Menn's Surface

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Definition

MennsSurface[a_][u_,v_]:={u,v,a u^4+u^2 v-v^2}

Equation

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

{eqn}=Factor/@GroebnerBasis[eqns,{x,y,z},{u,v},MonomialOrder->EliminationOrder]

eqn=a (x^4)+(x^2) y-(y^2)-z;

Plot

ParametricPlot3D[MennsSurface[2],{u,-.5,.5},{v,-.5,.5},Axes->None,Boxed->False]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["menn.m",]

Curvatures

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

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

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

FullSimplify[%,a>0]

Stationary Points

f=Last[MennsSurface[a]/.{u->x,v->y}]

extrema=Reduce[(∂_x)f==0&&(∂_y)f==0,{x,y},Reals]

Metric Properties

Metric Tensor

MetricTensor[MennsSurface[a][u,v],{u,v}]//FullSimplify

Christoffel Symbol of the Second Kind

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

Vector Properties

eqn={u,v,a u^4+u^2 v-v^2};
rules={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};
assum=a>0&&(u|v)∈Reals;

Vector Length

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

Normal Vector

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