Steinbach Screw

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Initialization

surf[u_,v_]:=a{u Cos[v],u Sin[v],v Cos[u]}

Plot

V5

<<Utilities`LiveGraphics3D`

WriteLiveForm["steinba.m",Show[ParametricPlot3D[{u Cos[v],u Sin[v],v Cos[u]},{u,-4,4},{v,0,6.25},Axes->False,Boxed->False]]]

V6

ParametricPlot3D[{u Cos[v],u Sin[v],v Cos[u]},{u,-4,4},{v,0,6.25},Axes->False,Boxed->False]

Equation

eqns=Thread[{x,y,z}==surf[u,v]]

Eliminate[eqns,{u,v}]

Curvatures

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

-((v (u (4 (u^2)+5) cos(u)-u cos(3 u)+2 (((v^2)+2) (u^2)+(2-(u^2) (v^2)) cos(2 u)+2) sin(u)))/(2 (Sqrt[2]) (((((v^2)+2) (u^2)+(1-(u^2) (v^2)) cos(2 u)+1))^(3/2))))

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

Metric Properties

Metric Tensor

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

Christoffel Symbol of the Second Kind

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

Vector Properties

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

Vector Length

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

Normal Vector

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