Corkscrew Surface

Author

Eric W. Weisstein
June 20, 2013

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

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Code

TwistedSphere[a_,b_]:={a Cos[u]Cos[v],a Sin[u]Cos[v],a Sin[v]+b u}

Graphic

ParametricPlot3D[Evaluate[TwistedSphere@@#],{u,0,2.2Pi},{v,-Pi,Pi},Axes->False,Boxed->False]&/@{{1,1},{1,.5},{1,2}}

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["corkscre.m",ParametricPlot3D[Evaluate[TwistedSphere[1,1]],{u,0,2Pi},{v,-Pi,Pi},Axes->False,Boxed->False]]

Properties

Curvatures[TwistedSphere[a,b],{u,v},FullSimplify[#,a>0&&b>0]&]

Fundamental Forms

FirstFundamentalForm[TwistedSphere[a,b],{u,v}]//FullSimplify//TrigSimplify//ColumnForm //TraditionalForm

FullSimplify[SecondFundamentalForm[TwistedSphere[a,b],{u,v}],a>0&&b>0]//TrigSimplify//ColumnForm//TraditionalForm

Gaussian Curvature

k=FullSimplify[GaussianCurvature[TwistedSphere[a,b],{u,v}]]//TrigSimplify//TraditionalForm

Mean Curvature

m=FullSimplify[MeanCurvature[TwistedSphere[a,b],{u,v}],a>0]//TrigSimplify

Simplify[Apart[m]/.(a^2) ((Cos[v])^2)+(b^2) ((Sin[v])^2)->c^2,c>0]

Principal Curvatures

Simplify[PrincipalCurvatures[TwistedSphere[a,b],{u,v}],a>0&&b>0&&0<v<2π&&u∈Reals]//Timing

FullSimplify[PrincipalCurvatures[TwistedSphere[a,b],{u,v}],a>0&&b>0&&0<v<2π&&u∈Reals]//Timing

Metric Properties

Metric Tensor

FullSimplify[MetricTensor[TwistedSphere[a,b],{u,v}],-π/2<u<π/2&&0<v<π&&a>0&&b>0]

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[TwistedSphere[a,b],{u,v}],-π/2<u<π/2&&0<v<π&&a>0&&b>0]

Vector Properties

eqn={a Cos[u]Cos[v],a Sin[u]Cos[v],a Sin[v]+b u};

Vector Length

FullSimplify[Norm[eqn]/.Abs[x_]^2:>x^2,(u|v)∈Reals&&a>0&&b>0]

Normal Vector

FullSimplify[NormalVector[eqn,{u,v}]/.Abs[x_]^2:>x^2,(u|v)∈Reals&&a>0&&b>0]