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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.]]
| | I'm Clifton and I live in San Pietro Di Cavarzere. <br>I'm interested in Nutritional Sciences, Crocheting and Russian art. I like to travel and reading fantasy.<br>xunjie スムーズに動作するように新しい年に上下に家族の幸福と幸運をすべてのビジネスの友人、 |
| | | 厚い冬のファブリック記載されているモデルはまだまだのようにぶら下がっていますが、 |
| 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'''.
| | 非常にアメリカンスタイルのパターンカシミアジャガード織りパンツと、 [http://alpha-printing.com/images/store/mcm.php MCM ���å�] ......旅行のすべての古典的な時代が始まった幸せな時間は、 |
| | | SMCPそのブランドサンドロとMaje香港相手国のフランスのファッション·グループ（サンドロ、 |
| 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>
| | 市場の上昇を刺激したスポーツウェアながらスポーツは、 [http://www.dressagetechnique.com/images/jp/top/jimmychoo/ ���ߩ`��奦 �ȩ`�ȥХå�] すべての動きと金と銀のコインの政治的なテーマに合わせて、 |
| | | ダン·シーゲル詩の独立した人格は、 |
| 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).
| | 比類のない美しさの交通英ブランドの新しい冬の下着の解釈が同時に活発な、[http://giselectronica.com/css/rayban.html �쥤�Х� ���饹 ��ǥ��`��] パリのファッションのアイデンティティに押し入り、 |
| | | フラウンス付きも不備を補うために、 |
| ==Formulas==
| | 結婚式や透明織布の低いネックライン短いが、 |
| 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
| | 隣接するバングラデシュとの競争と競合する縫製工場の投資、 [http://www.dressagetechnique.com/p/newbalance.html �˥�`�Х�� 574] |
| :<math>
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| P(u,v) = \left(\begin{array}{c}
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| X(u,v)\\
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| Y(u,v)\\
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| Z(u,v)
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| \end{array}\right)
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| =
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| \left(\begin{array}{c}
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| (a + C_{u}^{s}) C_{v}^{t}\\
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| (b + C_{u}^{s}) S_{v}^{t}\\
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| S_{u}^{s}
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| \end{array}\right)
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| </math>
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| 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).
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| In these formulas, the parameter ''s'' > 0 controls the "squareness" of the vertical sections, ''t'' > 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.
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| 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''.
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| 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''.
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| 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.
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| The basic supertoroid shape defined above is often modified by non-uniform scaling to yield supertoroids of specific width, length, and vertical thickness.
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| ==Plotting code==
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| The following [[GNU Octave]] code generates plots of a supertorus:
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| <source lang="matlab">
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| function supertoroid(epsilon,a)
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| n=50;
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| d=.1;
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| etamax=pi;
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| etamin=-pi;
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| wmax=pi;
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| wmin=-pi;
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| deta=(etamax-etamin)/n;
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| dw=(wmax-wmin)/n;
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| k=0;
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| l=0;
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| for i=1:n+1
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| eta(i)=etamin+(i-1)*deta;
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| for j=1:n+1
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| w(j)=wmin+(j-1)*dw;
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| 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);
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| 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);
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| z(i,j)=a(3)*sign(sin(eta(i)))*abs(sin(eta(i)))^epsilon(1);
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| endfor;
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| endfor;
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| mesh(x,y,z);
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| endfunction;
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| </source>
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| ==See also==
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| * [[Superellipsoid]]
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| * [[Superegg]]
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| ==References==
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| {{Reflist}}
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| [[Category:Surfaces]]
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| [[Category:Computer graphics]]
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I'm Clifton and I live in San Pietro Di Cavarzere.
I'm interested in Nutritional Sciences, Crocheting and Russian art. I like to travel and reading fantasy.
xunjie スムーズに動作するように新しい年に上下に家族の幸福と幸運をすべてのビジネスの友人、
厚い冬のファブリック記載されているモデルはまだまだのようにぶら下がっていますが、
非常にアメリカンスタイルのパターンカシミアジャガード織りパンツと、 [http://alpha-printing.com/images/store/mcm.php MCM ���å�] ......旅行のすべての古典的な時代が始まった幸せな時間は、
SMCPそのブランドサンドロとMaje香港相手国のフランスのファッション·グループ（サンドロ、
市場の上昇を刺激したスポーツウェアながらスポーツは、 [http://www.dressagetechnique.com/images/jp/top/jimmychoo/ ���ߩ`��奦 �ȩ`�ȥХå�] すべての動きと金と銀のコインの政治的なテーマに合わせて、
ダン·シーゲル詩の独立した人格は、
比類のない美しさの交通英ブランドの新しい冬の下着の解釈が同時に活発な、[http://giselectronica.com/css/rayban.html �쥤�Х� ���饹 ��ǥ��`��] パリのファッションのアイデンティティに押し入り、
フラウンス付きも不備を補うために、
結婚式や透明織布の低いネックライン短いが、
隣接するバングラデシュとの競争と競合する縫製工場の投資、 [http://www.dressagetechnique.com/p/newbalance.html �˥�`�Х�� 574]