Catalan's Surface

Author

Eric W. Weisstein
June 20, 2013

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

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

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

Initialization

Equation

catalan[a_][u_,v_]:=a{u-Sin[u]Cosh[v],1-Cos[u]Cosh[v],4 Sin[u/2]Sinh[v/2]}

Surface

V5

ParametricPlot3D[catalan[1][u,v],{u,0,4Pi},{v,-2,2},Boxed->False,Axes->False]

V6

ParametricPlot3D[catalan[1][u,v],{u,0,6π},{v,-2,2}]

Properties

Curvatures[catalan[a][u,v],{u,v},FullSimplify[#,a>0&&(u|v)∈Reals]&]

FullSimplify[PrincipalCurvatures[catalan[a][u,v],{u,v}],(u|v)∈Reals]

Metric Properties

Metric Tensor

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

Christoffel Symbol of the Second Kind

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

Vector Properties

eqn=a{u-Sin[u]Cosh[v],1-Cos[u]Cosh[v],4 Sin[u/2]Sinh[v/2]};

Vector Length

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

Normal Vector

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