Elliptic Paraboloid

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Plot

EllipticParaboloid[h_,a_]:=ParametricPlot3D[{a Sqrt[u] Cos[t], Sqrt[u] Sin[t],u},
  {t,0,2Pi},{u,0,h},Boxed->False,Axes->False]

LiveGraphics3D

<<Utilities`LiveGraphics3D`

WriteLiveForm["ellpara.m",Show[EllipticParaboloid[5,1.5],opts]]

Equations

GroebnerBasis[Append[Thread[{x,y,z}=={a u Cos[v],b u Sin[v],u^2}],Cos[v]^2+Sin[v]^2==1],{x,y,z},{u,Cos[v],Sin[v]},MonomialOrder->EliminationOrder]

{-(b^2) (x^2)-(a^2) (y^2)+(a^2) (b^2) z}

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

Properties

surf={a u Cos[v],b u Sin[v],u^2};

Curvatures[surf,{u,v},FullSimplify[#,a>0&&b>0&&0<v<2π&&u>0]&]

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

FullSimplify[%]

FullSimplify[PrincipalCurvatures[surf,{u,v}],a>0&b>0&&0<v<2π&&u>0]

Metric Properties

Metric Tensor

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

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[surf,{u,v}],a>0&&b>0&&u>0&&0<v<2π]

Vector Properties

eqn={a u Cos[v],b u Sin[v],u^2};
rules={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};
assum=a>0&&u>0&&0<v<2π;

Vector Length

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

Normal Vector

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