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Prochaine révision 		Révision précédente
projets:surfaces:accueil [2015/05/16 00:04]
alba [Formules] 		projets:surfaces:accueil [2016/02/01 14:19] (Version actuelle)
resonance
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 L'intention du projet était de produire une série d'impressions 3D des surfaces disponibles sur la galérie IMAGINARY de Herwig Hauser. L'intention du projet était de produire une série d'impressions 3D des surfaces disponibles sur la galérie IMAGINARY de Herwig Hauser.
-Ces surfaces devaient accompagner une exposition au Vieux Port, pour que les animateurs puissent s'en saisir et les utiliser dans les explications au public.+Ces surfaces devaient accompagner une exposition au Vieux Port de Marseille, pour que les animateurs puissent s'en saisir et les utiliser dans les explications au public.
  
 En tout, 7 surfaces ont été imprimées à temps pour l'exposition Mathématiques Vivantes et Visuelles, dont quatre sur l'Ultimaker 2 du fablab LFO :  En tout, 7 surfaces ont été imprimées à temps pour l'exposition Mathématiques Vivantes et Visuelles, dont quatre sur l'Ultimaker 2 du fablab LFO : 
Ligne 43: Ligne 43:
 ===== Formules ===== ===== Formules =====
  
-Dans la table ci-dessus, toutes les formules implicites proviennent de la galerie Herwig Hauser Classic sur Imaginary.org, avec éventuellement quelques constantes de rajoutées. Les paramétrisations correspondantes ont été calculées par Alba. L'intérêt d'une paramétrisation explicite pour la construction d'un modèle 3D est que les modèles qui en sortent sont plus propres, car approcher numériquement la solution d'une équation comporte des erreurs beaucoup plus grosses que d'évaluer une fonction. C'est particulièrement vrai dans le cas des équations polynomiales et l'évaluation des polynômes.+Dans la table ci-dessus, toutes les formules implicites proviennent des contributions de Herwig Hauser sur Imaginary.org (galerie Herwig Hauser classic et l'ensemble de surfaces algébriques de l'institut Forwiss), avec éventuellement quelques constantes de rajoutées. Les paramétrisations correspondantes ont été calculées par Alba. L'intérêt d'une paramétrisation explicite pour la construction d'un modèle 3D est que les modèles qui en sortent sont plus propres, car approcher numériquement la solution d'une équation comporte des erreurs beaucoup plus grosses que d'évaluer une fonction. C'est particulièrement vrai dans le cas des équations polynomiales et l'évaluation des polynômes.
  
 ^Nom^Équation polynomiale^Paramétrisation(s)^  ^Nom^Équation polynomiale^Paramétrisation(s)^ 
Ligne 49: Ligne 49:
 ^Limão^$x^2=y^3z^3$^$\left\{\begin{array}{rcl}x&=&u^3v^3\\y&=&\pm u^2\\z&=&\pm v^2\end{array}\right.$^  ^Limão^$x^2=y^3z^3$^$\left\{\begin{array}{rcl}x&=&u^3v^3\\y&=&\pm u^2\\z&=&\pm v^2\end{array}\right.$^ 
 ^Vis-à-vis^$x^2+y^2+y^4+z^3=x^3+z^4$^ ^ ^ ^Vis-à-vis^$x^2+y^2+y^4+z^3=x^3+z^4$^ ^ ^
-^Calypso^$x^2+y^2z=z^2$^$\left\{\begin{array}{rcl}x&=&v^2\sin(u)&\\y&=&v\cos(u)&\\z&=&v^2\end{array}\right.$^$\left\{\begin{array}{rcl}x&=&v^2\cos(u)&\\y&=&v\sqrt(2\sin(u))&\\z&=&v^2(\sin(u)-1)\end{array}\right.$^ +^Calypso^$x^2+y^2z=z^2$^$\left\{\begin{array}{rcl}x&=&v^2\sin(u)&\\y&=&v\cos(u)&\\z&=&v^2\end{array}\right.$^$\left\{\begin{array}{rcl}x&=&v^2\cos(u)&\\y&=&v\sqrt{2\sin(u)}&\\z&=&v^2(\sin(u)-1)\end{array}\right.$^ 
-^Calyx^$x^2+y^2z^3=z^4$^$\left\{\begin{array}{rcl}x&=&v^4\sin(u)&\\y&=&v\cos(u)&\\z&=&v^2\end{array}\right.$^$\left\{\begin{array}{rcl}x&=&v^4\sin(u)(\cos(u)-1)&\\y&=&v\sqrt(2\cos(u))&\\z&=&v^2(\cos(u)-1)\end{array}\right.$^+^Calyx^$x^2+y^2z^3=z^4$^$\left\{\begin{array}{rcl}x&=&v^4\sin(u)&\\y&=&v\cos(u)&\\z&=&v^2\end{array}\right.$^$\left\{\begin{array}{rcl}x&=&v^4\sin(u)(\cos(u)-1)&\\y&=&v\sqrt{2\cos(u)}&\\z&=&v^2(\cos(u)-1)\end{array}\right.$^
 ^Daisy^$(x^2-y^3)^2=(z^2-y^2)^3$^$\left\{\begin{array}{rcl}x&=&v^3(\sin(u)+(\sin^2(u)-1) (\sin(u)+\cos(u)))^2\\y&=&v^2 (\sin(u)+(\sin^2(u)-1) (\sin(u)+\cos(u))) \sin(u) \\z&=&v^2(\sin^3(u)-\cos^3(u))\end{array}\right.$^  ^Daisy^$(x^2-y^3)^2=(z^2-y^2)^3$^$\left\{\begin{array}{rcl}x&=&v^3(\sin(u)+(\sin^2(u)-1) (\sin(u)+\cos(u)))^2\\y&=&v^2 (\sin(u)+(\sin^2(u)-1) (\sin(u)+\cos(u))) \sin(u) \\z&=&v^2(\sin^3(u)-\cos^3(u))\end{array}\right.$^ 
 ^Diabolo^$x^2=(y^2+z^2)^2$^$\left\{\begin{array}{rcl}x&=&\pm(u^2+v^2)\\y&=&u\\z&=&v\end{array}\right.$^ ^Diabolo^$x^2=(y^2+z^2)^2$^$\left\{\begin{array}{rcl}x&=&\pm(u^2+v^2)\\y&=&u\\z&=&v\end{array}\right.$^
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 ^Herz^$x^2z^2+z^4=y^2+z^3$^$\left\{\begin{array}{rcl}x&=&\frac12v\sin(u)\\y&=&\pm\frac12\sqrt{v^2-1}(1+v\cos(u))\\z&=&\frac12(1+v\cos(u))\end{array}\right.$^  ^Herz^$x^2z^2+z^4=y^2+z^3$^$\left\{\begin{array}{rcl}x&=&\frac12v\sin(u)\\y&=&\pm\frac12\sqrt{v^2-1}(1+v\cos(u))\\z&=&\frac12(1+v\cos(u))\end{array}\right.$^ 
 ^Himmel & Hölle^$x^2=y^2z^2$^$\left\{\begin{array}{rcl}x&=&\pm uv\\y&=&u\\z&=&v\end{array}\right.$^  ^Himmel & Hölle^$x^2=y^2z^2$^$\left\{\begin{array}{rcl}x&=&\pm uv\\y&=&u\\z&=&v\end{array}\right.$^ 
-^Kolibri^$x^2=y^2z^2+z^3$^$\left\{\begin{array}{rcl}x&=&(u^2-v^2)\\y&=&v\\z&=&u^2-v^2\end{array}\right.$^ +^Kolibri^$x^2=y^2z^2+z^3$^$\left\{\begin{array}{rcl}x&=&u(u^2-v^2)\\y&=&v\\z&=&u^2-v^2\end{array}\right.$^ 
 ^Kreisel^$60(x^2+y^2)z^4=(60-x^2-y^2-z^2)^3$^$\left\{\begin{array}{rcl}x&=&\sqrt{\frac{60}{1+60(v^3+v^2)}}\cos(u)\\y&=&\sqrt{\frac{60}{1+60(v^3+v^2)}}\sin(u)\\z&=&\pm60\sqrt{\frac{v^3}{1+60(v^3+v^2)}}\end{array}\right.$^ ^Kreisel^$60(x^2+y^2)z^4=(60-x^2-y^2-z^2)^3$^$\left\{\begin{array}{rcl}x&=&\sqrt{\frac{60}{1+60(v^3+v^2)}}\cos(u)\\y&=&\sqrt{\frac{60}{1+60(v^3+v^2)}}\sin(u)\\z&=&\pm60\sqrt{\frac{v^3}{1+60(v^3+v^2)}}\end{array}\right.$^
 ^Miau^$x^2yz+x^2z^2+2y^3z+3y^3=0$^$ $^  ^Miau^$x^2yz+x^2z^2+2y^3z+3y^3=0$^$ $^ 
 ^Nepali^$(xy-z^3-1)^2=(1-x^2-y^2)^3$^$\left\{\begin{array}{rcl}x&=&\cos(v)\cos(u)\\y&=&\cos(v)\sin(u)\\z&=&\sqrt[3]{\frac12\cos^2(v)\sin(2u)-1-\sin^3(v)}\end{array}\right.$^  ^Nepali^$(xy-z^3-1)^2=(1-x^2-y^2)^3$^$\left\{\begin{array}{rcl}x&=&\cos(v)\cos(u)\\y&=&\cos(v)\sin(u)\\z&=&\sqrt[3]{\frac12\cos^2(v)\sin(2u)-1-\sin^3(v)}\end{array}\right.$^ 
-^ ^$ $^$\left\{\begin{array}{rcl}x&=&\\y&=&\\z&=&\end{array}\right.$^ +^Seepferdchen^$(x^2-y^3)^2=(x+y^2)z^3$^$ $^  
 +^Solitude^$x^2yz+xy^2+y^3+y^3z=x^2z^2$^$ $^  
 +^Tanz^$2x^4=x^2+y^2z^2$^$\left\{\begin{array}{rcl}x&=&\pm\frac12\sqrt{1+\sqrt{1+2u^2v^2}}\\y&=&\frac1{\sqrt{2}}u\\z&=&\frac1{\sqrt{2}}v\end{array}\right.$^  
 +^Taube^$256z^3 − 128x^2z^2+16x^4z+144xy^2z−4x^3y^2−27y^4=0$^$\left\{\begin{array}{rcl}x&=&3(u^2-v^2)\\y&=&\pm2v(3u^2-v^2)\\z&=&3u^2v^2\end{array}\right.$^$\left\{\begin{array}{rcl}x&=&-3(u^2+v^2)\\y&=&\pm2v(3u^2+v^2)\\z&=&-3u^2v^2\end{array}\right.$^  
 +^Tülle^$yz\cdot (x^2+y-z)=0$^$\left\{\begin{array}{rcl}x&=&u\\y&=&v\\z&=&u^2+v\end{array}\right.$^$\left\{\begin{array}{rcl}x&=&u\\y\text{ ou }z&=&v\\y\text{ ou }z&=&0\end{array}\right.$^  
 +^Zeck^$x^2+y^2=z^3\cdot(1-z)$^$\left\{\begin{array}{rcl}x&=&\sin(v)\cos^3(v)\cos(u)\\y&=&\sin(v)\cos^3(v)\sin(u)\\z&=&\cos^2(v)\end{array}\right.$^ 
 +^Spitz^$(y^2-x^2-z^2)^3=27x^2y^3z^2$^$\left\{\begin{array}{rcl}x&=&4v^3\cos(u)\\y&=&6v^2\sqrt[3]{\sin^2(2u)}-\sqrt{4v^2\sqrt[3]{sin(2u)^4}+16v^6}\\z&=&4v^3\sin(u)\end{array}\right.$^  
 +^Schneeflocke^$x^3+y^2z^3+yz^4=0$^$\left\{\begin{array}{rcl}x&=&\pm(u-v)\sqrt[3]{uv}\\y&=&u\\z&=&v-u\end{array}\right.$^  
 + 
 +===== Code ===== 
 +Ci-dessus, un script pour MathMod contenant toutes les surfaces listées dans la section Formules de ce wiki, ainsi qu'une paramétrisation pour la plupart d'entre elles.  
 + 
 +<code javascript>{ "MathModels": [  
 +{ "Iso3D": { "Component": [ "Zitrus" ], "Fxyz": [ "9*(x^2+z^2)-64*y^3*(1-y)^3" ], "Name": [ "Zitrus" ], "Xmax": [ "1/3" ], "Xmin": [ "-1/3" ], "Ymax": [ "1" ], "Ymin": [ "0" ], "Zmax": [ "1/3" ], "Zmin": [ "-1/3" ] } },  
 +{ "Param3D": { "Component": [ "Zitrus P" ], "Description": [ "Parametric Zitrus" ], "Fx": [ "(8/3)*sin(u)^3*cos(u)^3*cos(v)" ], "Fy": [ "cos(u)^2" ], "Fz": [ "(8/3)*sin(u)^3*cos(u)^3*sin(v)" ], "Name": [ "Zitrus P" ], "Umax": [ "pi/2" ], "Umin": [ "0" ], "Vmax": [ "2*pi" ], "Vmin": [ "0" ] } },  
 +{ "Iso3D": { "Component": [ "Limão" ], "Fxyz": [ "x^2-y^3*z^3" ], "Name": [ "Limão" ], "Xmax": [ "1" ], "Xmin": [ "-1" ], "Ymax": [ "1" ], "Ymin": [ "-1" ], "Zmax": [ "1" ], "Zmin": [ "-1" ] } },  
 +{ "Param3D": { "Component": ["Limão_+","Limão_-"], "Description": ["Parametric Limão"], "Fx": ["u^3*v^3","u^3*v^3"], "Fy": ["v^2","-v^2"], "Fz": ["u^2","-u^2"], "Name": ["Limão P"], "Umax": ["1","1"], "Umin": ["-1","-1"], "Vmax": ["1","1"], "Vmin": ["-1","-1"] } },  
 +{ "Iso3D": { "Component": [ "Vis-à-vis" ], "Fxyz": [ "x^2-x^3+y^2+y^4+z^3-z^4" ], "Name": [ "Vis-à-vis" ], "Xmax": [ "1.73" ], "Xmin": [ "-1.73" ], "Ymax": [ "2" ], "Ymin": [ "-2" ], "Zmax": [ "1.65" ], "Zmin": [ "-1.65" ] } },  
 +{ "Iso3D": { "Component": [ "Calypso" ], "Fxyz": [ "x^2+y^2*z-z^2" ], "Name": [ "Calypso" ], "Xmax": [ "2.55" ], "Xmin": [ "-2.55" ], "Ymax": [ "2.55" ], "Ymin": [ "-2.55" ], "Zmax": [ "2.55" ], "Zmin": [ "-2.55" ] } },  
 +{ "Param3D": { "Component": [ "CalypsoPolar_+", "CalypsoPolar_-" ], "Description": [ "Calypso parametrized polar" ], "Fx": [ "v^2*sin(u)", "v^2*cos(u)" ], "Fy": [ "v*cos(u)", "v*sqrt(2*sin(u))" ], "Fz": [ "v^2", "v^2*(sin(u)-1)" ], "Name": [ "Calypso polar" ], "Umax": [ "2*pi", "pi" ], "Umin": [ "0", "0" ], "Vmax": [ "1", "1" ], "Vmin": [ "0", "-1" ] } },  
 +{ "Iso3D": { "Component": [ "Calyx" ], "Fxyz": [ "x^2+y^2*z^3-z^4" ], "Name": [ "Calyx" ], "Xmax": [ "4" ], "Xmin": [ "-4" ], "Ymax": [ "2" ], "Ymin": [ "-2" ], "Zmax": [ "2" ], "Zmin": [ "-2" ] } },  
 +{ "Param3D": { "Component": [ "CalyxPolar_+", "CalyxPolar_-" ], "Description": [ "Calyx parametrized polar" ], "Fx": [ "v^4*sin(u)", "v^4*sin(u)*(cos(u)-1)" ], "Fy": [ "v*cos(u)", "v*sqrt(2*cos(u))" ], "Fz": [ "v^2", "v^2*(cos(u)-1)" ], "Name": [ "Calyx polar" ], "Umax": [ "2*pi", "pi/2" ], "Umin": [ "0", "-pi/2" ], "Vmax": [ "1", "1" ], "Vmin": [ "0", "-1" ] } },  
 +{ "Iso3D": { "Component": [ "Daisy" ], "Fxyz": [ "(x^2-y^3)^2-(z^2-y^2)^3" ], "Name": [ "Daisy" ], "Xmax": [ "0.1" ], "Xmin": [ "-0.1" ], "Ymax": [ "0.21" ], "Ymin": [ "-0.15" ], "Zmax": [ "0.21" ], "Zmin": [ "-0.21" ] } },  
 +{ "Param3D": { "Description": ["Parametric Daisy"], "Name": ["Daisy P"], "Component": ["Daisy_++","Daisy_+-","Daisy_-+","Daisy_--"], "Fx": ["sqrt((v+u)^3-u^3)","sqrt((v-u)^3+u^3)","-sqrt((v-u)^3+u^3)","-sqrt((v+u)^3-u^3)"], "Fy": ["-u","u","u","-u"], "Fz": ["sqrt((v+u)^2+u^2)","-sqrt((v-u)^2+u^2)","sqrt((v-u)^2+u^2)","-sqrt((v+u)^2+u^2)"], "Umax": [ "1", "1", "1", "1"], "Umin": ["-1","-1","-1","-1"], "Vmax": [ "1", "1", "1", "1"], "Vmin": [ "0", "0", "0", "0"] } },  
 +{ "Iso3D": { "Component": [ "Diabolo" ], "Fxyz": [ "x^2-(y^2+z^2)^2" ], "Name": [ "Diabolo" ], "Xmax": [ "1" ], "Xmin": [ "-1" ], "Ymax": [ "1" ], "Ymin": [ "-1" ], "Zmax": [ "1" ], "Zmin": [ "-1" ] } },  
 +{ "Param3D": { "Component": [ "DiaboloP_+","DiaboloP_-" ], "Description": [ "Parametric diabolo" ], "Fx": [ "u^2","-u^2" ], "Fy": [ "u*cos(v)","u*cos(v)" ], "Fz": [ "u*sin(v)","u*sin(v)" ], "Name": [ "Diabolo P" ], "Umax": [ "pi/2","0" ], "Umin": [ "0","-pi/2" ], "Vmax": [ "2*pi","2*pi" ], "Vmin": [ "0","0" ] } },  
 +{ "Iso3D": { "Component": [ "Ding Dong" ], "Fxyz": [ "x^2+y^2+z^3-z^2" ], "Name": [ "Ding Dong" ], "Xmax": [ "1.34" ], "Xmin": [ "-1.34" ], "Ymax": [ "1.34" ], "Ymin": [ "-1.34" ], "Zmax": [ "1" ], "Zmin": [ "-0.85" ] } },  
 +{ "Param3D": { "Component": [ "DingDongP_+", "DingDongP_-" ], "Description": [ "Parametric ding dong" ], "Fx": [ "2/(3*sqrt(3))*sin(3*v/2)*cos(u)", "-(v^6+1)/(3*sqrt(3)*v^3)*cos(u)" ], "Fy": [ "2/(3*sqrt(3))*sin(3*v/2)*sin(u)", "-(v^6+1)/(3*sqrt(3)*v^3)*sin(u)" ], "Fz": [ "(1+2*cos(v))/3", "-(1-v^2+v^4)/(3*v^2)" ], "Name": [ "Ding Dong P" ], "Umax": [ "2*pi", "2*pi" ], "Umin": [ "0", "0" ], "Vmax": [ "pi", "1.7" ], "Vmin": [ "0", "1"] } }, 
 +{ "Iso3D": { "Component": [ "Distel" ], "Fxyz": [ "x^2+y^2+z^2+1500*(x^2+y^2)*(x^2+z^2)*(y^2+z^2)-1" ], "Name": [ "Distel" ], "Xmax": [ "1" ], "Xmin": [ "-1" ], "Ymax": [ "1" ], "Ymin": [ "-1" ], "Zmax": [ "1" ], "Zmin": [ "-1" ] } },  
 +{ "Iso3D": { "Component": [ "Dullo" ], "Fxyz": [ "(x^2+y^2+z^2)^2-(x^2+y^2)" ], "Name": [ "Dullo" ], "Xmax": [ "1" ], "Xmin": [ "-1" ], "Ymax": [ "1" ], "Ymin": [ "-1" ], "Zmax": [ "0.5" ], "Zmin": [ "-0.5" ] } },  
 +{ "Param3D": { "Component": [ "Dullo P" ], "Description": [ "Parametric dullo" ], "Fx": [ "(1-cos(u))*cos(v)/2" ], "Fy": [ "(1-cos(u))*sin(v)/2" ], "Fz": [ "-sin(u)/2" ], "Name": [ "Dullo P" ], "Umax": [ "2*pi" ], "Umin": [ "0" ], "Vmax": [ "2*pi" ], "Vmin": [ "0" ] } },  
 +{ "Iso3D": { "Component": [ "Eistüte" ], "Fxyz": [ "(x^2+y^2)^3-4*x^2*y^2*(z^2+1)" ], "Name": [ "Eistüte" ], "Xmax": [ "10" ], "Xmin": [ "-10" ], "Ymax": [ "10" ], "Ymin": [ "-10" ], "Zmax": [ "10" ], "Zmin": [ "-10" ] } },  
 +{ "Param3D": { "Component": [ "Eistüte P" ], "Description": [ "Parametric Eistüte" ], "Fx": [ "sin(2*u)*sin(u)*sqrt(v^2+1)" ], "Fy": [ "sin(2*u)*cos(u)*sqrt(v^2+1)" ], "Fz": [ "v" ], "Name": [ "Eistüte P" ], "Umax": [ "2*pi" ], "Umin": [ "0" ], "Vmax": [ "10" ], "Vmin": [ "-10" ] } },  
 +{ "Iso3D": { "Component": [ "Helix" ], "Fxyz": [ "2*x^4+y^2*z^2-6*x^2" ], "Name": [ "Helix" ], "Xmax": [ "sqrt(3)" ], "Xmin": [ "-sqrt(3)" ], "Ymax": [ "3*sqrt(1.5)" ], "Ymin": [ "-3*sqrt(1.5)" ], "Zmax": [ "3*sqrt(1.5)" ], "Zmin": [ "-3*sqrt(1.5)" ] } },   
 +{ "Param3D": { "Component": [ "Helix_X++", "Helix_X+-", "Helix_X-+", "Helix_X--", "Helix_Y++", "Helix_Y+-", "Helix_Y-+", "Helix_Y--", "Helix_Z++", "Helix_Z+-", "Helix_Z-+", "Helix_Z--" ], "Description": [ "Parametric Helix" ], "Fx": [ "sqrt(1.5+sqrt(2.25-u^2*v^2))", "sqrt(1.5-sqrt(2.25-u^2*v^2))", "-sqrt(1.5+sqrt(2.25-u^2*v^2))", "-sqrt(1.5-sqrt(2.25-u^2*v^2))", "sqrt(1.5*(1+cos(u)))", "sqrt(1.5*(1+sin(u)))", "-sqrt(1.5*(1+sin(u)))", "-sqrt(1.5*(1+cos(u)))", "sqrt(1.5*(1+sin(u)))", "sqrt(1.5*(1+cos(u)))", "-sqrt(1.5*(1+cos(u)))", "-sqrt(1.5*(1+sin(u)))" ], "Fy": [ "v", "u", "u", "v", "sqrt(1.5)*v", "sqrt(1.5)*v", "sqrt(1.5)*v", "sqrt(1.5)*v", "sqrt(1.5)*cos(u)/v", "sqrt(1.5)*sin(u)/v", "sqrt(1.5)*sin(u)/v", "sqrt(1.5)*cos(u)/v" ], "Fz": [ "u", "v", "v", "u", "sqrt(1.5)*sin(u)/v", "sqrt(1.5)*cos(u)/v", "sqrt(1.5)*cos(u)/v", "sqrt(1.5)*sin(u)/v", "sqrt(1.5)*v", "sqrt(1.5)*v", "sqrt(1.5)*v", "sqrt(1.5)*v" ], "Name": [ "Helix P" ], "Umax": [ "sqrt(1.5)", "sqrt(1.5)", "sqrt(1.5)", "sqrt(1.5)", "2*pi", "2*pi", "2*pi", "2*pi", "2*pi", "2*pi", "2*pi", "2*pi" ], "Umin": [ "-sqrt(1.5)", "-sqrt(1.5)", "-sqrt(1.5)", "-sqrt(1.5)", "0", "0", "0", "0", "0", "0", "0", "0" ], "Vmax": [ "sqrt(1.5)", "sqrt(1.5)", "sqrt(1.5)", "sqrt(1.5)", "3", "-1", "3", "-1", "3", "-1", "3", "-1" ], "Vmin": [ "-sqrt(1.5)", "-sqrt(1.5)", "-sqrt(1.5)", "-sqrt(1.5)", "1", "-3", "1", "-3", "1", "-3", "1", "-3" ] } },  
 +{ "Iso3D": { "Component": [ "Herz" ], "Fxyz": [ "x^2*z^2+z^4-y^2-z^3" ], "Name": [ "Herz" ], "Xmax": [ "3" ], "Xmin": [ "-3" ], "Ymax": [ "2.7" ], "Ymin": [ "-2.7" ], "Zmax": [ "3.5" ], "Zmin": [ "-2.5" ] } },  
 +{ "Param3D": { "Component": [ "HerzPolar_+", "HerzPolar_-" ], "Description": [ "Herz parametrized polar" ], "Fx": [ "0.5*v*sin(u)", "0.5*v*cos(u)" ], "Fy": [ "0.5*sqrt(v^2-1)*(1+v*cos(u))", "-0.5*sqrt(v^2-1)*(1+v*sin(u))" ], "Fz": [ "0.5*(1+v*cos(u))", "0.5*(1+v*sin(u))" ], "Name": [ "Herz polar" ], "Umax": [ "2*pi", "2*pi" ], "Umin": [ "0", "0" ], "Vmax": [ "2", "2" ], "Vmin": [ "1", "1" ] } },  
 +{ "Iso3D": { "Component": [ "HimmelHolle_01", "HimmelHolle_02" ], "Fxyz": [ "x-y*z", "x+y*z" ], "Name": [ "Himmel & Hölle" ], "Xmax": [ "1", "1" ], "Xmin": [ "-1", "-1" ], "Ymax": [ "1", "1" ], "Ymin": [ "-1", "-1" ], "Zmax": [ "1", "1" ], "Zmin": [ "-1", "-1" ] } },  
 +{ "Param3D": { "Component": [ "HimmerHolleP_01", "HimmerHolleP_02" ], "Description": [ "parametric Himmer & Hölle" ], "Fx": [ "u*v", "-u*v" ], "Fy": [ "u", "u" ], "Fz": [ "v", "v" ], "Name": [ "Himmer & Hölle P" ], "Umax": [ "1.2", "1.2" ], "Umin": [ "-1.2", "-1.2" ], "Vmax": [ "1.2", "1.2" ], "Vmin": [ "-1.2", "-1.2" ] } },  
 +{ "Iso3D": { "Component": [ "Kolibri" ], "Fxyz": [ "y^2*z^2+z^3-x^2" ], "Name": [ "Kolibri" ], "Xmax": [ "1" ], "Xmin": [ "-1" ], "Ymax": [ "1" ], "Ymin": [ "-1" ], "Zmax": [ "1" ], "Zmin": [ "-1" ] } },  
 +{ "Param3D": { "Component": [ "KolibriP" ], "Description": [ "parametrized Kolibri" ], "Fx": [ "u*(u^2-v^2)" ], "Fy": [ "v" ], "Fz": [ "u^2-v^2" ], "Name": [ "Kolibri P" ], "Umax": [ "1" ], "Umin": [ "-1" ], "Vmax": [ "1" ], "Vmin": [ "-1" ] } },  
 +{ "Iso3D": { "Component": [ "Kreisel" ], "Fxyz": [ "(60-x^2-y^2-z^2)^3-60*(x^2+y^2)*z^4" ], "Name": [ "Kreisel" ], "Xmax": [ "sqrt(60)" ], "Xmin": [ "-sqrt(60)" ], "Ymax": [ "sqrt(60)" ], "Ymin": [ "-sqrt(60)" ], "Zmax": [ "sqrt(60)" ], "Zmin": [ "-sqrt(60)" ] } },  
 +{ "Param3D": { "Component": [ "Kreisel_03", "Kreisel_04", "Kreisel_02", "Kreisel_01" ], "Description": [ "Kreisel parametrized polar" ], "Fx": [ "sqrt(60/(1+60*(-v^3+v^2)))*cos(u)", "sqrt(60/(1+60*(v^3+v^2)))*cos(u)", "sqrt(-60*v^3/(-v^3+60*(1-v)))*cos(u)", "sqrt(60*v^3/(v^3+60*(1+v)))*cos(u)" ], "Fy": [ "sqrt(60/(1+60*(-v^3+v^2)))*sin(u)", "sqrt(60/(1+60*(v^3+v^2)))*sin(u)", "sqrt(-60*v^3/(-v^3+60*(1-v)))*sin(u)", "sqrt(60*v^3/(v^3+60*(1+v)))*sin(u)" ], "Fz": [ "60*sqrt(-v^3/(1+60*(v^2-v^3)))", "-60*sqrt(v^3/(1+60*(v^3+v^2)))", "-60/sqrt(-v^3+60*(1-v))", "60/sqrt(v^3+60*(1+v))" ], "Name": [ "Kreisel polar" ], "Umax": [ "2*pi", "2*pi", "2*pi", "2*pi" ], "Umin": [ "0", "0", "0", "0" ], "Vmax": [ "0", "0.25", "0", "4" ], "Vmin": [ "-0.25", "0", "-4", "0" ] } },{ "Iso3D": { "Component": [ "Miau" ], "Fxyz": [ "x^2*y*z+x^2*z^2+2*y^3*z+3*y^3" ], "Name": [ "Miau" ], "Xmax": [ "5" ], "Xmin": [ "-5" ], "Ymax": [ "5" ], "Ymin": [ "-5" ], "Zmax": [ "5" ], "Zmin": [ "-5" ] } },   
 +{ "Iso3D": { "Component": [ "Nepali" ], "Fxyz": [ "(x*y-z^3-1)^2-(1-x^2-y^2)^3" ], "Name": [ "Nepali" ], "Xmax": [ "1" ], "Xmin": [ "-1" ], "Ymax": [ "1" ], "Ymin": [ "-1" ], "Zmax": [ "0" ], "Zmin": [ "-1.26" ] } },  
 +{ "Param3D": { "Component": [ "Nepali Polar" ], "Description": [ "Nepali parametrized polar" ], "Fx": [ "sin(v)*cos(u)" ], "Fy": [ "sin(v)*sin(u)" ], "Fz": [ "-(-(sin(v)^2*sin(2*u)/2-1+cos(v)^3))^(1/3)" ], "Name": [ "Nepali polar" ], "Umax": [ "2*pi" ], "Umin": [ "0" ], "Vmax": [ "pi" ], "Vmin": [ "0" ] } }, 
 +{ "Iso3D": { "Component": [ "Seepferdchen" ], "Fxyz": [ "(x^2-y^3)^2-(x+y^2)*z^3" ], "Name": [ "Seepferdchen" ], "Xmax": [ "25" ], "Xmin": [ "-25" ], "Ymax": [ "15" ], "Ymin": [ "-15" ], "Zmax": [ "25" ], "Zmin": [ "-25" ] } },  
 +{ "Iso3D": { "Component": [ "Solitude" ], "Fxyz": [ "x^2*y*z+x*y^2+y^3+y^3*z-x^2*z^2" ], "Name": [ "Solitude" ], "Xmax": [ "25" ], "Xmin": [ "-25" ], "Ymax": [ "25" ], "Ymin": [ "-25" ], "Zmax": [ "25" ], "Zmin": [ "-25" ] } },  
 +{ "Iso3D": { "Component": [ "Tanz" ], "Fxyz": [ "2*x^4-x^2-y^2*z^2" ], "Name": [ "Tanz" ], "Xmax": [ "25" ], "Xmin": [ "-25" ], "Ymax": [ "25" ], "Ymin": [ "-25" ], "Zmax": [ "25" ], "Zmin": [ "-25" ] } },  
 +{ "Param3D": { "Component": [ "TanzP_+", "TanzP_-" ], "Description": [ "parametrized Tanz" ], "Fx": [ "sqrt(1+sqrt(1+2*u^2*v^2))/2", "-sqrt(1+sqrt(1+2*u^2*v^2))/2" ], "Fy": [ "u/sqrt(2)", "u/sqrt(2)" ], "Fz": [ "v/sqrt(2)", "v/sqrt(2)" ], "Name": [ "Tanz P" ], "Umax": [ "25", "25" ], "Umin": [ "-25", "-25" ], "Vmax": [ "25", "25" ], "Vmin": [ "-25", "-25" ] } }, 
 +{ "Iso3D": { "Component": [ "Taube" ], "Fxyz": [ "256*z^3-128*x^2*z^2+16*x^4*z+144*x*y^2*z-4*x^3*y^2-27*y^4" ], "Name": [ "Taube" ], "Xmax": [ "25" ], "Xmin": [ "-25" ], "Ymax": [ "25" ], "Ymin": [ "-25" ], "Zmax": [ "25" ], "Zmin": [ "-25" ] } },  
 +{ "Param3D": { "Component": [ "TaubePs_1", "TaubePs_2", "TaubePs_3", "TaubePs_4" ], "Description": [ "Parametric Taube" ], "Fx": [ "3*(u^2-v^2)", "-3*(u^2+v^2)", "-3*(u^2+v^2)", "3*(v^2-u^2)" ], "Fy": [ "2*v*(3*u^2-v^2)", "-2*v*(3*u^2+v^2)", "2*u*(3*v^2+u^2)", "-2*u*(3*v^2-u^2)" ], "Fz": [ "3*u^2*v^2", "-3*u^2*v^2", "-3*u^2*v^2", "3*u^2*v^2" ], "Name": [ "Taube Ps" ], "Umax": [ "2", "2", "2", "2" ], "Umin": [ "0", "0", "0", "0" ], "Vmax": [ "2", "2", "2", "2" ], "Vmin": [ "0", "0", "0", "0" ] } },  
 +{ "Iso3D": { "Component": [ "Tulle_01", "Tulle_02", "Tulle_03" ], "Fxyz": [ "x^2+y-z", "y", "z" ], "Name": [ "Tulle" ], "Xmax": [ "2", "2", "2" ], "Xmin": [ "-2", "-2", "-2" ], "Ymax": [ "2", "2", "2" ], "Ymin": [ "-2", "-2", "-2" ], "Zmax": [ "2", "2", "2" ], "Zmin": [ "-2", "-2", "-2" ] } },  
 +{ "Param3D": { "Component": [ "TulleP_01", "TulleP_02", "TulleP_03" ], "Description": [ "parametrized Tulle" ], "Fx": [ "v", "u", "u" ], "Fy": [ "0.5*(u-v^2)", "0", "v" ], "Fz": [ "0.5*(u+v^2)", "v", "0" ], "Name": [ "Tulle P" ], "Umax": [ "1", "1", "1" ], "Umin": [ "-1", "-1", "-1" ], "Vmax": [ "1", "1", "1" ], "Vmin": [ "-1", "-1", "-1" ] } }, 
 +{ "Iso3D": { "Component": [ "Zeek" ], "Fxyz": [ "x^2+y^2-z^3*(1-z)" ], "Name": [ "Zeek" ], "Xmax": [ "0.1875*sqrt(3)" ], "Xmin": [ "-0.1875*sqrt(3)" ], "Ymax": [ "0.1875*sqrt(3)" ], "Ymin": [ "-0.1875*sqrt(3)" ], "Zmax": [ "1" ], "Zmin": [ "0" ] } },  
 +{ "Param3D": { "Component": [ "Zeek Polar" ], "Description": [ "Parametric Zeek" ], "Fx": [ "sin(v)*cos(v)^3*cos(u)" ], "Fy": [ "sin(v)*cos(v)^3*sin(u)" ], "Fz": [ "cos(v)^2" ], "Name": [ "Zeek Polar" ], "Umax": [ "pi/2" ], "Umin": [ "-pi/2" ], "Vmax": [ "pi/2" ], "Vmin": [ "-pi/2" ] } },  
 +{ "Iso3D": { "Component": [ "Spitz" ], "Fxyz": [ "(y^2-x^2-z^2)^3-27*x^2*y^3*z^2" ], "Name": [ "Spitz" ], "Xmax": [ "2" ], "Xmin": [ "-2" ], "Ymax": [ "2" ], "Ymin": [ "-2" ], "Zmax": [ "2" ], "Zmin": [ "-2" ] } },  
 +{ "Param3D": { "Component": [ "SpitzPolar_+", "SpitzPolar_-" ], "Description": [ "Parametric Spitz" ], "Fx": [ "4*v^3*cos(u)", "4*v^3*cos(u)" ], "Fy": [ "6*v^2*(sin(2*u)^2)^(1/3)+sqrt(4*v^2*(sin(2*u)^4)^(1/3)+16*v^6)", "6*v^2*(sin(2*u)^2)^(1/3)-sqrt(4*v^2*(sin(2*u)^4)^(1/3)+16*v^6)" ], "Fz": [ "4*v^3*sin(u)", "4*v^3*sin(u)" ], "Name": [ "Spitz Polar" ], "Umax": [ "2*pi", "2*pi" ], "Umin": [ "0", "0" ], "Vmax": [ "1.5874", "1.5874" ], "Vmin": [ "0", "0" ] } },  
 +{ "Iso3D": { "Component": [ "Schneeflocke" ], "Fxyz": [ "x^3+y^2*z^3+y*z^4" ], "Name": [ "Schneeflocke" ], "Xmax": [ "1" ], "Xmin": [ "-1" ], "Ymax": [ "1" ], "Ymin": [ "-1" ], "Zmax": [ "1" ], "Zmin": [ "-1" ] } },  
 +{ "Param3D": { "Component": [ "SchneeflockePX_1", "SchneeflockePX_2", "SchneeflockePX_3", "SchneeflockePX_4" ], "Description": [ "Parametric Schneeflocke" ], "Fx": [ "(u-v)*(u*v)^(1/3)", "-(u-v)*(-u*v)^(1/3)", "(u-v)*(u*v)^(1/3)", "-(u-v)*(-u*v)^(1/3)" ], "Fy": [ "u", "u", "u", "u" ], "Fz": [ "v-u", "v-u", "v-u", "v-u" ], "Name": [ "Schneeflocke P X" ], "Umax": [ "2", "0", "0", "2" ], "Umin": [ "0", "-2", "-2", "0" ], "Vmax": [ "2", "2", "0", "0" ], "Vmin": [ "0", "0", "-2", "-2" ] } }  
 +] } </code> 
 +Voici un script pour MathMod contenant les surface accompagnées ou bien d'une sphère avec laquelle les intersecter, ou bien d'une sphère les contenant. 
 +<code javascript>{ "MathModels": [  
 +{ "Param3D": { "Component": [ "Zitrus P", "Sphere" ], "Description": [ "Parametric Zitrus with sphere" ], "Fx": [ "(8/3)*sin(u)^3*cos(u)^3*cos(v)", "1/2*cos(v)*cos(u)" ], "Fy": [ "cos(u)^2", "1/2*cos(v)*sin(u)+1/2" ], "Fz": [ "(8/3)*sin(u)^3*cos(u)^3*sin(v)", "1/2*sin(v)" ], "Name": [ "Zitrus⊂Sphere½" ], "Umax": [ "pi/2", "2*pi" ], "Umin": [ "0", "0" ], "Vmax": [ "2*pi", "pi/2" ], "Vmin": [ "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "Limão_+", "Limão_-", "Sphere" ], "Description": [ "Parametric Limão" ], "Fx": [ "u^3*v^3", "u^3*v^3", "cos(v)*cos(u)" ], "Fy": [ "v^2", "-v^2", "cos(v)*sin(u)" ], "Fz": [ "u^2", "-u^2", "sin(v)" ], "Name": [ "Limão+Sphere1" ], "Umax": [ "1", "1", "2*pi" ], "Umin": [ "-1", "-1", "0" ], "Vmax": [ "1", "1", "pi/2" ], "Vmin": [ "-1", "-1", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "CalypsoPolar_+", "CalypsoPolar_-", "Sphere" ], "Description": [ "Calypso parametrized polar" ], "Fx": [ "v^2*sin(u)", "v^2*cos(u)", "sqrt(2)*cos(v)*cos(u)" ], "Fy": [ "v*cos(u)", "v*sqrt(2*sin(u))", "sqrt(2)*cos(v)*sin(u)" ], "Fz": [ "v^2", "v^2*(sin(u)-1)", "sqrt(2)*sin(v)" ], "Name": [ "Calypso+Sphere√2" ], "Umax": [ "2*pi", "pi", "2*pi" ], "Umin": [ "0", "0", "0" ], "Vmax": [ "1", "1", "pi/2" ], "Vmin": [ "0", "-1", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "CalyxPolar_+", "CalyxPolar_-", "Sphere" ], "Description": [ "Calyx parametrized polar" ], "Fx": [ "v^4*sin(u)", "v^4*sin(u)*(cos(u)-1)", "sqrt(2)*cos(v)*cos(u)" ], "Fy": [ "v*cos(u)", "v*sqrt(2*cos(u))", "sqrt(2)*cos(v)*sin(u)" ], "Fz": [ "v^2", "v^2*(cos(u)-1)", "sqrt(2)*sin(v)" ], "Name": [ "Calyx+Sphere√2" ], "Umax": [ "2*pi", "pi/2", "2*pi" ], "Umin": [ "0", "-pi/2", "0" ], "Vmax": [ "1", "1.121", "pi/2" ], "Vmin": [ "0", "-1.121", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "Daisy_++", "Daisy_+-", "Daisy_-+", "Daisy_--", "Sphere" ], "Description": [ "Parametric Daisy" ], "Fx": [ "sqrt((v+u)^3-u^3)", "sqrt((v-u)^3+u^3)", "-sqrt((v-u)^3+u^3)", "-sqrt((v+u)^3-u^3)", "cos(v)*sin(u)" ], "Fy": [ "-u", "u", "u", "-u", "cos(v)*cos(u)" ], "Fz": [ "sqrt((v+u)^2+u^2)", "-sqrt((v-u)^2+u^2)", "sqrt((v-u)^2+u^2)", "-sqrt((v+u)^2+u^2)", "sin(v)" ], "Name": [ "Daisy + Sphere 1" ], "Umax": [ "1/sqrt(3)", "1/2*(sqrt(5)-1)", "1/2*(sqrt(5)-1)", "1/sqrt(3)", "2*pi" ], "Umin": [ "1/2*(1-sqrt(5))", "-1/sqrt(3)", "-1/sqrt(3)", "1/2*(1-sqrt(5))", "0" ], "Vmax": [ "1.01881", "1.0181", "1.0181", "1.0181", "pi/2" ], "Vmin": [ "0", "0", "0", "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "DiaboloP_+", "DiaboloP_-", "Sphere" ], "Description": [ "Parametric diabolo" ], "Fx": [ "u^2", "-u^2", "sqrt(2)*cos(v)*cos(u)" ], "Fy": [ "u*cos(v)", "u*cos(v)", "sqrt(2)*cos(v)*sin(u)" ], "Fz": [ "u*sin(v)", "u*sin(v)", "sqrt(2)*sin(v)" ], "Name": [ "Diabolo+Sphere√2" ], "Umax": [ "1", "0", "2*pi" ], "Umin": [ "0", "-1", "0" ], "Vmax": [ "2*pi", "2*pi", "pi/2" ], "Vmin": [ "0", "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "DingDongP_+", "DingDongP_-", "Sphere" ], "Description": [ "Parametric ding dong" ], "Fx": [ "2/(3*sqrt(3))*sin(3*v/2)*cos(u)", "-(v^6+1)/(3*sqrt(3)*v^3)*cos(u)", "cos(v)*cos(u)" ], "Fy": [ "2/(3*sqrt(3))*sin(3*v/2)*sin(u)", "-(v^6+1)/(3*sqrt(3)*v^3)*sin(u)", "cos(v)*sin(u)" ], "Fz": [ "(1+2*cos(v))/3", "-(1-v^2+v^4)/(3*v^2)", "sin(v)" ], "Name": [ "Ding Dong + Sphere1" ], "Umax": [ "2*pi", "2*pi", "2*pi" ], "Umin": [ "0", "0", "0" ], "Vmax": [ "pi", "1.564", "pi/2" ], "Vmin": [ "0", "1", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "Dullo P", "Sphere" ], "Description": [ "Parametric dullo" ], "Fx": [ "(1-cos(u))*cos(v)/2", "cos(v)*cos(u)" ], "Fy": [ "(1-cos(u))*sin(v)/2", "cos(v)*sin(u)" ], "Fz": [ "-sin(u)/2", "sin(v)" ], "Name": [ "Dullo⊂Sphere1" ], "Umax": [ "2*pi", "2*pi" ], "Umin": [ "0", "0" ], "Vmax": [ "2*pi", "pi/2" ], "Vmin": [ "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "Eistüte P", "Sphere" ], "Description": [ "Parametric Eistüte" ], "Fx": [ "sin(2*u)*sin(u)*cosh(v)", "sinh(3)*cos(v)*cos(u)" ], "Fy": [ "sin(2*u)*cos(u)*cosh(v)", "sinh(3)*cos(v)*sin(u)" ], "Fz": [ "sinh(v)", "sinh(3)*sin(v)" ], "Name": [ "Eistüte+SphereSH3" ], "Umax": [ "2*pi", "2*pi" ], "Umin": [ "0", "0" ], "Vmax": [ "3", "pi/2" ], "Vmin": [ "-3", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "Helix_X++", "Helix_X+-", "Helix_X-+", "Helix_X--", "Helix_Y++", "Helix_Y+-", "Helix_Y-+", "Helix_Y--", "Helix_Z++", "Helix_Z+-", "Helix_Z-+", "Helix_Z--", "Sphere" ], "Description": [ "Parametric Helix" ], "Fx": [ "sqrt(1.5+sqrt(2.25-u^2*v^2))", "sqrt(1.5-sqrt(2.25-u^2*v^2))", "-sqrt(1.5+sqrt(2.25-u^2*v^2))", "-sqrt(1.5-sqrt(2.25-u^2*v^2))", "sqrt(1.5*(1+cos(u)))", "sqrt(1.5*(1+sin(u)))", "-sqrt(1.5*(1+sin(u)))", "-sqrt(1.5*(1+cos(u)))", "sqrt(1.5*(1+sin(u)))", "sqrt(1.5*(1+cos(u)))", "-sqrt(1.5*(1+cos(u)))", "-sqrt(1.5*(1+sin(u)))", "3*sqrt(1.5)*cos(v)*cos(u)" ], "Fy": [ "v", "u", "u", "v", "sqrt(1.5)*v", "sqrt(1.5)*v", "sqrt(1.5)*v", "sqrt(1.5)*v", "sqrt(1.5)*cos(u)/v", "sqrt(1.5)*sin(u)/v", "sqrt(1.5)*sin(u)/v", "sqrt(1.5)*cos(u)/v", "3*sqrt(1.5)*cos(v)*sin(u)" ], "Fz": [ "u", "v", "v", "u", "sqrt(1.5)*sin(u)/v", "sqrt(1.5)*cos(u)/v", "sqrt(1.5)*cos(u)/v", "sqrt(1.5)*sin(u)/v", "sqrt(1.5)*v", "sqrt(1.5)*v", "sqrt(1.5)*v", "sqrt(1.5)*v", "3*sqrt(1.5)*sin(v)" ], "Name": [ "Helix + Sphere3" ], "Umax": [ "sqrt(1.5)", "sqrt(1.5)", "sqrt(1.5)", "sqrt(1.5)", "2*pi", "2*pi", "2*pi", "2*pi", "2*pi", "2*pi", "2*pi", "2*pi", "2*pi" ], "Umin": [ "-sqrt(1.5)", "-sqrt(1.5)", "-sqrt(1.5)", "-sqrt(1.5)", "0", "0", "0", "0", "0", "0", "0", "0", "0" ], "Vmax": [ "sqrt(1.5)", "sqrt(1.5)", "sqrt(1.5)", "sqrt(1.5)", "3", "-1", "3", "-1", "3", "-1", "3", "-1", "pi/2" ], "Vmin": [ "-sqrt(1.5)", "-sqrt(1.5)", "-sqrt(1.5)", "-sqrt(1.5)", "1", "-3", "1", "-3", "1", "-3", "1", "-3", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "HerzPolar", "Sphere" ], "Description": [ "Herz parametrized polar" ], "Fx": [ "0.5*cosh(v)*sin(u)", "cos(v)*sin(u)" ], "Fy": [ "0.5*sinh(v)*(1+cosh(v)*cos(u))", "cos(v)*cos(u)" ], "Fz": [ "0.5*(1+cosh(v)*cos(u))", "sin(v)" ], "Name": [ "Herz+Sphere1" ], "Umax": [ "2*pi", "2*pi" ], "Umin": [ "0", "0" ], "Vmax": [ "1.47", "pi/2" ], "Vmin": [ "-1.47", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "HerzPolar", "Sphere" ], "Description": [ "Herz parametrized polar" ], "Fx": [ "0.5*cosh(v)*sin(u)", "sqrt(2)*cos(v)*sin(u)" ], "Fy": [ "0.5*sinh(v)*(1+cosh(v)*cos(u))", "sqrt(2)*cos(v)*cos(u)" ], "Fz": [ "0.5*(1+cosh(v)*cos(u))", "sqrt(2)*sin(v)" ], "Name": [ "Herz+Sphere√2" ], "Umax": [ "2*pi", "2*pi" ], "Umin": [ "0", "0" ], "Vmax": [ "1.77", "pi/2" ], "Vmin": [ "-1.77", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "HerzPolar", "Sphere" ], "Description": [ "Herz parametrized polar" ], "Fx": [ "0.5*cosh(v)*sin(u)", "2*cos(v)*sin(u)" ], "Fy": [ "0.5*sinh(v)*(1+cosh(v)*cos(u))", "2*cos(v)*cos(u)" ], "Fz": [ "0.5*(1+cosh(v)*cos(u))", "2*sin(v)" ], "Name": [ "Herz+Sphere2" ], "Umax": [ "2*pi", "2*pi" ], "Umin": [ "0", "0" ], "Vmax": [ "2.1", "pi/2" ], "Vmin": [ "-2.1", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "HimmerHolleP_01", "HimmerHolleP_02", "Sphere" ], "Description": [ "parametric Himmer & Hölle" ], "Fx": [ "u*v", "-u*v", "cos(v)*sin(u)" ], "Fy": [ "u", "u", "cos(v)*cos(u)" ], "Fz": [ "v", "v", "sin(v)" ], "Name": [ "Himmer & Hölle + Sphere 1" ], "Umax": [ "1", "1", "2*pi" ], "Umin": [ "-1", "-1", "0" ], "Vmax": [ "1", "1", "pi/2" ], "Vmin": [ "-1", "-1", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "KolibriP", "Sphere" ], "Description": [ "parametrized Kolibri" ], "Fx": [ "u*(u^2-v^2)", "cos(v)*cos(u)" ], "Fy": [ "v", "cos(v)*sin(u)" ], "Fz": [ "u^2-v^2", "sin(v)" ], "Name": [ "Kolibri + Sphere 1" ], "Umax": [ "5^(1/4)/sqrt(2)", "2*pi" ], "Umin": [ "-5^(1/4)/sqrt(2)", "0" ], "Vmax": [ "5^(1/4)/sqrt(2)", "pi/2" ], "Vmin": [ "-5^(1/4)/sqrt(2)", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "KolibriP", "Sphere" ], "Description": [ "parametrized Kolibri" ], "Fx": [ "u*(u^2-v^2)", "sqrt(2)*cos(v)*cos(u)" ], "Fy": [ "v", "sqrt(2)*cos(v)*sin(u)" ], "Fz": [ "u^2-v^2", "sqrt(2)*sin(v)" ], "Name": [ "Kolibri + Sphere √2" ], "Umax": [ "sqrt(1/2+sqrt(5/2))", "2*pi" ], "Umin": [ "-sqrt(1/2+sqrt(5/2))", "0" ], "Vmax": [ "sqrt(1/2+sqrt(5/2))", "pi/2" ], "Vmin": [ "-sqrt(1/2+sqrt(5/2))", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "KolibriP", "Sphere" ], "Description": [ "parametrized Kolibri" ], "Fx": [ "u*(u^2-v^2)", "2*cos(v)*cos(u)" ], "Fy": [ "v", "2*cos(v)*sin(u)" ], "Fz": [ "u^2-v^2", "2*sin(v)" ], "Name": [ "Kolibri + Sphere 2" ], "Umax": [ "sqrt(1/2*(3+sqrt(26)))", "2*pi" ], "Umin": [ "-sqrt(1/2*(3+sqrt(26)))", "0" ], "Vmax": [ "sqrt(1/2*(3+sqrt(26)))", "pi/2" ], "Vmin": [ "-sqrt(1/2*(3+sqrt(26)))", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "Kreisel_03", "Kreisel_04", "Kreisel_02", "Kreisel_01", "Kreisel_01_01" ], "Description": [ "Kreisel parametrized polar" ], "Fx": [ "sqrt(60/(1+60*(-v^3+v^2)))*cos(u)", "sqrt(60/(1+60*(v^3+v^2)))*cos(u)", "sqrt(-60*v^3/(-v^3+60*(1-v)))*cos(u)", "sqrt(60*v^3/(v^3+60*(1+v)))*cos(u)", "2*sqrt(15)*cos(v)*cos(u)" ], "Fy": [ "sqrt(60/(1+60*(-v^3+v^2)))*sin(u)", "sqrt(60/(1+60*(v^3+v^2)))*sin(u)", "sqrt(-60*v^3/(-v^3+60*(1-v)))*sin(u)", "sqrt(60*v^3/(v^3+60*(1+v)))*sin(u)", "2*sqrt(15)*cos(v)*sin(u)" ], "Fz": [ "60*sqrt(-v^3/(1+60*(v^2-v^3)))", "-60*sqrt(v^3/(1+60*(v^3+v^2)))", "-60/sqrt(-v^3+60*(1-v))", "60/sqrt(v^3+60*(1+v))", "2*sqrt(15)*sin(v)" ], "Name": [ "Kreisel⊂Sphere√60" ], "Umax": [ "2*pi", "2*pi", "2*pi", "2*pi", "2*pi" ], "Umin": [ "0", "0", "0", "0", "0" ], "Vmax": [ "0", "0.25", "0", "4", "pi/2" ], "Vmin": [ "-0.25", "0", "-4", "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "Nepali Polar", "NepaliPolar_01" ], "Description": [ "Nepali parametrized polar" ], "Fx": [ "sin(v)*cos(u)", "(2+2^(1/3))/(2*2^(2/3))*cos(v)*cos(u)" ], "Fy": [ "sin(v)*sin(u)", "(2+2^(1/3))/(2*2^(2/3))*cos(v)*sin(u)" ], "Fz": [ "-(-(sin(v)^2*sin(2*u)/2-1+cos(v)^3))^(1/3)", "(2+2^(1/3))/(2*2^(2/3))*(sin(v)-1)" ], "Name": [ "Nepali⊂Sphere" ], "Umax": [ "2*pi", "2*pi" ], "Umin": [ "0", "0" ], "Vmax": [ "pi", "pi/2" ], "Vmin": [ "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "TanzP_+", "TanzP_-", "Sphere" ], "Description": [ "parametrized Tanz" ], "Fx": [ "sqrt(1+sqrt(1+2*u^2*v^2))/2", "-sqrt(1+sqrt(1+2*u^2*v^2))/2", "sqrt(313)*cos(v)*cos(u)" ], "Fy": [ "u/sqrt(2)", "u/sqrt(2)", "sqrt(313)*cos(v)*sin(u)" ], "Fz": [ "v/sqrt(2)", "v/sqrt(2)", "sqrt(313)*sin(v)" ], "Name": [ "Tanz + Sphere √313" ], "Umax": [ "25", "25", "2*pi" ], "Umin": [ "-25", "-25", "0" ], "Vmax": [ "25", "25", "pi/2" ], "Vmin": [ "-25", "-25", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "TaubePs_1", "TaubePs_2", "TaubePs_3", "TaubePs_4", "Sphere" ], "Description": [ "Parametric Taube" ], "Fx": [ "3*(u^2-v^2)", "-3*(u^2+v^2)", "-3*(u^2+v^2)", "3*(v^2-u^2)", "cos(v)*cos(u)" ], "Fy": [ "2*v*(3*u^2-v^2)", "-2*v*(3*u^2+v^2)", "2*u*(3*v^2+u^2)", "-2*u*(3*v^2-u^2)", "cos(v)*sin(u)" ], "Fz": [ "3*u^2*v^2", "-3*u^2*v^2", "-3*u^2*v^2", "3*u^2*v^2", "sin(v)" ], "Name": [ "Taube + Sphere 1" ], "Umax": [ "0.615", "0", "0.734145", "0.734145", "2*pi" ], "Umin": [ "0", "-0.734145", "0", "0", "0" ], "Vmax": [ "0.734145", "0.734145", "0", "0.615", "pi/2" ], "Vmin": [ "0", "0", "-0.734145", "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "TaubePs_1", "TaubePs_2", "TaubePs_3", "TaubePs_4", "Sphere" ], "Description": [ "Parametric Taube" ], "Fx": [ "3*(u^2-v^2)", "-3*(u^2+v^2)", "-3*(u^2+v^2)", "3*(v^2-u^2)", "sqrt(2)*cos(v)*cos(u)" ], "Fy": [ "2*v*(3*u^2-v^2)", "-2*v*(3*u^2+v^2)", "2*u*(3*v^2+u^2)", "-2*u*(3*v^2-u^2)", "sqrt(2)*cos(v)*sin(u)" ], "Fz": [ "3*u^2*v^2", "-3*u^2*v^2", "-3*u^2*v^2", "3*u^2*v^2", "sqrt(2)*sin(v)" ], "Name": [ "Taube + Sphere √2" ], "Umax": [ "0.692", "0", "0.84172", "0.84172", "2*pi" ], "Umin": [ "0", "-0.84172", "0", "0", "0" ], "Vmax": [ "0.84172", "0.84172", "0", "0.692", "pi/2" ], "Vmin": [ "0", "0", "-0.84172", "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "TaubePs_1", "TaubePs_2", "TaubePs_3", "TaubePs_4", "Sphere" ], "Description": [ "Parametric Taube" ], "Fx": [ "3*(u^2-v^2)", "-3*(u^2+v^2)", "-3*(u^2+v^2)", "3*(v^2-u^2)", "2*cos(v)*cos(u)" ], "Fy": [ "2*v*(3*u^2-v^2)", "-2*v*(3*u^2+v^2)", "2*u*(3*v^2+u^2)", "-2*u*(3*v^2-u^2)", "2*cos(v)*sin(u)" ], "Fz": [ "3*u^2*v^2", "-3*u^2*v^2", "-3*u^2*v^2", "3*u^2*v^2", "2*sin(v)" ], "Name": [ "Taube + Sphere 2" ], "Umax": [ "0.82", "0", "0.963985", "0.963985", "2*pi" ], "Umin": [ "0", "-0.963985", "0", "0", "0" ], "Vmax": [ "0.963985", "0.963985", "0", "0.82", "pi/2" ], "Vmin": [ "0", "0", "-0.963985", "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "TaubePs_1", "TaubePs_2", "TaubePs_3", "TaubePs_4", "Sphere" ], "Description": [ "Parametric Taube" ], "Fx": [ "3*(u^2-v^2)", "-3*(u^2+v^2)", "-3*(u^2+v^2)", "3*(v^2-u^2)", "6*cos(v)*cos(u)" ], "Fy": [ "2*v*(3*u^2-v^2)", "-2*v*(3*u^2+v^2)", "2*u*(3*v^2+u^2)", "-2*u*(3*v^2-u^2)", "6*cos(v)*sin(u)" ], "Fz": [ "3*u^2*v^2", "-3*u^2*v^2", "-3*u^2*v^2", "3*u^2*v^2", "6*sin(v)" ], "Name": [ "Taube + Sphere 6" ], "Umax": [ "1.42", "0", "1.45475", "1.45475", "2*pi" ], "Umin": [ "0", "-1.45475", "0", "0", "0" ], "Vmax": [ "1.45475", "1.45475", "0", "1.42", "pi/2" ], "Vmin": [ "0", "0", "-1.45475", "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "TaubePs_1", "TaubePs_2", "TaubePs_3", "TaubePs_4", "Sphere" ], "Description": [ "Parametric Taube" ], "Fx": [ "3*(u^2-v^2)", "-3*(u^2+v^2)", "-3*(u^2+v^2)", "3*(v^2-u^2)", "12*cos(v)*cos(u)" ], "Fy": [ "2*v*(3*u^2-v^2)", "-2*v*(3*u^2+v^2)", "2*u*(3*v^2+u^2)", "-2*u*(3*v^2-u^2)", "12*cos(v)*sin(u)" ], "Fz": [ "3*u^2*v^2", "-3*u^2*v^2", "-3*u^2*v^2", "3*u^2*v^2", "12*sin(v)" ], "Name": [ "Taube + Sphere 12" ], "Umax": [ "2", "0", "1.85398", "1.85398", "2*pi" ], "Umin": [ "0", "-2", "0", "0", "0" ], "Vmax": [ "1.85398", "1.85398", "0", "2", "pi/2" ], "Vmin": [ "0", "0", "-2", "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "TulleP_01", "TulleP_02", "TulleP_03", "Sphere" ], "Description": [ "parametrized Tulle" ], "Fx": [ "v", "u", "u", "cos(v)*cos(u)" ], "Fy": [ "0.5*(u-v^2)", "0", "v", "cos(v)*sin(u)" ], "Fz": [ "0.5*(u+v^2)", "v", "0", "sin(v)" ], "Name": [ "Tulle + Sphere 1" ], "Umax": [ "sqrt(2)", "1", "1", "2*pi" ], "Umin": [ "-sqrt(2)", "-1", "-1", "0" ], "Vmax": [ "1", "1", "1", "pi/2" ], "Vmin": [ "-1", "-1", "-1", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "TulleP_01", "TulleP_02", "TulleP_03", "Sphere" ], "Description": [ "parametrized Tulle" ], "Fx": [ "v", "u", "u", "sqrt(2)*cos(v)*cos(u)" ], "Fy": [ "0.5*(u-v^2)", "0", "v", "sqrt(2)*cos(v)*sin(u)" ], "Fz": [ "0.5*(u+v^2)", "v", "0", "sqrt(2)*sin(v)" ], "Name": [ "Tulle + Sphere √2" ], "Umax": [ "2", "sqrt(2)", "sqrt(2)", "2*pi" ], "Umin": [ "-2", "-sqrt(2)", "-sqrt(2)", "0" ], "Vmax": [ "sqrt(2)", "sqrt(2)", "sqrt(2)", "pi/2" ], "Vmin": [ "-sqrt(2)", "-sqrt(2)", "-sqrt(2)", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "TulleP_01", "TulleP_02", "TulleP_03", "Sphere" ], "Description": [ "parametrized Tulle" ], "Fx": [ "v", "u", "u", "2*cos(v)*cos(u)" ], "Fy": [ "0.5*(u-v^2)", "0", "v", "2*cos(v)*sin(u)" ], "Fz": [ "0.5*(u+v^2)", "v", "0", "2*sin(v)" ], "Name": [ "Tulle + Sphere 2" ], "Umax": [ "2*sqrt(2)", "2", "2", "2*pi" ], "Umin": [ "-2*sqrt(2)", "-2", "-2", "0" ], "Vmax": [ "2", "2", "2", "pi/2" ], "Vmin": [ "-2", "-2", "-2", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "Zeek Polar", "Sphere" ], "Description": [ "Parametric Zeek" ], "Fx": [ "sin(v)*cos(v)^3*cos(u)", "cos(v)*cos(u)/2" ], "Fy": [ "sin(v)*cos(v)^3*sin(u)", "cos(v)*sin(u)/2" ], "Fz": [ "cos(v)^2", "(sin(v)+1)/2" ], "Name": [ "Zeek⊂Sphere½" ], "Umax": [ "2*pi", "2*pi" ], "Umin": [ "0", "0" ], "Vmax": [ "pi/2", "pi/2" ], "Vmin": [ "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "SpitzPolar_+", "SpitzPolar_-", "Sphere" ], "Description": [ "Parametric Spitz" ], "Fx": [ "4*v^3*cos(u)", "4*v^3*cos(u)", "cos(v)*cos(u)" ], "Fy": [ "6*v^2*(sin(2*u)^2)^(1/3)+sqrt(4*v^2*(sin(2*u)^4)^(1/3)+16*v^6)", "6*v^2*(sin(2*u)^2)^(1/3)-sqrt(4*v^2*(sin(2*u)^4)^(1/3)+16*v^6)", "cos(v)*sin(u)" ], "Fz": [ "4*v^3*sin(u)", "4*v^3*sin(u)", "sin(v)" ], "Name": [ "Spitz +Sphere 1" ], "Umax": [ "2*pi", "2*pi", "2*pi" ], "Umin": [ "0", "0", "0" ], "Vmax": [ "1/2^(5/6)", "0.63", "pi/2" ], "Vmin": [ "0", "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "SpitzPolar_+", "SpitzPolar_-", "Sphere" ], "Description": [ "Parametric Spitz" ], "Fx": [ "4*v^3*cos(u)", "4*v^3*cos(u)", "sqrt(2)*cos(v)*cos(u)" ], "Fy": [ "6*v^2*(sin(2*u)^2)^(1/3)+sqrt(4*v^2*(sin(2*u)^4)^(1/3)+16*v^6)", "6*v^2*(sin(2*u)^2)^(1/3)-sqrt(4*v^2*(sin(2*u)^4)^(1/3)+16*v^6)", "sqrt(2)*cos(v)*sin(u)" ], "Fz": [ "4*v^3*sin(u)", "4*v^3*sin(u)", "sqrt(2)*sin(v)" ], "Name": [ "Spitz +Sphere √2" ], "Umax": [ "2*pi", "2*pi", "2*pi" ], "Umin": [ "0", "0", "0" ], "Vmax": [ "1/2^(2/3)", "0.71", "pi/2" ], "Vmin": [ "0", "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "SpitzPolar_+", "SpitzPolar_-", "Sphere" ], "Description": [ "Parametric Spitz" ], "Fx": [ "4*v^3*cos(u)", "4*v^3*cos(u)", "2*cos(v)*cos(u)" ], "Fy": [ "6*v^2*(sin(2*u)^2)^(1/3)+sqrt(4*v^2*(sin(2*u)^4)^(1/3)+16*v^6)", "6*v^2*(sin(2*u)^2)^(1/3)-sqrt(4*v^2*(sin(2*u)^4)^(1/3)+16*v^6)", "2*cos(v)*sin(u)" ], "Fz": [ "4*v^3*sin(u)", "4*v^3*sin(u)", "2*sin(v)" ], "Name": [ "Spitz +Sphere 2" ], "Umax": [ "2*pi", "2*pi", "2*pi" ], "Umin": [ "0", "0", "0" ], "Vmax": [ "1/sqrt(2)", "0.8", "pi/2" ], "Vmin": [ "0", "0", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "SchneeflockePX_1", "SchneeflockePX_2", "SchneeflockePX_3", "SchneeflockePX_4", "Sphere" ], "Description": [ "Parametric Schneeflocke" ], "Fx": [ "(u-v)*(u*v)^(1/3)", "-(u-v)*(-u*v)^(1/3)", "(u-v)*(u*v)^(1/3)", "-(u-v)*(-u*v)^(1/3)", "cos(v)*cos(u)" ], "Fy": [ "u", "u", "u", "u", "cos(v)*sin(u)" ], "Fz": [ "v-u", "v-u", "v-u", "v-u", "sin(v)" ], "Name": [ "Schneeflocke +Sphere 1" ], "Umax": [ "1", "0", "0", "1", "2*pi" ], "Umin": [ "0", "-1", "-1", "0", "0" ], "Vmax": [ "1.23", "1.23", "0", "0", "pi/2" ], "Vmin": [ "0", "0", "-1.23", "-1.23", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "SchneeflockePX_1", "SchneeflockePX_2", "SchneeflockePX_3", "SchneeflockePX_4", "Sphere" ], "Description": [ "Parametric Schneeflocke" ], "Fx": [ "(u-v)*(u*v)^(1/3)", "-(u-v)*(-u*v)^(1/3)", "(u-v)*(u*v)^(1/3)", "-(u-v)*(-u*v)^(1/3)", "sqrt(2)*cos(v)*cos(u)" ], "Fy": [ "u", "u", "u", "u", "sqrt(2)*cos(v)*sin(u)" ], "Fz": [ "v-u", "v-u", "v-u", "v-u", "sqrt(2)*sin(v)" ], "Name": [ "Schneeflocke +Sphere √2" ], "Umax": [ "sqrt(2)", "0", "0", "sqrt(2)", "2*pi" ], "Umin": [ "0", "-sqrt(2)", "-sqrt(2)", "0", "0" ], "Vmax": [ "1.67", "1.67", "0", "0", "pi/2" ], "Vmin": [ "0", "0", "-1.67", "-1.67", "-pi/2" ] } }, 
 +{ "Param3D": { "Component": [ "SchneeflockePX_1", "SchneeflockePX_2", "SchneeflockePX_3", "SchneeflockePX_4", "Sphere" ], "Description": [ "Parametric Schneeflocke" ], "Fx": [ "(u-v)*(u*v)^(1/3)", "-(u-v)*(-u*v)^(1/3)", "(u-v)*(u*v)^(1/3)", "-(u-v)*(-u*v)^(1/3)", "2*cos(v)*cos(u)" ], "Fy": [ "u", "u", "u", "u", "2*cos(v)*sin(u)" ], "Fz": [ "v-u", "v-u", "v-u", "v-u", "2*sin(v)" ], "Name": [ "Schneeflocke +Sphere 2" ], "Umax": [ "2", "0", "0", "2", "2*pi" ], "Umin": [ "0", "-2", "-2", "0", "0" ], "Vmax": [ "2.27", "2.27", "0", "0", "pi/2" ], "Vmin": [ "0", "0", "-2.27", "-2.27", "-pi/2" ] } } 
 +] } </code>
 ===== Tutoriel : Comment construire le modèle stl d'une surface algébrique ===== ===== Tutoriel : Comment construire le modèle stl d'une surface algébrique =====
 Voyons comment construire un modèle à partir de sa formule en utilisant l'exemple de Kolibri. Voyons comment construire un modèle à partir de sa formule en utilisant l'exemple de Kolibri.
/home/resonancg/www/wiki/data/attic/projets/surfaces/accueil.1431727487.txt.gz · Dernière modification: 2015/05/16 00:04 de alba

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