|
|
| Line 1: |
Line 1: |
| [[File:Ex_sq.png|right|320px|thumb|Some superquadrics.]]
| | == 秦Yuは直接おめでとうを見て微笑んだ == |
| In [[mathematics]], the '''superquadrics''' or '''super-quadrics''' (also '''superquadratics''') are a family of [[geometry|geometric shapes]] defined by formulas that resemble those of [[elipsoid]]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 [[Lamé curve]]s ("[[Superellipse]]s").
| |
|
| |
|
| The superquadrics include many shapes that resemble [[cube]]s, [[octahedron|octahedra]], [[Cylinder (geometry)|cylinders]], [[lozenge]]s and [[spindle torus|spindle]]s, with rounded or sharp corners. Because of their flexibility and relative simplicity, they are popular [[geometric model]]ing tools, especially in [[computer graphics]].
| | Forberは今ドアのところにあります。<br><br>雲府中の神のしもべたちは、非常に規則です。隠れ家パート、パトロール、ガードなどにいくつかがあります [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン パンプス]。全体の雲は、政府が現れる神々の中に歩いていないどこまで見ることができます。<br>秦ゆうの神に会っ<br>は、敬意を敬礼である [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_11.php クリスチャンルブタン ブーツ]。これは確かに定命の世界を感じる秦ゆうへの回帰の一種である。<br><br>年時定命の世界での宮殿 [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_6.php クリスチャンルブタン 銀座]。そのメイド公務員を見た人は、また丁重にサーバントが今だけ神である、「3殿下」と呼ばれるが、いくつかの致命的なサーバントを持っていたされている [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_11.php クリスチャンルブタン ブーツ]。<br><br>「出口の所有者におめでとう。」Forberが出ています。秦Yuは直接おめでとうを見て微笑んだ [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_9.php クリスチャンルブタン セール]。<br><br>秦ゆう華聯は、そのバリエーションのアバターを考え、その後、空間の理解を向上させるためにすべての時間を感じて、笑ってはいられませんでした：」の右ハハ、はいおめでとうに値する。 '<br>第XVIに設定<br>候補<br><br>ホール、秦Yuはメインシートに座っている。 Forberと秋ストレッチデュオは、2つの神々は、メイドが紅茶で送られ、ゲスト錯体の両側に座っている。 |
| | | 相关的主题文章： |
| Some authors, such as [[Alan H. Barr|Alan Barr]], define "superquadrics" as including both the [[superellipsoid]]s and the [[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, the (proper) supertoroids are not superquadrics as defined above; and, while some superquadrics are superellipsoids, neither family is contained in the other.
| | <ul> |
| | | |
| == Formulas ==
| | <li>[http://water.xilejia.com/plus/feedback.php?aid=596 http://water.xilejia.com/plus/feedback.php?aid=596]</li> |
| === Implicit equation ===
| | |
| The basic superquadric has the formula
| | <li>[http://hero-hk.freebbs.com.tw/viewthread.php?tid=338474&extra= http://hero-hk.freebbs.com.tw/viewthread.php?tid=338474&extra=]</li> |
| :<math> \left|x\right|^r + \left|y\right|^s + \left|z\right|^t =1</math>
| | |
| where ''r'', ''s'', and ''t'' are positive real numbers that determine the main features of the superquadric. Namely:
| | <li>[http://www.macaustamp.cn/home.php?mod=space&uid=27754 http://www.macaustamp.cn/home.php?mod=space&uid=27754]</li> |
| | | |
| * less than 1: a pointy octahedron with [[concave polygon|concave]] [[face (geometry)|faces]] and sharp [[edge (geometry)|edges]].
| | </ul> |
| * exactly 1: a regular octahedron.
| |
| * between 1 and 2: an octahedron with convex faces, blunt edges and blunt corners.
| |
| * exactly 2: a sphere
| |
| * greater than 2: a cube with 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 \leq 1</math>
| |
| | |
| === Parametric description ===
| |
| Parametric equations in terms of surface parameters ''u'' and ''v'' (longitude and latitude) are
| |
| :<math>\begin{align}
| |
| x(u,v) &{}= A c\left(v,\frac{2}{r}\right) c\left(u,\frac{2}{r}\right) \\
| |
| y(u,v) &{}= B c\left(v,\frac{2}{s}\right) s\left(u,\frac{2}{s}\right) \\
| |
| z(u,v) &{}= C s\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 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>
| |
| | |
| <!-- Fix this for superquadrics:
| |
| ==Properties==
| |
| ===Volume and area===
| |
| 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 = \frac{2}{3} A B C e n \beta \left( \frac{1}{r},\frac{1}{r} \right) \beta \left({2}{t},\frac{1}{t} \right) . </math>
| |
| -->
| |
| <!--
| |
| == Plots ==
| |
| [[Image:Sh.svg|right|thumb|320px|Superhyperboloids]]
| |
| -->
| |
| | |
| == Plotting code == | |
| | |
| The following [[GNU Octave]] code generates a mesh approximation of a superquadric:
| |
| <!-- Is this code worth including? The equation is enough! -->
| |
| <source lang="matlab">
| |
| function retval=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);
| |
| endfunction;
| |
| </source> | |
| | |
| <!--The hyperboloid images generated by this code are not very good. --> | |
| <!--
| |
| 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>
| |
| -->
| |
| | |
| == References ==
| |
| | |
| {{reflist}}
| |
| | |
| *Jaklič, A., Leonardis, A., ''Solina, F., Segmentation and Recovery of Superquadrics''. Kluwer Academic Publishers, Dordrecht, 2000.
| |
| | |
| == See also ==
| |
| | |
| * [[Quadric]]
| |
| * [[Superellipse]]
| |
| * [[Supertoroid]]
| |
| * [[Superellipsoid]]
| |
| * [[Superegg]]
| |
| | |
| == 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]
| |
| | |
| | |
| | |
| [[Category:Computer graphics]]
| |