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[[File:Supertorus_table.png|thumb|400px|right|Supertoroids with a=b=2, and different combinations for the parameters s and t.]]
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In [[geometry]] and [[computer graphics]], a '''supertoroid''' or '''supertorus''' is usually understood to be a family of [[doughnut]]-like [[surface (geometry)|surfaces]] (technically, a [[topology|topological]] [[torus (mathematics)|torus]]) whose shape is defined by mathematical formulas similar to those that define the [[superquadric]]s. The plural of "supertorus" is either '''supertori''' or '''supertoruses'''.
 
The family was described and named by [[Alan H.Barr|Alan Barr]] in 1994.<ref name=barr>Alan H. Barr (1981) ''Superquadrics and Angle-Preserving Transformations''. IEEE Computer Graphics and Applications, volume 1 issue 1. pp. 11-23.</ref>
 
Barr's supertoroids have been fairly popular in computer graphics as a convenient model for many objects, such as smooth frames for rectangular things. One quarter of a supertoroid can provide a smooth and seamless 90-degree joint between two superquadric [[cylinder (geometry)|cylinder]]s. However they are not [[algebraic surface]]s (except in special cases).
 
==Formulas==
Alan Barr's supertoroids are defined by parametric equations similar to the [[trigonometry|trigonometric]] equations of the torus, except that the [[sine]] and [[cosine]] terms are raised to arbitrary [[exponentiation|powers]]. Namely, the generic point ''P''(''u'', ''v'') of the surface is given by
:<math>
P(u,v) = \left(\begin{array}{c}
  X(u,v)\\
  Y(u,v)\\
  Z(u,v)
\end{array}\right)
=
\left(\begin{array}{c}
  (a + C_{u}^{s}) C_{v}^{t}\\
  (b + C_{u}^{s}) S_{v}^{t}\\
  S_{u}^{s}
\end{array}\right)
</math>
where <math>C_{\theta}^{\epsilon} = \operatorname{sgn}(\cos \theta)\left|\cos\theta\right|^\epsilon</math>, <math>S_{\theta}^{\epsilon} = \operatorname{sgn}(\sin \theta)\left|\sin\theta\right|^\epsilon</math>, and the parameters ''u'' and ''v'' range from 0 to 360 degrees (0 to 2''π'' [[radian]]s).
 
In these formulas, the parameter ''s'' &gt; 0 controls the "squareness" of the vertical sections, ''t'' &gt; 0 controls the squareness of the horizontal sections, and ''a'', ''b'' ≥ 1 are  the major radii in the ''X'' and ''Y'' directions. With ''s''=''t''=1 and ''a''=''b''=''R'' one obtains the ordinary torus with major radius ''R'' and minor radius 1, with the center at the origin and [[symmetry|rotational symmetry]] about the ''Z'' axis.
 
In general, the supertorus defined as above spans the [[interval (mathematics)|intervals]] <math>[-(a+1),+(a+1)]</math> in ''X'', <math>[-(b+1),+(b+1)]</math> in ''Y'', and <math>[-1,+1]</math> in ''Z''.  The whole shape is symmetric about the planes ''X''=0, ''Y''=0, and ''Z''=0.  The hole runs in the ''Z'' direction and spans the intervals <math>[-(a-1),+(a-1)]</math> in ''X'' and  <math>[-(b-1),+(b-1)]</math> in ''Y''.
 
A curve of constant ''u'' on this surface is an horizontal [[Lamé curve]] with exponent 2/''t'', scaled in ''X'' and ''Y'' and displaced in ''Z''.  A curve of constant ''v'', projected on the plane ''X''=0 or ''Y''=0, is a Lamé curve with exponent 2/''s'', scaled and horizontally shifted. If ''v'' is 0, the curve is planar and spans the interval <math>[a-1,a+1]</math> in ''X'', and <math>[-1,+1]</math> in ''Z''; and similarly if ''v'' is 90, 180, or 270 degrees. The curve is planar also if ''a'' = ''b''.
 
In general, if ''a''≠''b'' and ''v'' is not a multiple of 90 degrees, the curve of constant ''v'' will not be planar; and, conversely, a vertical plane section of the supertorus will not be a Lamé curve.
 
The basic supertoroid shape defined above is often modified by non-uniform scaling to yield supertoroids of specific width, length, and vertical thickness.
 
==Plotting code==
 
The following [[GNU Octave]] code generates plots of a supertorus:
 
<source lang="matlab">
function supertoroid(epsilon,a)
  n=50;
  d=.1;
  etamax=pi;
  etamin=-pi;
  wmax=pi;
  wmin=-pi;
  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)*(a(4)+sign(cos(eta(i)))*abs(cos(eta(i)))^epsilon(1))*sign(cos(w(j)))*abs(cos(w(j)))^epsilon(2);
      y(i,j)=a(2)*(a(4)+sign(cos(eta(i)))*abs(cos(eta(i)))^epsilon(1))*sign(sin(w(j)))*abs(sin(w(j)))^epsilon(2);
      z(i,j)=a(3)*sign(sin(eta(i)))*abs(sin(eta(i)))^epsilon(1);
    endfor;
  endfor;
  mesh(x,y,z);
  endfunction;
</source>
 
==See also==
* [[Superellipsoid]]
* [[Superegg]]
 
==References==
{{Reflist}}
 
[[Category:Surfaces]]
[[Category:Computer graphics]]

Revision as of 22:32, 28 February 2014

If you are looking for a specific plugin, then you can just search for the name of the plugin. Good luck on continue learning how to make a wordpress website. Step-4 Testing: It is the foremost important of your Plugin development process. Dead links are listed out simply because it will negatively have an influence on the website's search engine rating. In the most current edition you can customize your retailer layout and display hues and fonts similar to your site or blog.

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