Two-Sheeted Hyperboloid

Author

Eric W. Weisstein
June 10, 2014

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Surfaces/Two-SheetedHyperboloid.nb.

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

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

Equation

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

surf2[a_,c_][u_,v_]:={a Sinh[u]Cos[v],a Sinh[u]Sin[v],c u/Sqrt[u^2]Cosh[u]}

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

Simplify[Cosh[u]^2-Sinh[u]^2]

eqns=Join[Thread[{x,y,z}^2==surf[a,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)+(y^2))/(a^2))-((z^2)/(c^2))+1]]

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

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},RegionFunction->(#1^2+#2^2+#3^2<3^2&)]]

Parametric

V5

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}]

ParametricPlot3D[surf2[1,1][u,v],{u,-2,2},{v,0,π}]

ParametricPlot3D[surf2[1,1][u,v],{u,-2,2},{v,0,2π}]

ParametricPlot3D[(surf2@@#)[u,v],{u,-2,2},{v,0,2π}]&/@{{2,1},{1,1},{1,2}}

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

Top Sheet Parametrization

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

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

SortBy[GaussianCurvature[eqn,{x,y,z}]//FullSimplify,ByteCount]

Both Sheets Parametrization

Curvatures[surf2[a,c][u,v],{u,v},FullSimplify[#,a>0&&c>0&&u∈Reals]&]

FullSimplify[PrincipalCurvatures[surf2[a,c][u,v],{u,v}],a>0&&c>0&&(u|v)∈Reals&&u∈Reals]/.Sinh[u]<0:>u<0

SortBy[GaussianCurvature[eqn,{x,y,z}]//FullSimplify,ByteCount]

Vector Properties

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

Vector Length

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

Normal Vector

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