years later [[Donald Knuth]] would build the ability to choose between true ellipses and superellipses (both approximated by [[cubic spline]]s) into his [[Computer Modern]] type family.
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| Though he is often credited with its invention, the [[Denmark|Danish]] poet and scientist [[Piet Hein (Denmark)|Piet Hein]] (1905–1996) did not discover the super-ellipse. In 1959, city planners in [[Stockholm]], [[Sweden]] announced a design challenge for a [[roundabout]] in their city square [[Sergels Torg]]. Piet Hein's winning proposal was based on a superellipse with ''n'' = 2.5 and ''a''/''b'' = 6/5.<ref name=gardner>{{Citation | last=Gardner | first=Martin | author-link=Martin Gardner | chapter=Piet Hein’s Superellipse | year=1977 | title=Mathematical Carnival. A New Round-Up of Tantalizers and Puzzles from Scientific American | location=New York | publisher=[[Vintage Press]] | pages=240–254 | isbn=978-0-394-72349-5}}</ref> As he explained it:
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| :''Man is the animal that draws lines which he himself then stumbles over. In the whole pattern of civilization there have been two tendencies, one toward straight lines and rectangular patterns and one toward circular lines. There are reasons, mechanical and psychological, for both tendencies. Things made with straight lines fit well together and save space. And we can move easily — physically or mentally — around things made with round lines. But we are in a straitjacket, having to accept one or the other, when often some intermediate form would be better. To draw something freehand — such as the patchwork traffic circle they tried in Stockholm — will not do. It isn't fixed, isn't definite like a circle or square. You don't know what it is. It isn't esthetically satisfying. The super-ellipse solved the problem. It is neither round nor rectangular, but in between. Yet it is fixed, it is definite — it has a unity.''
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| Sergels Torg was completed in 1967. Meanwhile Piet Hein went on to use the superellipse in other artifacts, such as beds, dishes, tables, etc.<ref name=bbc>[http://www.bbc.co.uk/dna/h2g2/A1053884 ''The Superellipse''], in ''The Guide to Life, The Universe and Everything'' by [[British Broadcasting Corporation|BBC]] (27th June 2003)</ref> By rotating a superellipse around the longest axis, he created the [[superegg]], a solid egg-like shape that could stand upright on a flat surface, and was marketed as a [[novelty item|novelty toy]].
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| In 1968, when negotiators in [[Paris]] for the [[Vietnam War]] could not agree on the shape of the negotiating table, Balinski, [[Kieron Underwood]] and Holt suggested a superelliptical table in a letter to the [[New York Times]].<ref name=gardner/> The superellipse was used for the shape of the 1968 [[Estadio Azteca|Azteca Olympic Stadium]], in [[Mexico City]].
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| [[Waldo R. Tobler]] developed a [[map projection]], the [[Tobler hyperelliptical projection]], published in 1973,<ref>{{Citation|
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| last=Tobler|
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| first=Waldo|
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| title=The hyperelliptical and other new pseudocylindrical equal area map projections|
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| journal=Journal of Geophysical Research|
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| volume=78|
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| issue=11|
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| pages=1753–1759|
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| year=1973|
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| doi=10.1029/JB078i011p01753|
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| postscript=.
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| |bibcode=1973JGR....78.1753T
| |
| }}
| |
| </ref> in which the [[Meridian (geography)|meridians]] are arcs of superellipses. | |
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| Three connected superellipses are used in the logo of the [[Pittsburgh Steelers]].
| |
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| == See also ==
| |
| * [[Astroid]], the superellipse with ''n'' = {{fraction|2|3}} and ''a'' = ''b'', is a hypocycloid with four cusps.
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| ** [[Deltoid curve]], the hypocycloid of ''three'' cusps.
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| * [[Squircle]], the superellipse with ''n'' = 4 and ''a'' = ''b'', looks like "The Four-Cornered Wheel."
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| ** [[Reuleaux triangle]], "The Three-Cornered Wheel."
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| * [[Superformula]], a generalization of the superellipse.
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| * [[Superquadrics]], the three-dimensional "relatives" of superellipses.
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| * [[Superelliptic curve]], equation of the form ''Y''<sup>''n''</sup> = ''f''(''X'').
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| * [[Lp_space|L<sup>p</sup> spaces]]
| |
| | |
| ==References==
| |
| {{Reflist}}
| |
| * {{citation | last=Barr | first=Alan H. | title=Geometric Modeling and Fluid Dynamic Analysis of Swimming Spermatozoa | year=1983 | publisher=[[Rensselaer Polytechnic Institute]] }} (Ph.D. dissertation using superellipsoids)
| |
| * {{citation | last=Barr | first=Alan H. | contribution=Rigid Physically Based Superquadrics | title=[[Graphics Gems]] III | year=1992 | editor-last=Kirk | editor-first=David | publisher=[[Academic Press]] | isbn=978-0-12-409672-1 | pages=137–159 ([http://www.graphicsgems.org/gemsiii/sqfinal.c code]: 472–477)}}
| |
| * {{citation | last=Gielis | first=Johan | title=Inventing the Circle: The Geometry of Nature | year=2003 | location=Antwerp | publisher=Geniaal Press | isbn=978-90-807756-1-9}}
| |
| | |
| ==External links==
| |
| *{{SpringerEOM|title=Lamé curve|id=Lam%C3%A9_curve&oldid=11566|last=Sokolov|first=D.D.}}
| |
| *[http://www.mathcurve.com/courbes2d/lame/lame.shtml "Courbe de Lamé"] at Encyclopédie des Formes Mathématiques Remarquables (in French)
| |
| *{{MathWorld|title=Superellipse|urlname=Superellipse}}
| |
| *{{MacTutor|class=history/Curves|id=Lame|title=Lame Curves}}
| |
| * [http://www.2dcurves.com/power/powerse.html "Super Ellipse"] on 2dcurves.com
| |
| * [http://www.procato.com/superellipse/ Superellipse Calculator & Template Generator]
| |
| * [http://users.cs.cf.ac.uk/Paul.Rosin/fitting.html C code for fitting superellipses]
| |
| | |
| [[Category:Curves]]
| |
overwhelmed win over the soul of the house
Kind of near-blind reverence, like pious believers like.
when cumulative prestige to some level, but also in the hearts quietly sublimation, and no doubt, these alliance members fear for Xiao Yan, also in this カシオ 腕時計 スタンダード sublimation, the more カシオ電波ソーラー腕時計レディース fanatical, in this case, 'medicine' old, who are somewhat surprised, but it did not stop, people always need to have a faith, because this belief, they will have a high degree of cohesion, and Tianfu alliance among the ability to be this belief, apparently with only Xiao Yan
league carnival, which lasted two weeks later, just gradually subsided, but although unusual member of the excitement カシオ 電波時計 腕時計 a little crazy, but league executives, but it is still maintained a casio 腕時計 メンズ calm, overwhelmed カシオ ソーラー 腕時計 win over the soul of the house, for the prestigious Union does has immeasurable benefits, but since the top of this living place, it will inevitably be many slings and arrows, the past trend of strong soul of the house, in
相关的主题文章：
'Come on
Tsing Yi with the girls in the juvenile behind.
'Come on, Ling old, say good-bye.'
Xiao Yan raised his head, looked before the mountain, he could feel the breath of a large number of forceful, where fast cohesion, and, in the カシオ腕時計 g-shock other two directions Mountains, also came bursts of spatial fluctuations, it カシオ ソーラー 腕時計 seems Yan Lei II strong family is dispatched
voice down, Xiao Yan is no longer hesitated, holding the hand Kaoru children, stature flash, is swept カシオ スタンダード 腕時計 into the sky.
'Xiao Yan Master, I wish triumphant 腕時計 メンズ casio return!'
looked back duo Xiao Yan Ling shadow is smile, muttered to himself.
when Xiao Yan horizon when the two appear in the first mountain, where the air is already shadows scattered, many of the ancient tribe of the strong launch into the sky, forming a neat formation in the sky, the vast surging between breath, like a remote one, that like power and influence, even into the territory of the soul of Emperor casio 腕時計 g-shock Xiao Yan, are
相关的主题文章：
and under this acid itch
Gu fine blood into the body, Xiao Yan is covered with the skin immediately harnessed into a bright Jinmang, casio 時計 the vast energy in the body as カシオ 腕時計 チタン the storm swept in general open, had exhausted the meridians, almost in an instant, that is once again being gurgle vindictive, quickly filled.
Moreover, in this energy crisis sweeping the room, Xiao Yan also feel the body's bones, muscles and even the meridians, etc., are among faint heard a strange acid itch, and under this acid itch, as if bones, muscles, etc., are slowly becoming more and more tough and 時計 カシオ strong
'This is the essence and blood cologne energy カシオ gショック 腕時計 inherent in it, really amazing.'
feel the change in the body, Xiao Yan could not help but come to mind is some amazing promise of 'color', this ability, almost all can, and some higher-order Dan 'medicine' compared to this really worthy カシオ腕時計 g-shock cologne Void Top-like existence of Warcraft community, just fine blood force, is that it has
相关的主题文章：