Elliptic Helicoid

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Plot

EllipticHelicoid[a_,b_,c_]:={a v Cos[u],b v Sin[u],c u}

g=ParametricPlot3D[EllipticHelicoid[1,2,1.5],{u,0,4Pi},{v,0,4},
  Boxed->False,Axes->False,PlotPoints->{40,10}]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["ellhel.m",g]

Properties

surf=EllipticHelicoid[a,b,c]

Curvatures[surf,{u,v},FullSimplify]

FullSimplify[PrincipalCurvatures[surf,{u,v}],a>0&&b>0&&c>0&&(u|v)∈Reals]

Metric Properties

Metric Tensor

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

Christoffel Symbol of the Second Kind

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

Vector Properties

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

Vector Length

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

Normal Vector

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