Monkey Saddle

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Definition

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

Equation

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

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

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

Plot

3D

Plot3D[x(x^2-3y^2),{x,-10,10},{y,-10,10},Boxed->False,Axes->False,ColorFunction->"Aquamarine"]

Plot3D[x(x^2-3y^2),{x,-10,10},{y,-10,10},Boxed->False,Axes->False]

ParametricPlot3D[{u,v,(u^3)-3u (v^2)},{u,-.5,.5},{v,-.5,.5}]

Along Axes

Plot[x(x^2-3y^2)/.y->0,{x,-4,4},PlotStyle->{Red,Thickness[.005]}]

Plot[x(x^2-3y^2)/.x->0,{y,-4,4},PlotStyle->{Red,Thickness[.005]}]

LiveGraphics3D

WriteLiveForm["monkey.m",Show[Plot3D[x(x^2-3y^2),{x,-10,10},{y,-10,10},Boxed->False,Axes->False]]]

POVRay

<<Utilities`POVray`

MonkeySaddle=Expand[x(x^2-3y^2)-z];
POVCoefficients[MonkeySaddle,{x,y,z}]

Curvatures

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

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

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

-((36 (a^4) ((x^2)+(y^2)))/((((a^4)+9 (x^4)+18 (x^2) (y^2)+9 (y^4)))^2))/.{x->0,y->0}

Stationary Points

f=x^2-3x 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

Saddle Point

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

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

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

Metric Properties

Metric Tensor

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

Christoffel Symbol of the Second Kind

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

Vector Properties

eqn=a {u,v,u^3-3u 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]]