Catenoid

Author

Eric W. Weisstein
June 20, 2013

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

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Plot

cat1=ParametricPlot3D[{x,Cosh[x]Cos[t],Cosh[x]Sin[t]},{t,0,2Pi},{x,-1,1},Axes->False,Boxed->False]

cat2=ParametricPlot3D[{Cosh[u] Cos[v],Cosh[u] Sin[v],u},{v,0,2 Pi},{u,-1,1},Axes->False,ViewPoint->{3,2,.5},Boxed->False]

LiveGraphics3D

WriteLiveForm["catenoi1.m",cat1]

WriteLiveForm["catenoi2.m",cat2]

Properties

surf={c Cosh[v/c]Cos[u],c Cosh[v/c]Sin[u],v};

Simplify[Total[{a Cosh[v/a]Cos[u],a Cosh[v/a]Sin[u]}^2]]

FullSimplify[x^2+y^2-(a^2) ((Cosh[(z/a)])^2)/.Thread[{x,y,z}->surf]]

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

ParametricPlot3D[catenoid/.a->1,{u,0,2π},{v,-1.5,1.5}]

Properties

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

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

FullSimplify[GaussianCurvature[x^2+y^2-(a^2) ((Cosh[(z/a)])^2),{x,y,z}],a>0]

Metric Properties

Metric Tensor

FullSimplify[MetricTensor[surf,{u,v}],-π/2<u<π/2&&0<v<π]

Christoffel Symbol of the Second Kind

FullSimplify[ChristoffelSymbols[surf,{u,v}],-π/2<u<π/2&&0<v<π&&a>0]

Minimal Surface

catenaryConstants[{x1_,y1_},{x2_,y2_}]:=Module[{i,a,b},
{a,b} /. FindRoot[
{y1==a Cosh[(x1-b)/a],
y2==a Cosh[(x2-b)/a]},
{a,1},{b,0},MaxIterations->100]
]

CatenaryConstants[{x1_,y1_},{x2_,y2_}]:=Check[catenaryConstants[{x1,y1},{x2,y2}],$Goldschmidt]

Catenoid[{x1_,y1_},{x2_,y2_}]:=Module[
		{a,b,i,r,cc=CatenaryConstants[{x1,y1},{x2,y2}]},
		If[cc==$Goldschmidt,
		Show[Graphics3D[{ParametricPlot3D[{x1,r Cos[t],r Sin[t]},{r,0,y1},{t,0,2Pi},	PlotPoints->{7,15},Boxed->False,DisplayFunction->Identity,Axes->False][[1]],
							{PointSize[.05],Point[{x1,0,y1}],Point[{x2,0,y2}]},
				ParametricPlot3D[{x2,r Cos[t],r Sin[t]},{r,0,y2},{t,0,2Pi},
						PlotPoints->{7,15},Boxed->False,DisplayFunction->Identity,Axes->False][[1]]}],
			DisplayFunction->Identity,Boxed->False],,
			(*Else*)
			{a,b}=cc;
		line=ParametricPlot3D[{x,0,a Cosh[(x-b)/a] },{x,x1,x2},DisplayFunction->Identity];
		Show[Graphics3D[{ParametricPlot3D[{x,a Cosh[(x-b)/a] Cos[t],a Cosh[(x-b)/a]Sin[t]},{t,0,2Pi},{x,x1,x2},
Boxed->False,Axes->False,DisplayFunction->Identity][[1]],
	Table[{Thickness[.02],line[[i]]},{i,Length[line]-1}],
					{PointSize[.05],Point[{x1,0,y1}],Point[{x2,0,y2}]}}],Boxed->False,
		DisplayFunction->Identity]
			]
	]

<<Graphics`ImplicitPlot`

Plot[a Cosh[1/(2a)]-1,{a,.15,2}]

A[a_]:=Pi a^2(Sinh[1/a]+1/a)

FullSimplify[4Pi a Integrate[Cosh[x/a]^2,{x,0,x0}]]

a1=a /.FindRoot[a Cosh[1/(2a)]-1,{a,.2}]

a2=a /.FindRoot[a Cosh[1/(2a)]-1,{a,.8}]

N[A/@{a1,a2},10]

{CatenaryConstants[{-.5,1},{.5,1}],CatenaryConstants[{-.6,1},{.6,1}],
CatenaryConstants[{-.66,1},{.66,1}],CatenaryConstants[{-.67,1},{.67,1}]}

CatenaryConstants[{-1,1},{1,2}]

Show[Catenoid[{-1.,2},{1.5,2}],DisplayFunction->$DisplayFunction]

Show[GraphicsArray[{Catenoid[{-1.2,2},{1.2,2}],
			Catenoid[{1,1},{3,2.5}],
		Catenoid[{-2,1},{2,2}]}],
		DisplayFunction->$DisplayFunction]

Vector Properties

eqn={a Cosh[v/a] Cos[u],a Cosh[v/a] Sin[u],v};

Vector Length

FullSimplify[Norm[eqn]/.Abs[x_]^(n_/;Mod[n,2]==0):>x^n,(u|v)∈Reals&&a>0]

Normal Vector

FullSimplify[NormalVector[eqn,{u,v}]/.Abs[x_]^(n_/;Mod[n,2]==0):>x^n,(u|v)∈Reals&&a>0]