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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. 11–23</ref><ref name="barr92">Barr, A.H. (1992), ''Rigid Physically Based Superquadrics''. Chapter III.8 of ''Graphics Gems III'', edited by D. Kirk, pp. 137–159</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'' = ''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'' = ''u'' cos ''θ'' and ''y'' = ''u'' sin ''θ'', for a fixed ''θ'', 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 −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'' = 2, ''t'' = 2.5, ''A'' = ''B'' = 3, ''C'' = 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'') = Γ(''m'')Γ(''n'')/Γ(''m'' + ''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. 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]] |
Revision as of 12:55, 23 January 2014
In mathematics, a super-ellipsoid or superellipsoid is a solid whose horizontal sections are 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 supertoroids).[2][3] However, while some super-ellipsoids are superquadrics, neither family is contained in the other.
Piet Hein's supereggs are special cases of super-ellipsoids.
Formulas
Basic shape
The basic super-ellipsoid is defined by the implicit equation
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 = 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 :
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 = u cos θ and y = u sin θ, for a fixed θ, then
where
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 −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
Setting r = 2, t = 2.5, A = B = 3, C = 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):[3]
where the auxiliary functions are
and the sign function sgn(x) is
The volume inside this surface can be expressed in terms of beta functions, β(m,n) = Γ(m)Γ(n)/Γ(m + n), as
See also
References
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- 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. 648–657
External links
- Bibliography: SuperQuadric Representations
- Superquadric Tensor Glyphs
- SuperQuadric Ellipsoids and Toroids, OpenGL Lighting, and Timing
- Superquadratics by Robert Kragler, The Wolfram Demonstrations Project.
- ↑ http://www.povray.org/documentation/view/3.6.1/285/
- ↑ Barr, A.H. (January 1981), Superquadrics and Angle-Preserving Transformations. IEEE_CGA vol. 1 no. 1, pp. 11–23
- ↑ 3.0 3.1 Barr, A.H. (1992), Rigid Physically Based Superquadrics. Chapter III.8 of Graphics Gems III, edited by D. Kirk, pp. 137–159