Helicoid

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Graphic

View 1

V5

g=ParametricPlot3D[{u Cos[v],u Sin[v],v/10},{u,0,1},{v,0,4Pi},Boxed->False,Axes->False,PlotPoints->{5,50}]

<<Utilities`LiveGraphics3D`

WriteLiveForm["helicoid.m",g]

V6

ParametricPlot3D[{u Cos[v],u Sin[v],v/10},{u,0,1},{v,0,4Pi},Boxed->False,Axes->False,ImageSize->Small]

ParametricPlot3D[{u Cos[v],u Sin[v],v/10},{u,0,1},{v,0,6Pi},Boxed->False,Mesh->{7,40},Axes->False,ImageSize->Small]

View 2

v5

ParametricPlot3D[{u Cos[v],u Sin[v],v/5},{u,0,1},{v,0,4Pi},
PlotPoints->{10,50},Ticks->None]

V6

ParametricPlot3D[{u Cos[v],u Sin[v],v/5},{u,0,1},{v,0,4Pi},ImageSize->Small]

ParametricPlot3D[{u Cos[v],u Sin[v],v/5},{u,0,1},{v,0,4Pi},ImageSize->Small,Mesh->{7,40}]

Animation

HelicoidCatenoid[t_]:=Cos[t]{Sinh[v]Sin[u],-Sinh[v]Cos[u],u}+Sin[t]{Cosh[v]Cos[u],Cosh[v]Sin[u],v}

Show/@Table[
  ParametricPlot3D[HelicoidCatenoid[t],{u,0,2Pi},{v,-2,2},
Boxed->False,Axes->False],
    {t,0,2Pi,Pi/10}]

Show[GraphicsArray[Partition[Table[
  ParametricPlot3D[HelicoidCatenoid[t],{u,0,2Pi},{v,-2,2},
    DisplayFunction->Identity,Boxed->False,Axes->False],
    {t,0,Pi/2,Pi/10}],3]],DisplayFunction->$DisplayFunction]

LiveGraphics3D

WriteLiveAnimationForm["cathel.m",
	Table[
  ParametricPlot3D[Evaluate[HelicoidCatenoid[t]],{u,0,2Pi},{v,-2,2},
  Boxed->False,Axes->False,DisplayFunction->Identity,
PlotPoints->{20,8}],
 {t,0,2Pi,Pi/10}]]

Equation

eqns=Thread[{x,y,z}=={u Cos[v],u Sin[v],c v}]

Eliminate[eqns,{u,v}]

eqn=y-x Tan[z/c];

eqn/.Thread[{x,y,z}->{u Cos[v],u Sin[v],c v}]

Curvature

Curvatures[{u Cos[v],u Sin[v],c v},{u,v},Simplify[#,c>0]&]

FullSimplify[PrincipalCurvatures[{u Cos[v],u Sin[v],c v},{u,v}],c>0&&u>0&&v>0]

FullSimplify[PowerExpand[Integrate[(Sqrt[(c^2)+(u^2)]),{v,0,θ},{u,0,r}]]]

FullSimplify[GaussianCurvature[eqn,{x,y,z}],c>0]

FullSimplify[MeanCurvature[eqn,{x,y,z}],c>0]

Metric Properties

surf={u Cos[v],u Sin[v],c v};

Metric Tensor

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

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[surf,{u,v}],c>0]

Vector Properties

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

Vector Length

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

Normal Vector

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