Catenary

Note

This Mathematica notebook is from the WWW project A Visual Dictionary of Special Plane Curves. Copyright © 1995-2000 by Xah Lee. (http://xahlee.org/)

Packages and Initializations

Needs["ParaPlot`"]

Needs["PlaneCurveFormulas`"]

Needs["PlaneCurveGenerator`"]

Needs["Graphics`ImplicitPlot`"]
(*MathSource Item:0200-529*)

$HistoryLength = 5;

Description

Catenary

Catenary (aka chainette or alysoid, hyperbolic cosine) is defined as the curve of the function Cosh[x], or (E^x + E^(-x))/2.

Catenary[][t]

ParaPlot[Catenary[][t], {t, -3, 3}, 
        PlotRange -> {{-1, 1} 3, {0, 5}}, 
        AspectRatio -> Automatic, PlotDot -> False];

Properties

catacoustic

The Catacaustic of the expontial curve with light rays parallel
to the y axes is the catenary.

ParaPlot[{t, (E^t)}, {t, -5, 5, 9/50}, 
        CausticOrigin -> {0, 20000}, CausticLineLength -> 21000, 
        PlotRange -> {{-5, 2.5}, {-2, 8}}, 
        AspectRatio -> Automatic, Axes -> False, 
        Background -> Hue[0, 0, 0], CausticLineStyle -> {Hue[0]}, 
        CurveStyle -> {Thickness[0.007]}, 
        PlotDot -> False];

involute

The Involute of Catenary is Tractrix

ParaPlot[Catenary[][t], {t, -2, 2.5}, Range[0, 5,  .25], 
        PlotInvolute -> True, AspectRatio -> Automatic, 
        PlotRange -> {{-2.05, 4.5}, {-0.05, 6.05}}, 
        CurveColorFunction -> {({Hue[ .7,  .6, 1], 
                  Thickness[ .009]} &)}];

ParaPlot[{Catenary[][t], 
          Involute[Catenary[][t], t, 0]}, {t, -4, 4}, 
        AspectRatio -> Automatic, 
        PlotStyle -> {Hue[ .65], Hue[0], Hue[ .35]}, 
        PlotRange -> {{-3.5, 3.5}, {-0.05, 6.05}}];

Radial

Radial of Catenary is Kampyle of Eudoxus.

ParaPlot[{Evolute[Catenary[][t], t], Catenary[][
            t]}, {t, -3, 3,  .1}, PlotDot -> False, 
        PlotOsculatingCircle -> False, AspectRatio -> Automatic, 
        Background -> Hue[0, 0, 0]];

ParaPlot[{Catenary[][t], 
          Radial[Catenary[][t], t]}, {t, -3, 3}, 
        AspectRatio -> Automatic, 
        PlotStyle -> {Hue[ .65], Hue[0], Hue[ .35]}, 
        PlotRange -> {{-10, 10}, {-1, 10}}];

ParaPlot[Catenary[][t], {t, -0, 3, 3/20}, 
        PlotRadial -> True, PlotEvolute -> True, 
        AspectRatio -> Automatic, Background -> Hue[0, 0, 0], 
        LineStyle -> Table[Hue[x], {x, 0, 1, 1/30}], 
        PlotRange -> {{-10, 10}, {-1, 10}}];

Graphics gallery

modern moonlight

evolute = 
        First@ParametricPlot[
            Evaluate@Radial[Catenary[][t], t], {t, -4, 4}, 
            AspectRatio -> Automatic, 
            PlotStyle -> {Hue[ .65], Hue[0], Hue[ .35]}, 
            PlotRange -> {{-10, 10}, {-1, 10}}, 
            DisplayFunction -> Identity];

ParaPlot[Catenary[][t], {t, -0, 4, 4/30}, 
        PlotRadial -> True, PlotEvolute -> True, 
        LineStyle -> {Thickness[ .003]}, Axes -> False, 
        PlotDot -> False, DotStyle -> {PointSize[ .004]}, 
        PlotOsculatingCircle -> True, 
        CurveColorFunction -> {{Hue[0, 1, 0.8], Thickness[ .004]} &}, 
        PlotRange -> {{-1, 1}, {-1, 1}}*{1.5, 1}*20, 
        AspectRatio -> Automatic, Background -> Hue[0, 0, 0], 
        Prolog -> {evolute}];

Clear[evolute];