Quartoid

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Definition

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

Equation

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

Plot

V5

Plot3D[If[x^2+y^2>4,-20,-(x^2+y^2)^2],{x,-2,2},{y,-2,2},PlotPoints->40,
PlotRange->{-15,3},Axes->False,ClipFill->None]

V6

Plot3D[-(x^2+y^2)^2,{x,-1,1},{y,-1,1},RegionFunction->(#1^2+#2^2<1&)]

Curvatures

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

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&&(u|v)∈Reals]

Vector Properties

eqn={u,v,-((u^2+v^2)^2/a^3)};
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]]