Superquadrics

{{Short description|Family of geometric shapes}} [[File:Ex_sq.png|right|300px|thumb|Some superquadrics.]] In [[mathematics]], the '''superquadrics''' or '''super-quadrics''' (also '''superquadratics''') are a family of [[geometry|geometric shapes]] defined by formulas that resemble those of [[ellipsoid]]s and other [[quadric]]s, except that the [[square (algebra)|squaring]] operations are replaced by arbitrary powers. They can be seen as the three-dimensional relatives of the [[superellipse]]s. The term may refer to the solid object or to its [[Surface (mathematics)|surface]], depending on the context. The equations below specify the surface; the solid is specified by replacing the equality signs by less-than-or-equal signs.

The superquadrics include many shapes that resemble [[cube]]s, [[octahedron|octahedra]], [[Cylinder (geometry)|cylinders]], [[Lozenge (shape)|lozenge]]s and [[spindle torus|spindle]]s, with rounded or sharp corners.<ref name="barr81" /> Because of their flexibility and relative simplicity, they are popular [[geometric model]]ing tools, especially in [[computer graphics]]. It becomes an important [[geometric primitive]] widely used in [[computer vision]],<ref name=":1" /><ref name=":0" /> robotics,<ref>{{Cite journal |last1=Ruan |first1=Sipu |last2=Wang |first2=Xiaoli |last3=Chirikjian |first3=Gregory S. |date=2022 |title=Collision Detection for Unions of Convex Bodies With Smooth Boundaries Using Closed-Form Contact Space Parameterization |journal=IEEE Robotics and Automation Letters |volume=7 |issue=4 |pages=9485–9492 |doi=10.1109/LRA.2022.3190629 |bibcode=2022IRAL....7.9485R |s2cid=250543506 |issn=2377-3766|doi-access=free }}</ref> and physical simulation.<ref>{{Cite journal |last1=Lu |first1=G. |last2=Third |first2=J. R. |last3=Müller |first3=C. R. |date=2012-08-20 |title=Critical assessment of two approaches for evaluating contacts between super-quadric shaped particles in DEM simulations |url=https://www.sciencedirect.com/science/article/pii/S0009250912003223 |journal=Chemical Engineering Science |language=en |volume=78 |pages=226–235 |doi=10.1016/j.ces.2012.05.041 |bibcode=2012ChEnS..78..226L |issn=0009-2509|url-access=subscription }}</ref>

Some authors, such as [[Alan H. Barr|Alan Barr]], define "superquadrics" as including both the [[superellipsoid]]s and the [[supertoroid]]s.<ref name="barr81">{{cite journal|last1=Barr|title=Superquadrics and Angle-Preserving Transformations|journal=IEEE Computer Graphics and Applications|issn=0272-1716 |pages=11–23|volume=1|issue=1|date=1 January 1981|doi=10.1109/MCG.1981.1673799|s2cid=9389947 }}</ref><ref name="barr92">Alan H. Barr (1992), ''Rigid Physically Based Superquadrics''. Chapter III.8 of ''Graphics Gems III'', edited by D. Kirk, pp. 137–159</ref> In modern computer vision literatures, superquadrics and [[Superellipsoid|superellipsoids]] are used interchangeably, since [[Superellipsoid|superellipsoids]] are the most representative and widely utilized shape among all the superquadrics.<ref name=":1">{{Cite book |last1=Paschalidou |first1=Despoina |last2=Ulusoy |first2=Ali Osman |last3=Geiger |first3=Andreas |title=2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) |chapter=Superquadrics Revisited: Learning 3D Shape Parsing Beyond Cuboids |date=2019 |pages=10336–10345 |doi=10.1109/CVPR.2019.01059|arxiv=1904.09970 |isbn=978-1-7281-3293-8 |s2cid=128265641 }}</ref><ref name=":0" /> Comprehensive coverage of geometrical properties of superquadrics and methods of their recovery from [[range image]]s and [[Point cloud|point clouds]] are covered in several computer vision literatures.<ref name="barr81" /><ref name=":0">{{Cite book |last1=Liu |first1=Weixiao |last2=Wu |first2=Yuwei |last3=Ruan |first3=Sipu |last4=Chirikjian |first4=Gregory S. |title=2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR) |chapter=Robust and Accurate Superquadric Recovery: A Probabilistic Approach |date=2022 |pages=2666–2675 |doi=10.1109/CVPR52688.2022.00270|arxiv=2111.14517 |isbn=978-1-6654-6946-3 |s2cid=244715106 }}</ref><ref>Aleš Jaklič, Aleš Leonardis, Franc Solina (2000) ''Segmentation and Recovery of Superquadrics''. Kluwer Academic Publishers, Dordrecht</ref><ref>{{Cite book |last1=Wu |first1=Yuwei |last2=Liu |first2=Weixiao |last3=Ruan |first3=Sipu |last4=Chirikjian |first4=Gregory S. |title=Computer Vision – ECCV 2022 |chapter=Primitive-Based Shape Abstraction via Nonparametric Bayesian Inference |date=2022 |editor-last=Avidan |editor-first=Shai |editor2-last=Brostow |editor2-first=Gabriel |editor3-last=Cissé |editor3-first=Moustapha |editor4-last=Farinella |editor4-first=Giovanni Maria |editor5-last=Hassner |editor5-first=Tal |chapter-url=https://link.springer.com/chapter/10.1007/978-3-031-19812-0_28 |series=Lecture Notes in Computer Science |volume=13687 |language=en |location=Cham |publisher=Springer Nature Switzerland |pages=479–495 |doi=10.1007/978-3-031-19812-0_28 |arxiv=2203.14714 |isbn=978-3-031-19812-0}}</ref>

* less than 1: a pointy octahedron modified to have [[concave polygon|concave]] [[face (geometry)|faces]] and sharp [[edge (geometry)|edges]]. * exactly 1: a regular [[octahedron]]. * between 1 and 2: an octahedron modified to have convex faces, blunt edges and blunt corners. * exactly 2: a sphere * greater than 2: a cube modified to have rounded edges and corners. * [[infinity (mathematics)|infinite]] (in the [[limit (mathematics)|limit]]): a cube

Each exponent can be varied independently to obtain combined shapes. For example, if ''r''=''s''=2, and ''t''=4, one obtains a [[solid of revolution]] which resembles an ellipsoid with round cross-section but flattened ends. This formula is a special case of the superellipsoid's formula if (and only if) ''r'' = ''s''.

If any exponent is allowed to be negative, the shape extends to infinity. Such shapes are sometimes called '''super-hyperboloids'''.

The basic shape above spans from -1 to +1 along each coordinate axis. The general superquadric is the result of [[similarity (geometry)|scaling]] this basic shape by different amounts ''A'', ''B'', ''C'' along each axis. Its general equation is :<math> \left|\frac{x}{A}\right|^r + \left|\frac{y}{B}\right|^s + \left|\frac{z}{C}\right|^t = 1.</math>

=== Parametric description === Parametric equations in terms of surface parameters ''u'' and ''v'' (equivalent to longitude and latitude if m equals 2) are :<math>\begin{align} x(u,v) &{}= A g\left(v,\frac{2}{r}\right) g\left(u,\frac{2}{r}\right) \\ y(u,v) &{}= B g\left(v,\frac{2}{s}\right) f\left(u,\frac{2}{s}\right) \\ z(u,v) &{}= C f\left(v,\frac{2}{t}\right) \\ & -\frac{\pi}{2} \le v \le \frac{\pi}{2}, \quad -\pi \le u < \pi , \end{align}</math> where the [[auxiliary function]]s are :<math>\begin{align} f(\omega,m) &{}= \sgn(\sin \omega) \left|\sin \omega \right|^m \\ g(\omega,m) &{}= \sgn(\cos \omega) \left|\cos \omega \right|^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>

===Spherical product===

Barr introduces the ''spherical product'' which given two plane curves produces a 3D surface. If <math display="block">f(\mu)=\begin{pmatrix}f_1(\mu) \\ f_2(\mu)\end{pmatrix},\quad g(\nu)=\begin{pmatrix}g_1(\nu)\\g_2(\nu)\end{pmatrix}</math> are two plane curves then the spherical product is <math display="block">h(\mu,\nu) = f(\mu)\otimes g(\nu) = \begin{pmatrix} f_1(\mu)\ g_1(\nu) \\ f_1(\mu)\ g_2(\nu) \\ f_2(\mu) \end{pmatrix}</math> This is similar to the typical parametric equation of a [[sphere]]: <math display="block">\begin{align} x&=x_{0}+r\sin \theta \;\cos \varphi \\ y&=y_{0}+r\sin \theta \;\sin \varphi \qquad (0\leq \theta \leq \pi ,\;0\leq \varphi <2\pi )\\ z&=z_{0}+r\cos \theta \end{align}</math> which give rise to the name spherical product.

Barr uses the spherical product to define quadric surfaces, like [[ellipsoid]]s, and [[hyperboloid]]s as well as the [[torus]], [[superellipsoid]], superquadric hyperboloids of one and two sheets, and supertoroids.<ref name="barr81"/>

== Plotting code ==

The following [[GNU Octave]] code generates a mesh approximation of a superquadric:  <syntaxhighlight lang="matlab"> function superquadric(epsilon, a) n = 50; etamax = pi / 2; etamin = -pi / 2; wmax = pi; wmin = -pi; deta = (etamax - etamin) / n; dw = (wmax - wmin) / n; [i, j] = meshgrid(1:n+1, 1:n+1) eta = etamin + (i - 1) * deta; w = wmin + (j - 1) * dw; x = a(1) .* sign(cos(eta)) .* abs(cos(eta)).^epsilon(1) .* sign(cos(w)) .* abs(cos(w)).^epsilon(1); y = a(2) .* sign(cos(eta)) .* abs(cos(eta)).^epsilon(2) .* sign(sin(w)) .* abs(sin(w)).^epsilon(2); z = a(3) .* sign(sin(eta)) .* abs(sin(eta)).^epsilon(3);

mesh(x, y, z); end </syntaxhighlight>

<!-- The following code plots superhyperboloids:

<code> function superhyper(epsilon,a) n=50; d=.1; etamax=pi/2-d; etamin=-pi/2+d; wmax=3*pi/2-d; wmin=pi/2+d; deta=(etamax-etamin)/n; dw=(wmax-wmin)/n; k=0; l=0; for i=1:n+1 eta(i)=etamin+(i-1)*deta; for j=1:n+1 w(j)=wmin+(j-1)*dw; x(i,j)=a(1)*sign(sec(eta(i)))*abs(sec(eta(i)))^epsilon(1)*sign(sec(w(j)))*abs(sec(w(j)))^epsilon(2); y(i,j)=a(2)*sign(sec(eta(i)))*abs(sec(eta(i)))^epsilon(1)*sign(tan(w(j)))*abs(tan(w(j)))^epsilon(2); z(i,j)=a(3)*sign(tan(eta(i)))*abs(tan(eta(i)))^epsilon(1); endfor; endfor; mesh(x,y,z); endfunction; </code> -->

== See also == * [[Superegg]] * [[Superellipsoid]] * [[Ellipsoid]]

== References ==

== 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/ Superquadrics] by Robert Kragler, [[The Wolfram Demonstrations Project]]. * [https://github.com/pratikmallya/Superquad Superquadrics in Python] * [https://github.com/bmlklwx/EMS-superquadric_fitting Superquadrics recovery algorithm in Python and MATLAB]

[[Category:Computer graphics]] [[Category:Computer vision]] [[Category:Geometry]] [[Category:Geometry in computer vision]] [[Category:Robotics engineering]]