Fresnel's Elasticity Surface

Author

Eric W. Weisstein
January 20, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Surfaces/FresnelsElasticitySurface.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/FresnelsElasticitySurface.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Initialization

Graphic

Love's Definition

plane[x_,y_,z_]:=l x+m y+n z-c

Envelope of

l x+m y+n z==c

where k, m, and n are direction cosines and c is given by the solution to a complicated determinant, subject to

((l^2)/(c^2-LL))+((m^2)/(c^2-MM))+((n^2)/(c^2-NN))==0

von Seggern's Definition

David von Seggern
13 Jan 2008

Manipulate[x=Cos[v] Sin[u];
y=Sin[v] Sin[u];
z=Cos[u];
rsq=a^2*x^2+b^2*y^2+c^2*z^2;
SphericalPlot3D[Sqrt[rsq],{u,0,π},{v,0,2 π},PlotRange->{{-2,2},{-2,2},{-2,2}},Boxed->False,Axes->False],{{a,1,"X scale"},0.1,1.0},{{b,1,"Y scale"},0.1,1.0},{{c,0.5,"Z scale"},0.1,1.0}]

JavaView's Definition, Fresnel1

cos(u)*cos(v)/(-2.*sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))*cos((acos(-(-0.941/6.+0.374*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4)-1.309/6.*((cos(u)^6+sin(u)^6)*cos(v)^6+sin(v)^6)-1.221*cos(u)^2*cos(v)^4*sin(u)^2*sin(v)^2)/sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))^3)+pi)/3.)+0.8)

sin(u)*cos(v)/(-2.*sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))*cos((acos(-(-0.941/6.+0.374*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4)-1.309/6.*((cos(u)^6+sin(u)^6)*cos(v)^6+sin(v)^6)-1.221*cos(u)^2*cos(v)^4*sin(u)^2*sin(v)^2)/sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))^3)+pi)/3.)+0.8)

sin(v)/(-2.*sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))*cos((acos(-(-0.941/6.+0.374*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4)-1.309/6.*((cos(u)^6+sin(u)^6)*cos(v)^6+sin(v)^6)-1.221*cos(u)^2*cos(v)^4*sin(u)^2*sin(v)^2)/sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))^3)+pi)/3.)+0.8)

xx1[u_,v_]:=Cos[u]Cos[v]/(-2.Sqrt[0.965/3.-0.935/3.((Cos[u]^4+Sin[u]^4)Cos[v]^4+Sin[v]^4)]Cos[(ArcCos[-(-0.941/6.+0.374((Cos[u]^4+Sin[u]^4)Cos[v]^4+Sin[v]^4)-1.309/6.((Cos[u]^6+Sin[u]^6)Cos[v]^6+Sin[v]^6)-1.221Cos[u]^2Cos[v]^4Sin[u]^2Sin[v]^2)/Sqrt[0.965/3.-0.935/3.((Cos[u]^4+Sin[u]^4)Cos[v]^4+Sin[v]^4)]^3]+Pi)/3.]+0.8);
yy1[u_,v_]:=Sin[u]Cos[v]/(-2.Sqrt[0.965/3.-0.935/3.((Cos[u]^4+Sin[u]^4)Cos[v]^4+Sin[v]^4)]Cos[(ArcCos[-(-0.941/6.+0.374((Cos[u]^4+Sin[u]^4)Cos[v]^4+Sin[v]^4)-1.309/6.((Cos[u]^6+Sin[u]^6)Cos[v]^6+Sin[v]^6)-1.221Cos[u]^2Cos[v]^4Sin[u]^2Sin[v]^2)/Sqrt[0.965/3.-0.935/3.((Cos[u]^4+Sin[u]^4)Cos[v]^4+Sin[v]^4)]^3]+Pi)/3.]+0.8);
zz1[u_,v_]:=Sin[v]/(-2.Sqrt[0.965/3.-0.935/3.((Cos[u]^4+Sin[u]^4)Cos[v]^4+Sin[v]^4)]Cos[(ArcCos[-(-0.941/6.+0.374((Cos[u]^4+Sin[u]^4)Cos[v]^4+Sin[v]^4)-1.309/6.((Cos[u]^6+Sin[u]^6)Cos[v]^6+Sin[v]^6)-1.221Cos[u]^2Cos[v]^4Sin[u]^2Sin[v]^2)/Sqrt[0.965/3.-0.935/3.((Cos[u]^4+Sin[u]^4)Cos[v]^4+Sin[v]^4)]^3]+Pi)/3.]+0.8);

g1=ParametricPlot3D[Evaluate[Through[{xx1,yy1,zz1}[u,v]]],{u,0,2Pi},{v,-Pi/2,Pi/2},Boxed->False,Axes->False]

LiveGraphics3D

WriteLiveForm["fresnel.m",Show[ParametricPlot3D[Evaluate[Through[{x,y,z}[u,v]]],{u,0,2Pi},{v,-Pi/2,Pi/2},DisplayFunction->Identity,Boxed->False,Axes->False],DisplayFunction->$DisplayFunction,opts]]

JavaView Frensel2

cos(u)*cos(v)/(-2.*sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))*cos((acos(-(-0.941/6.+0.374*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4)-1.309/6.*((cos(u)^6+sin(u)^6)*cos(v)^6+sin(v)^6)-1.221*cos(u)^2*cos(v)^4*sin(u)^2*sin(v)^2)/sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))^3)-pi)/3.)+0.8)
sin(u)*cos(v)/(-2.*sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))*cos((acos(-(-0.941/6.+0.374*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4)-1.309/6.*((cos(u)^6+sin(u)^6)*cos(v)^6+sin(v)^6)-1.221*cos(u)^2*cos(v)^4*sin(u)^2*sin(v)^2)/sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))^3)-pi)/3.)+0.8)
sin(v)/(-2.*sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))*cos((acos(-(-0.941/6.+0.374*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4)-1.309/6.*((cos(u)^6+sin(u)^6)*cos(v)^6+sin(v)^6)-1.221*cos(u)^2*cos(v)^4*sin(u)^2*sin(v)^2)/sqrt(0.965/3.-0.935/3.*((cos(u)^4+sin(u)^4)*cos(v)^4+sin(v)^4))^3)-pi)/3.)+0.8)

xx2[u_,v_]:=Cos[u]*Cos[v]/(-2.*Sqrt[0.965/3.-0.935/3.*((Cos[u]^4+Sin[u]^4)*Cos[v]^4+Sin[v]^4)]*Cos[(ArcCos[-(-0.941/6.+0.374*((Cos[u]^4+Sin[u]^4)*Cos[v]^4+Sin[v]^4)-1.309/6.*((Cos[u]^6+Sin[u]^6)*Cos[v]^6+Sin[v]^6)-1.221*Cos[u]^2*Cos[v]^4*Sin[u]^2*Sin[v]^2)/Sqrt[0.965/3.-0.935/3.*((Cos[u]^4+Sin[u]^4)*Cos[v]^4+Sin[v]^4)]^3]-Pi)/3.]+0.8);
yy2[u_,v_]:=Sin[u]*Cos[v]/(-2.*Sqrt[0.965/3.-0.935/3.*((Cos[u]^4+Sin[u]^4)*Cos[v]^4+Sin[v]^4)]*Cos[(ArcCos[-(-0.941/6.+0.374*((Cos[u]^4+Sin[u]^4)*Cos[v]^4+Sin[v]^4)-1.309/6.*((Cos[u]^6+Sin[u]^6)*Cos[v]^6+Sin[v]^6)-1.221*Cos[u]^2*Cos[v]^4*Sin[u]^2*Sin[v]^2)/Sqrt[0.965/3.-0.935/3.*((Cos[u]^4+Sin[u]^4)*Cos[v]^4+Sin[v]^4)]^3]-Pi)/3.]+0.8);
zz2[u_,v_]:=Sin[v]/(-2.*Sqrt[0.965/3.-0.935/3.*((Cos[u]^4+Sin[u]^4)*Cos[v]^4+Sin[v]^4)]*Cos[(ArcCos[-(-0.941/6.+0.374*((Cos[u]^4+Sin[u]^4)*Cos[v]^4+Sin[v]^4)-1.309/6.*((Cos[u]^6+Sin[u]^6)*Cos[v]^6+Sin[v]^6)-1.221*Cos[u]^2*Cos[v]^4*Sin[u]^2*Sin[v]^2)/Sqrt[0.965/3.-0.935/3.*((Cos[u]^4+Sin[u]^4)*Cos[v]^4+Sin[v]^4)]^3]-Pi)/3.]+0.8)

g2=ParametricPlot3D[Evaluate[Through[{xx2,yy2,zz2}[u,v]]],{u,0,2Pi},{v,-Pi/2,Pi/2},Boxed->False,Axes->False]

GraphicsGrid[{{g1,g2}},Spacings->-20,ImageSize->500]