Kuen Surface

Author

Eric W. Weisstein
June 10, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Plot

Defaults

ParametricPlot3D[{
2(Cos[u]+u Sin[u])Sin[v]/(1+u^2 Sin[v]^2),2(Sin[u]-u Cos[u])Sin[v]/(1+u^2 Sin[v]^2),Log[Tan[v/2]]+2 Cos[v]/(1+u^2 Sin[v]^2)},
{u,0,4.5},{v,0,Pi},
Boxed->False,Axes->False]

Half

ParametricPlot3D[{
2(Cos[u]+u Sin[u])Sin[v]/(1+u^2 Sin[v]^2),2(Sin[u]-u Cos[u])Sin[v]/(1+u^2 Sin[v]^2),Log[Tan[v/2]]+2 Cos[v]/(1+u^2 Sin[v]^2)},
{u,0,4.5},{v,0,Pi},
Boxed->False,Axes->False,MaxRecursion->4,PlotPoints->{30,40}]

Full

ParametricPlot3D[{
  2(Cos[u]+u Sin[u])Sin[v]/(1+u^2 Sin[v]^2),
  2(Sin[u]-u Cos[u])Sin[v]/(1+u^2 Sin[v]^2),
  Log[Tan[v/2]]+2 Cos[v]/(1+u^2 Sin[v]^2)},{u,-4.5,4.5},
  {v,0,2 Pi},Boxed->False,Axes->False,MaxRecursion->4,PlotPoints->{30,40}]

Wraparound

ParametricPlot3D[{
2(Cos[u]+u Sin[u])Sin[v]/(1+u^2 Sin[v]^2),2(Sin[u]-u Cos[u])Sin[v]/(1+u^2 Sin[v]^2),Log[Tan[v/2]]+2 Cos[v]/(1+u^2 Sin[v]^2)},
{u,-6,6},{v,0,Pi},
Boxed->False,Axes->False,MaxRecursion->4,PlotPoints->{30,40}]

For Calculate

ParametricPlot3D[{
2(Cos[u]+u Sin[u])Sin[v]/(1+u^2 Sin[v]^2),2(Sin[u]-u Cos[u])Sin[v]/(1+u^2 Sin[v]^2),Log[Tan[v/2]]+2 Cos[v]/(1+u^2 Sin[v]^2)},
{u,0,4.5},{v,0,Pi},
Boxed->False,Axes->False,Mesh->Automatic,MaxRecursion->5,PlotPoints->10]

Hidden Mesh

ParametricPlot3D[{2(Cos[u]+u Sin[u])Sin[v]/(1+u^2 Sin[v]^2),2(Sin[u]-u Cos[u])Sin[v]/(1+u^2 Sin[v]^2),Log[Tan[v/2]]+2 Cos[v]/(1+u^2 Sin[v]^2),EdgeForm[]},{u,-4,4},{v,0.01,Pi-.01},Boxed->False,Axes->False,
PlotPoints->{50,40}]

ParametricPlot3D[{2(Cos[u]+u Sin[u])Sin[v]/(1+u^2 Sin[v]^2),2(Sin[u]-u Cos[u])Sin[v]/(1+u^2 Sin[v]^2),Log[Tan[v/2]]+2 Cos[v]/(1+u^2 Sin[v]^2),EdgeForm[]},{u,-4,4},{v,0.01,Pi-.01},Boxed->False,Axes->False,
PlotRange->All,
PlotPoints->{1,1}70]

ParametricPlot3D[{2(Cos[u]+u Sin[u])Sin[v]/(1+u^2 Sin[v]^2),2(Sin[u]-u Cos[u])Sin[v]/(1+u^2 Sin[v]^2),Log[Tan[v/2]]+2 Cos[v]/(1+u^2 Sin[v]^2),EdgeForm[]},{u,-4,4},{v,0.05,Pi-.05},Boxed->False,Axes->False,
PlotPoints->{50,40}]

LiveGraphics3D

<<LiveGraphics3D`

WriteLiveForm["kuen.m",ParametricPlot3D[{2(Cos[u]+u Sin[u])Sin[v]/(1+u^2 Sin[v]^2),2(Sin[u]-u Cos[u])Sin[v]/(1+u^2 Sin[v]^2),Log[Tan[v/2]]+2 Cos[v]/(1+u^2 Sin[v]^2)},{u,-4,4},{v,0.01,Pi-.01},Boxed->False,Axes->False]]

Curvature

kuen=a{2(Cos[u]+u Sin[u])Sin[v]/(1+u^2 Sin[v]^2),2(Sin[u]-u Cos[u])Sin[v]/(1+u^2 Sin[v]^2),Log[Tan[v/2]]+2 Cos[v]/(1+u^2 Sin[v]^2)};

Curvatures[kuen,{u,v},FullSimplify[#,Assumptions->a>0&&0<v<π&&0<u<2π]&]

FullSimplify[{-((4 a u Abs[2-(u^2)+(u^2) Cos[2 v]] Sin[v])/(((2+(u^2)-(u^2) Cos[2 v]))^2)),0,((4 a u Abs[2-(u^2)+(u^2) Cos[2 v]] Csc[v])/(((2+(u^2)-(u^2) Cos[2 v]))^2))}/.Abs[x_]:>x]

FullSimplify[-((Abs[2-(u^2)+(u^2) Cos[2 v]] (-6 (u^2) Csc[v]+((Csc[v])^3)+(u^4) Sin[v]))/(8 a u (((Csc[v]-(u^2) Sin[v]))^2)))/.Abs[x_]:>x]

Simplify[PrincipalCurvatures[kuen,{u,v}],a>0&&0<v<π&&0<u<2π]

a=Simplify[Together[-((Csc[v])/(4 u))+(1/4) u (1+(8/(2-(u^2)+(u^2) Cos[2 v]))) Sin[v]]]

b=PowerExpand[Sqrt[Simplify[Together[Expand[1+(((((Csc[v])/(4 u))-(1/4) u (1+(8/(2-(u^2)+(u^2) Cos[2 v]))) Sin[v]))^2)]]]]]

Simplify[{a+b,a-b}]

Metric Properties

Metric Tensor

MetricTensor[kuen,{u,v}]//FullSimplify

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[kuen,{u,v}],a>0]

FullSimplify[ChristoffelSymbols[kuen,{u,v}],a>0&&(u|v)∈Reals]

Vector Properties

eqn=a{2 (Cos[u]+u Sin[u]) Sin[v]/(1+u^2 Sin[v]^2),2 (Sin[u]-u Cos[u]) Sin[v]/(1+u^2 Sin[v]^2),Log[Tan[v/2]]+2 Cos[v]/(1+u^2 Sin[v]^2)};
rules={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};
assum=a>0&&u∈Reals&&0<v<Pi;

Vector Length

Assuming[assum,FullSimplify[Norm[eqn]/.rules]]

Normal Vector

Assuming[assum,FullSimplify[NormalVector[eqn,{u,v}]/.rules]]