Hyperbolic Cylinder

Author

Eric W. Weisstein
June 3, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Definition

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

Equation

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

eqns=Append[Thread[{x,y,z}==surf[a,b][u,v]],Cosh[u]^2-Sinh[u]^2==1]

{eqn}=Factor/@GroebnerBasis[eqns,{x,y},{Cosh[u],Sinh[u],v}]

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

((#)/((a^2)(b^2)))&/@eqn

Plot

V5

<<Graphics`ContourPlot3D`

ContourPlot3D[x^2-y^2+1,{x,-2,2},{y,-2,2},{z,-1,1}]

ParametricPlot3D[{Sinh[u],Cosh[u],v},{u,-1,1},{v,0,1}]

ParametricPlot3D[{Sinh[u],-Cosh[u],v},{u,-1,1},{v,0,1}]

WriteLiveForm["hypercyl.m",Show[ParametricPlot3D[{{x,Sqrt[x^2+1],t},{x,-Sqrt[x^2+1],t}},{x,-1,1},{t,0,2},Boxed->False,Ticks->None,AspectRatio->Automatic,Axes->False],opts]]

V6

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

ContourPlot3D[Evaluate[With[{a=1,b=1},((x^2)/(a^2))-((y^2)/(b^2))==-1]],{x,-2,2},{y,-2,2},{z,-1,1}]

Fundamental Forms

Curvatures[{a Sinh[u],b Cosh[u],v},{u,v}]

SecondFundamentalForm[{a Sinh[u],b Cosh[u],v},{u,v}]//FullSimplify

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

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

GaussianCurvature[eqn/.{x->y,y->z,z->x},{x,y,z}]

MeanCurvature[{a Sinh[u],b Cosh[u],v},{u,v}]//FullSimplify

Area Element

AreaElement[{a Sinh[u],b Cosh[u],v},{u,v}]

Integrate[AreaElement[{a Sinh[u],b Cosh[u],v},{u,v}],{u,-∞,∞},{v,0,h},Assumptions->a>0&&b>0]//Timing