Equichordal Point

Author

Eric W. Weisstein
March 16, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/EquichordalPoint.nb.

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

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

Initialization

EquichordalCurve[r_,t_]:={
PolarPlot[r,{t,0,Pi},DisplayFunction->Identity][[1]],
PolarPlot[r-1,{t,0,Pi},DisplayFunction->Identity][[1]]
}

EquichordalLengths[r_,t_,n_]:=
Table[Sqrt[#.#]&[r{Cos[t],Sin[t]}-(r-1){Cos[t],Sin[t]}],{t,0,Pi,Pi/n}]

EquichordalCurvePlot[r_,t_,n_]:=Graphics[{
EquichordalCurve[r,t],
Table[Line[{r{Cos[t],Sin[t]},(r-1){Cos[t],Sin[t]}}],{t,0,Pi,Pi/n}],
{RGBColor[1,0,0],PointSize[.08],Point[{0,0}]}
},
Ticks->None,AspectRatio->Automatic]

f[0]=1-f[Pi]

Plots

Show[GraphicsArray[Table[EquichordalCurvePlot[x+(.5-x)Cos[2t],t,10],
{x,.3,.5,.05}]]]

Plot[x+(.5-x)Cos[2t]/.x->.35,{t,0,Pi}]

Show[EquichordalCurvePlot[1-t/Pi,t,10]]

Simplify[EquichordalLengths[t^2/Pi^2,t,20]]

Table[Show[EquichordalCurvePlot[t^2/Pi^2,t,10]],{x,.1,.3,.1}]

Plot[1-t/Pi,{t,0,Pi}]

Show[EquichordalCurvePlot[x+(.5-x)Cos[4t] /. x->.35 ,t,10]]

Plot[x+(.5-x)Cos[4t] /. x->.35,{t,0,Pi}]

{x+(.5-x)Cos[4t]/.t->0 ,x+(.5-x)Cos[4t]/.t->Pi}

{x+(.5-x)Cos[4t]/.t->0 ,x+(.5-x)Cos[4t]/.t->Pi}

{1-t/Pi/.t->0,1-t/Pi/.t->Pi}

{t^2/Pi^2/.t->0,t^2/Pi^2/.t->Pi}