Hyperbolic Paraboloid

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Definition

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

Equation

eqns=Numerator[Factor[{x,y,z}-surf[a,b][u,v]]]

GroebnerBasis[eqns,{x,y,z},{u,v}]

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

First Parametrization

ParametricPlot3D[{x,y,y^2-x^2},{x,-1,1},{y,-1,1},Boxed->False,Ticks->None,AspectRatio->Automatic,Axes->False]

h1=ParametricPlot3D[{x,y,y^2-x^2},{x,-1,1},{y,-1,1},Boxed->False,Ticks->None,AspectRatio->Automatic,Axes->False,RegionFunction->(#1^2+#2^2+#2^2<1^2&)]

WriteLiveForm["hyperpa1.m",h1]

Curvatures

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

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

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

FullSimplify[%]

Second Paramaterization

ParametricPlot3D[{u,v,u v},{u,-1,1},{v,-1,1},Boxed->False,Axes->False]

h2=ParametricPlot3D[{u,v,u v},{u,-3,3},{v,-3,3},Boxed->False,Axes->False,RegionFunction->(#1^2+#2^2+#2^2<2^2&)]

WriteLiveForm["hyperpa2.m",h2]

ParametricPlot3D[{u Cosh[v], u Sinh[v], u^2},
  {u,-1,1},{v,-1,1},Boxed->False,Ticks->None,
  AspectRatio->Automatic,Axes->False]

Curvatures

Curvatures[{u,v,u v},{u,v}]

FullSimplify[Integrate[(Sqrt[1+(u^2)+(v^2)]),{u,0,uu},{v,0,vv},Assumptions->{uu>0&&uu∈Reals&&vv>0&&vv∈Reals}]]

Metric Properties

Metric Tensor

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

Christoffel Symbol of the Second Kind

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

Vector Properties

eqn={u,v,v^2/b^2-u^2/a^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]]