Convection–diffusion equation: Difference between revisions

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[[File:Superellipsoid collection.png|right|400px|thumb|Superellipsoid collection with exponent parameters, created using [[POV-Ray]]. Here, e = 2/r, and n = 2/t (equivalently, r = 2/e and t = 2/n).<ref>http://www.povray.org/documentation/view/3.6.1/285/</ref> The [[cube]], [[cylinder (geometry)|cylinder]], [[sphere]], [[Steinmetz solid]], [[bicone]] and regular [[octahedron]] can all be seen as special cases.]]
In [[mathematics]], a '''super-ellipsoid''' or '''superellipsoid''' is a solid whose horizontal sections are [[super ellipse|super-ellipses]] (Lamé curves) with the same [[exponent]] ''r'', and whose vertical sections through the center are super-ellipses with the same exponent ''t''.
 
Super-ellipsoids as [[computer graphics]] primitives were popularized by [[Alan H. Barr]] (who used the name "superquadrics" to refer to both superellipsoids and [[supertoroid]]s).<ref name="barr81">Barr, A.H. (January 1981), ''Superquadrics and Angle-Preserving Transformations''. IEEE_CGA vol. 1 no. 1, pp. 11&ndash;23</ref><ref name="barr92">Barr, A.H. (1992), ''Rigid Physically Based Superquadrics''. Chapter III.8 of ''Graphics Gems III'', edited by D. Kirk, pp. 137&ndash;159</ref> However, while some super-ellipsoids are [[superquadric]]s, neither family is contained in the other.
 
[[Piet Hein (Denmark)|Piet Hein]]'s [[superegg]]s are special cases of super-ellipsoids.
 
==Formulas==
 
===Basic shape===
The basic super-ellipsoid is defined by the [[implicit function|implicit equation]]
:<math> \left( \left|x\right|^{r} + \left|y\right|^{r} \right)^{t/r} + \left|z\right|^{t} \leq 1</math>
The parameters ''r'' and ''t'' are positive real numbers that control the amount of flattening at the tips and at the equator. Note that the formula becomes a special case of the superquadric's equation if (and only if) ''t''&nbsp;=&nbsp;''r''.
 
Any "[[parallel of latitude]]" of the superellipsoid (a horizontal section at any constant ''z'' between -1 and +1) is a Lamé curve with exponent ''r'', scaled by <math> a = (1 - \left|z\right|^{t})^{1/t}</math>:
 
: <math> \left|\frac{x}{a}\right|^{r} + \left|\frac{y}{a}\right|^{r} \leq 1</math>
 
Any "[[meridian of longitude]]" (a section by any vertical plane through the origin) is a Lamé curve with exponent ''t'', stretched horizontally by a factor ''w'' that depends on the sectioning plane. Namely, if ''x''&nbsp;=&nbsp;''u''&nbsp;cos&nbsp;''&theta;'' and ''y''&nbsp;=&nbsp;''u''&nbsp;sin&nbsp;''&theta;'', for a fixed ''&theta;'', then
 
: <math> \left|\frac{u}{w}\right|^t + \left|z\right|^t \leq 1</math>
 
where
 
:<math>w = (\left|\cos \theta\right|^r + \left|\sin\theta\right|^r)^{-1/r}.</math>
 
In particular, if ''r'' is 2, the horizontal cross-sections are circles, and the horizontal stretching ''w'' of the vertical sections is 1 for all planes.  In that case, the super-ellipsoid is a [[solid of revolution]], obtained by rotating the Lamé curve with exponent ''t'' around the vertical axis.
 
The basic shape above extends from &minus;1 to +1 along each coordinate axis. The general super-ellipsoid is obtained by scaling the basic shape along each axis by factors ''A'', ''B'', ''C'', the semi-diameters of the resulting solid. The implicit equation is
 
:<math> \left( \left|\frac{x}{A}\right|^r + \left|\frac{y}{B}\right|^r \right)^{t/r} + \left|\frac{z}{C}\right|^{t} \leq 1</math>
 
Setting ''r''&nbsp;=&nbsp;2, ''t''&nbsp;=&nbsp;2.5, ''A''&nbsp;=&nbsp;''B''&nbsp;=&nbsp;3, ''C''&nbsp;=&nbsp;4 one obtains Piet Hein's superegg.
 
The general superellipsoid has a [[parametric representation]] in terms of surface parameters ''u'' and ''v'' (longitude and latitude):<ref name="barr92"/>
 
:<math>\begin{align}
x(u,v) &{}= A c\left(v,\frac{2}{t}\right) c\left(u,\frac{2}{r}\right) \\
y(u,v) &{}= B c\left(v,\frac{2}{t}\right) s\left(u,\frac{2}{r}\right) \\
z(u,v) &{}= C s\left(v,\frac{2}{t}\right) \\
& -\pi/2 \le v \le \pi/2, \quad -\pi \le u < \pi ,
\end{align}</math>
 
where the auxiliary functions are
 
:<math>\begin{align}
c(\omega,m) &{}= \sgn(\cos \omega) |\cos \omega|^m \\
s(\omega,m) &{}= \sgn(\sin \omega) |\sin \omega|^m
\end{align}</math>
and the [[sign function]] sgn(''x'') is  
:<math> \sgn(x) = \begin{cases}
-1, & x < 0 \\
  0, & x = 0 \\
+1, & x > 0 .
\end{cases}</math>
 
The volume inside this surface can be expressed in terms of [[beta function]]s, β(''m'',''n'')&nbsp;=&nbsp;Γ(''m'')Γ(''n'')/Γ(''m''&nbsp;+&nbsp;''n''), as
 
:<math> V = \frac23 A B C \frac{4}{r t} \beta \left( \frac{1}{r},\frac{1}{r} \right) \beta \left(\frac{2}{t},\frac{1}{t} \right). </math><!--WILL SOMEONE PLEASE CHECK THIS FORMULA?-->
 
== See also ==
 
* [[Super ellipse]]
 
== References ==
{{Reflist}}
 
*Jaklič, A., Leonardis, A.,Solina, F., ''Segmentation and Recovery of Superquadrics''. Kluwer Academic Publishers, Dordrecht, 2000.
*Aleš Jaklič and Franc Solina (2003) Moments of Superellipsoids and their Application to Range Image Registration. IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, 33 (4). pp.&nbsp;648–657
 
==External links==
* [http://iris.usc.edu/Vision-Notes/bibliography/describe461.html Bibliography: SuperQuadric Representations]
* [http://www.cs.utah.edu/~gk/papers/vissym04/ Superquadric Tensor Glyphs]
* [http://www.gamedev.net/reference/articles/article1172.asp SuperQuadric Ellipsoids and Toroids, OpenGL Lighting, and Timing]
* [http://demonstrations.wolfram.com/Superquadrics/ Superquadratics] by Robert Kragler, [[The Wolfram Demonstrations Project]].
 
[[Category:Computer graphics]]

Latest revision as of 19:34, 12 October 2014

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