Plücker's Conoid

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Initialization

surf[c_,n_][r_,t_]:={r Cos[t],r Sin[t],c Sin[n t]}

Curvature

surf[c,2][r,t]

Graphic

ParametricPlot3D[{r Cos[t],r Sin[t],Cos[t]Sin[t]},{r,0,Sqrt[2]},{t,0,2Pi},Boxed->False,Axes->False,PlotPoints->{10,30}]

Do[
WriteLiveForm["plucker"<>ToString[n]<>".m",Show[ParametricPlot3D[PluckersConoid[n]//Evaluate,{r,0,Sqrt[2.]},{t,0,2Pi}, DisplayFunction->Identity,Boxed->False,Axes->False,PlotPoints->{10,50}],opts]],
  {n,2,4}]

V5

Show[GraphicsArray[Table[ParametricPlot3D[PluckersConoid[n]//Evaluate,{r,0,Sqrt[2.]},{t,0,2Pi},DisplayFunction->Identity,Boxed->False,Axes->False,PlotPoints->{10,30}],{n,2,4}]],DisplayFunction->$DisplayFunction]

V6

GraphicsGrid[{Table[ParametricPlot3D[surf[1,n][r,t],{r,0,Sqrt[2.]},{t,0,2Pi},Boxed->False,Axes->False,Mesh->False],{n,2,4}]}]

Properties

Curvatures[surf[c,n][r,t],{r,t},FullSimplify[#,c>0&]&]

FullSimplify[PrincipalCurvatures[surf[c,n][r,t],{r,t}],c>0&&n≥0&&n∈Integers]

Metric Properties

Metric Tensor

MetricTensor[surf[c,n][r,t],{r,t}]//FullSimplify

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[surf[c,n][r,t],{r,t}],c>0&&n>0]

Vector Properties

eqn={r Cos[t],r Sin[t],c Sin[n t]};
rules={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};
assum=c>0&&n>0&&0<r<Sqrt[2]&&0<t<2π;

Vector Length

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

Normal Vector

Assuming[assum,FullSimplify[NormalVector[eqn,{r,t}]/.rules]]