Quadratrix of Hippias

Author

Eric W. Weisstein
June 6, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Plot

V5

<<Utilities`Plot`

With[{a=1},
PolarPlot[2a θ/(π Sin[θ]),{θ,-100,100},Ticks->None,PlotStyle->Red,
PlotRange->{{-5,5},{-5,5}},AspectRatio->Automatic]
]

With[{a=1},
PolarPlot[2a θ/(π Sin[θ]),{θ,-π,π},Ticks->None,PlotStyle->Red,
PlotRange->{{-5,5},{-2,2}},
AspectRatio->Automatic]
]

With[{a=1},
ParametricPlot[{y Cot[π y/(2a)] ,y},{y,-60,60},Ticks->None,PlotStyle->Red,PlotRange->{{-5,1},{-3,3}},AspectRatio->Automatic]
]

With[{a=1},
Show[
PlotDiscontinuous[y Cot[π y/(2a)] ,Evaluate[{y,Sequence@@Range[-10,10,2]}],PlotStyle->Red,
VerticalAsymptotes->Range[-10,10,2],
DisplayFunction->Identity]/.Line[x_]:>Line[Reverse/@x],
DisplayFunction->$DisplayFunction,
Axes->True,
Ticks->None,
PlotRange->{{-15,15},{-8,8}},
AspectRatio->Automatic
]
]

V6

PolarPlot[2 /(π Sinc[θ]),{θ,-100,100},Ticks->None,PlotStyle->Red,PlotRange->{{-10,10},{-8,8}},AspectRatio->Automatic,Exclusions->(Sinc[θ]==0),ExclusionsStyle->{Dashing[.02]}]

Reduce[Sin[t]==0&&-100<t<100,t]

ParametricPlot[2 /(π Sinc[t]){Cos[t],Sin[t]},{t,-20,20},PlotStyle->Red,PlotRange->{{-10,10},{-8,8}},AspectRatio->Automatic,Exclusions->Range[-100,100]π,ExclusionsStyle->{Dashing[.02]}]

Equation

Eliminate[{x==2 a θ/(π Sin[θ])Cos[θ],y==2 a θ/(π Sin[θ])Sin[θ],Sin[θ]^2+Cos[θ]^2==1},θ]

FullSimplify[%]

Properties

quadratrix=2 a/(Pi Sinc[t]) {Cos[t],Sin[t]};

Arc Length

v4.2.1

ArcLength[quadratrix,t]//Timing

v5

ArcLength[quadratrix,t]//Timing

v6

ArcLength[quadratrix,t]//Timing

Curvature

Parametric

FullSimplify[Curvature[quadratrix,t]]

FullSimplify[%,a>0&&0<t<π]

Polar

FullSimplify[Curvature[2 a/(Pi Sinc[q]),q],a>0&&q∈Reals]

Slope

Parametric

Slope[2 a/(Pi Sinc[t]) {Cos[t],Sin[t]},t]//FullSimplify

Polar

Slope[2 a/(Pi Sinc[q]),q]//FullSimplify

Slope[,{x,y}]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[quadratrix,t],a>0&&t∈Reals]

Tangential Angle

v4.2.1

FullSimplify[TangentialAngle[quadratrix,t],0<t<π]//Timing

v5

FullSimplify[TangentialAngle[quadratrix,t],0<t<π]//Timing

FullSimplify[%,0<t<π]

v6

FullSimplify[TangentialAngle[quadratrix,t],0<t<π]//Timing

Plot[(1/2) (t+ArcCot[Cot[t]-t ((Csc[t])^2)]+ArcTan[(t/(1-t Cot[t]))]),{t,-10,10}]

Vector Properties

parm=2 a/(Pi Sinc[t]) {Cos[t],Sin[t]};
assum=a>0&&t∈Reals;
xform={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};

VectorLength

Assuming[assum,FullSimplify[Norm[parm]/.xform]]

NormalVector

Assuming[assum,FullSimplify[NormalVector[parm,t]/.xform]]

TangentVector

Assuming[assum,FullSimplify[TangentVector[parm,t]/.xform]]