Handkerchief Surface

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Definition

surf[u_,v_]:={u,v,u^3/3+u v^2+2(u^2-v^2)}

Equation

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

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

eqn=6 (x^2)+(x^3)-6 (y^2)+3 x (y^2)-3 z;

Solve[eqn==0,z]//Simplify

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

Graphics

ParametricPlot3D[{u,v,u^3/3+u v^2+2(u^2-v^2)},{u,-6,6},{v,-6,6},Axes->None,Boxed->False]

ParametricPlot3D[{u,v,u^3/3+u v^2+2(u^2-v^2)},{u,-6,6},{v,-6,6},Axes->None,Boxed->False,BoxRatios->{1,1,1}]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

g=ParametricPlot3D[surf[u,v],{u,-6,6},{v,-6,6},Axes->None,Boxed->False,PlotRange->All,BoxRatios->{1,1,1}]

WriteLiveForm["handker.m",g]

Properties

Curvatures[surf[u,v],{u,v},FullSimplify[#,(u|v)∈Reals]&]

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

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

FullSimplify[%]

Stationary Points

f=x^3/3+x y^2+2(x^2-y^2);

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

({(∂_(x,x))f,(∂_(x,x))f(∂_(y,y))f-((((∂_(x,y))f))^2)}/.#)&/@extrema

Maximum

FindMaximum[f,{x,-4},{y,0}]

FindMaximum[f,{x,-4.1},{y,0}]

Saddle Point

FindMaximum[f,{x,.01},{y,.01}]

FindMinimum[f,{x,.01},{y,.01}]

Singular Points

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

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,v,u^3/3+u v^2+2 (u^2-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]]