Plateau Curves

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Initialization

plateau[{a_,m_,n_},t_]:={a Sin[(m+n)t]/Sin[(m-n)t],2a Sin[m t]Sin[n t]/Sin[(m-n)t]}

Plot

ParametricPlot[plateau[{1,5,3},t],{t,0,Pi},PlotRange->{{-3.5,4},{-4,4}},Ticks->None,AspectRatio->Automatic,PlotStyle->Red]

ParametricPlot[plateau[{##},t],{t,0,2Pi},Ticks->None,AspectRatio->Automatic,PlotStyle->Red,PlotRange->10]&@@@{{1,5,3},{1,3,1},{1,3,4},{1,2,3},{1,1,2},{1,5,4}}

Special Case

FullSimplify[plateau[{a,2n,n},t],n∈Integers]

ParametricPlot[plateau[1,6,3],{t,0,Pi/3},PlotRange->{{-3.5,4},{-4,4}},Ticks->None,AspectRatio->Automatic,Epilog->{Red,PointSize[.03],Point[{1,0}]},PlotStyle->Red]

Properties

Arc Length

v5

ArcLength[plateau[{a,m,n},t],t]

v6

ArcLength[plateau[{a,m,n},t],t]//Timing

...crash...

Curvature

v5

FullSimplify[Curvature[plateau[{a,m,n},t],t],a>0&&(m|n)∈Integers&&m>0&&n>0]//Timing

v6

FullSimplify[Curvature[plateau[{a,m,n},t],t],a>0&&(m|n)∈Integers&&m>0&&n>0]

Tangential Angle

v5

FullSimplify[TangentialAngle[plateau[{a,m,n},t],t]]//Timing

...ran overnight...

v6

FullSimplify[TangentialAngle[plateau[{a,m,n},t],t]]

...crash become of arc length...