One-Sheeted Hyperboloid

Author

Eric W. Weisstein
June 10, 2014

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

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

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

Additional Authors

Wires connected two rings by
Primitivo Ramos

Equation

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

eqns=Append[Thread[{x,y,z}^2==surf[a,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)+(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);

Parametric Representations

Simplify[(1+u^2)Cos[v]^2+(1+u^2)Sin[v]^2-u^2]

V5

WriteLiveForm["hyperb11.m",Show[ParametricPlot3D[{Sqrt[1+u^2]Cos[v],Sqrt[1+u^2]Sin[v],u},{v,0,2Pi},{u,-2,2},Boxed->False,Axes->False],opts]]

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

WriteLiveForm["hyperb12.m",Show[ParametricPlot3D[{Cos[u]-v Sin[u],Sin[u]+v Cos[u],v},{u,0,2Pi},{v,-2,2},Boxed->False,Axes->False],opts]]

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

WriteLiveForm["hyperb13.m",Show[ParametricPlot3D[{Cosh[v]Cos[u],Cosh[v]Sin[u],Sinh[v]},{u,0,2Pi},{v,-1.5,1.5},Boxed->False,Axes->False],opts]]

V6

Quit

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

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

Volume

v5.1

HoldForm[π (a^2)h]Apart[Cancel[(1/(π (a^2)h))Integrate[c π a^2(1+(u^2)),{u,-h/(2c),h/(2c)}]]]

Simplify[Solve[((R^2)/(a^2))-(((((h/2)))^2)/(c^2))==1,#]&/@{R,h},c>0]

Simplify[Expand[Integrate[c π a^2(1+(u^2)),{u,-h/(2c),h/(2c)}]/.h^3->(((2 c))^2) (-1+((R^2)/(a^2)))h]]

Properties

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]

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

Surface Area

{ℰ,ℱ,𝒢}=FullSimplify[FirstFundamentalForm[surf[a,c][u,v],{u,v}]]

V5.0

Refine[Integrate[(Sqrt[ℰ 𝒢-(ℱ^2)]),{u,-h/(2c),h/(2c)},{v,0,2π},Assumptions->h>0&&a>0&&c>0],
h>0&&a>0&&c>0]//Timing

V5.1

Refine[Integrate[(Sqrt[ℰ 𝒢-(ℱ^2)]),{u,-h/(2c),h/(2c)},{v,0,2π},Assumptions->h>0&&a>0&&c>0],
h>0&&a>0&&c>0]//Timing

V8

FullSimplify[(Sqrt[ℰ 𝒢-(ℱ^2)]),a>0&&c>0]

(A=Assuming[h>0&&a>0&&c>0,Refine[Integrate[(Sqrt[ℰ 𝒢-(ℱ^2)]),{u,-h/(2c),h/(2c)},{v,0,2π}]]])//Timing

Assuming[h>0&&a>0&&c>0,FullSimplify[A]]

Limit

Assuming[h>0&&a>0,Limit[2 a π (((h (Sqrt[4 (c^4)+((a^2)+(c^2)) (h^2)]))/(4 (c^2)))+(((c^2) ArcSinh[(((Sqrt[(a^2)+(c^2)]) h)/(2 (c^2)))])/(Sqrt[(a^2)+(c^2)]))),c->∞]]

Connecting Two Rings

<<Utilities`LiveGraphics3D`

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

HyperboloidWireframe2[n_,off_]:=Module[{k,u,v,g1,g2,g3,g4},
g1=Line[Table[{Cos[((off π k)/n)],Sin[((off π k)/n)],1+(((-1))^k)},{k,0,n}]];
g2=Line[Table[{Cos[((off π k)/n)],Sin[((off π k)/n)],1-(((-1))^k)},{k,0,n}]];
g3=ParametricPlot3D[torus[1,.04,0][u,v]//Evaluate,{u,0,2Pi},{v,0,2Pi},PlotPoints->{33,8},Axes->None,Boxed->False,DisplayFunction->Identity][[1]];
g4=ParametricPlot3D[torus[1,.04,2][u,v]//Evaluate,{u,0,2Pi},{v,0,2Pi},PlotPoints->{33,8},Axes->None,Boxed->False,DisplayFunction->Identity][[1]];
{g1,g2,g3,g4}
]

WriteLiveForm["hyperwir.m",Show[Graphics3D[HyperboloidWireframe2[32,22]],Boxed->False]]

Metric Tensor

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

Christoffel Symbol of the Second Kind

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

Vector Properties

eqn={a Sqrt[1+u^2]Cos[v],a Sqrt[1+u^2]Sin[v],c 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]]