Elliptic Hyperboloid

Author

Eric W. Weisstein
June 10, 2014

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

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

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

One-Sheeted

surf[a_,b_,c_][u_,v_]:={a  Sqrt[1+u^2] Cos[v],b  Sqrt[1+u^2] Sin[v],c u}

Equation

eqns=Append[Thread[{x,y,z}^2==surf[a,b,c][u,v]^2],Sin[v]^2+Cos[v]^2==1]

GroebnerBasis[eqns,{x,y,z},{u,Cos[v],Sin[v]},MonomialOrder->EliminationOrder]

Numerator[-Factor[((x^2)/(a^2))+((y^2)/(b^2))-((z^2)/(c^2))-1]]

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

Plot

V5

EllipticHyperboloid[a_,b_,c_]:={a(Cos[u]-v Sin[u]),b(Sin[u]+v Cos[u]),c v}

Simplify[(Cos[u]-v Sin[u])^2+(Sin[u]+v Cos[u])^2-v^2]

<<Utilities`LiveGraphics3D`

WriteLiveForm["ellhyper.m",
Show[ParametricPlot3D[EllipticHyperboloid[1,2,3]//Evaluate,
{u,0,2Pi},{v,-1.5,1.5},
  Boxed->False,Axes->False],opts]]

V6

ParametricPlot3D[surf[Sequence@@#][u,v],{u,-1,1},{v,0,2π}]&/@{{1,2,3},{1,1,2},{1,1,1}}

Properties

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

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

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

surf={a Sqrt[1+u^2] Cos[v],b Sqrt[1+u^2] Sin[v],c u};

Metric Tensor

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

Christoffel Symbol of the Second Kind

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

Vector Properties

eqn={a Sqrt[1+u^2] Cos[v],b Sqrt[1+u^2] Sin[v],c u};
rules={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};
assum=a>0&&c>0&&0u∈Reals&&0<v<2π;

Vector Length

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

Normal Vector

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

Two-Sheeted

surf[a_,b_,c_][u_,v_]:={a Sinh[u]Cos[v],b Sinh[u]Sin[v],c Cosh[u]}

Equation

eqns=Join[Thread[{x,y,z}^2==surf[a,b,c][u,v]^2],{Sin[v]^2+Cos[v]^2==1,Cosh[u]^2-Sinh[u]^2==1}]

GroebnerBasis[eqns,{x,y,z},{u,Cos[v],Sin[v],Sinh[u],Cosh[u]},MonomialOrder->EliminationOrder]

Numerator[-Factor[((x^2)/(a^2))+((y^2)/(b^2))-((z^2)/(c^2))+1]]

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

Cartesian

<<Graphics`ContourPlot3D`

V5

With[{a=1,c=1},
ContourPlot3D[x^2/a^2+y^2/a^2-z^2/c^2+1,{x,-2,2},{y,-2,2},{z,-3,3},
PlotPoints->5,Boxed->False]
]

V6

ContourPlot3D[x^2/#1^2+y^2/#2^2-z^2/#3^2==-1,{x,-3,3},{y,-3,3},{z,-3,3},RegionFunction->(#1^2+#2^2+#3^2<3^2&)]&@@@{{1,2,.5},{1,2,1},{1,1,1}}

Parametric

V6

ParametricPlot3D[{{Sinh[u]Cos[v],Sinh[u]Sin[v],Cosh[u]},{Sinh[u]Cos[v],Sinh[u]Sin[v],-Cosh[u]}},{v,0,Pi},{u,-2,2},Boxed->False,Axes->False]

V6

ParametricPlot3D[surf[1,1][u,v],{u,-2,2},{v,0,π}]

LiveGraphics3D

WriteLiveForm["hyperb2.m",Show[ParametricPlot3D[{{Sinh[u]Cos[v],Sinh[u]Sin[v],Cosh[u]},{Sinh[u]Cos[v],Sinh[u]Sin[v],-Cosh[u]}},{v,0,Pi},{u,-2,2},Boxed->False,Axes->False],opts]]

Check

Simplify[(Sinh[u]Cos[v])^2+(Sinh[u]Sin[v])^2-Cosh[u]^2]

Properties

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