Peano Surface

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Initialization

Definition

PeanoSurface[a_][x_,y_]:=(2x^2-a y)(a y- x^2)
surf[a_][u_,v_]:={u,v,(2u^2-a v)(a v-u^2)}

Equation

Expand[PeanoSurface[a][x,y]-z]

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

Graphic

V5

Plot3D[PeanoSurface[x,y],{x,-5,5},{y,-5,5},Boxed->False,Axes->False,PlotRange->{-200,50},ClippingStyle->None]

V6

Plot3D[PeanoSurface[x,y],{x,-5,5},{y,-5,5},Boxed->False,Axes->False,PlotRange->All]

ParametricPlot3D[surf[1][x,y],{x,-.5,.5},{y,-.5,.5},Boxed->False,Axes->False]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["peano.m"]

Maximum

Plot[{PeanoSurface[x,0],PeanoSurface[0,x]},{x,-1,1}]

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

Properties

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

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

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={u,v,(2 u^2-a v) (a v-u^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]]