Crossed Trough

Author

Eric W. Weisstein
June 20, 2013

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

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

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

Definition

surf[a_][u_,v_]:=a{u,v, u^2 v^2}

Equation

{eqn}=GroebnerBasis[Thread[{x,y,z}==surf[a][u,v]],{x,y,z},{u,v}]

eqn=(x^2) (y^2)-(a^3) z;

AlgebraicDegree[eqn,{x,y,z}]

Surface

V5

cross=Plot3D[x^2y^2,{x,-5,5},{y,-5,5},Boxed->False,Ticks->None,Axes->False]

WriteLiveForm["crosstr.m",cross]

V6

ParametricPlot3D[surf[1][u,v],{u,-2,2},{v,-2,2},BoxRatios->{1,1,1},PlotRange->All]

Properties

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

FullSimplify[PrincipalCurvatures[{x,y,x^2y^2},{x,y}],(x|y)∈Reals]

GaussianCurvature[eqn,{x,y,z}]

Principal Curvature

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

Metric Properties

Metric Tensor

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

Christoffel Symbol of the Second Kind

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

Vector Properties

eqn=a {u,v,u^2 v^2};

Vector Length

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

Normal Vector

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