Kampyle of Eudoxus

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Definition

curve=a Sec[t]^2{Cos[t],Sin[t]};

Equations

eqns=Append[Numerator[Factor[{x,y}-curve/.{Sec[t]->1/cos,Tan[t]->sin/cos}]],sin^2+cos^2-1]

Factor/@GroebnerBasis[eqns,{x,y},{cos,sin}]

eqn=((a^2) (x^2)-(x^4)+(a^2) (y^2));

Plot

Implicit

With[{a=1},(a^2) (x^2)-(x^4)+(a^2) (y^2)]

ContourPlot[(x^2)-(x^4)+(y^2)==0,{x,-3,3},{y,-3,3}]

Paramatric

{ParametricPlot[Sec[t]^2{Cos[t],Sin[t]},{t,-1,1},PlotStyle->Red,AxesOrigin->{0,0},ImageSize->Small,Ticks->None],
ParametricPlot[Sec[t]^2{Cos[t],Sin[t]},{t,0,π},PlotStyle->Red,AxesOrigin->{0,0},ImageSize->Small,Ticks->None]
ParametricPlot[Sec[t]^2{Cos[t],Sin[t]},{t,π,2π},PlotStyle->Red,AxesOrigin->{0,0},ImageSize->Small,Ticks->None]}

ParametricPlot[Sec[t]^2{Cos[t],Sin[t]},{t,-π,π},PlotStyle->Red,AxesOrigin->{0,0},ImageSize->Small]

Polar

PolarPlot[Sec[t]^2,{t,-π,π},PlotStyle->Red,AxesOrigin->{0,0},Ticks->None,ImageSize->Small]

PolarPlot[Sec[t]^2,{t,0,2π},PlotStyle->Red,AxesOrigin->{0,0},Ticks->None,ImageSize->Small]

Properties

curve=a Sec[t]^2{Cos[t],Sin[t]};

Arc Length Functiopn

v4.2

ArcLength[Sec[t]^2,t,Assumptions->t>0]//Timing

v5

ArcLength[Sec[t]^2,t,Assumptions->t>0]//Timing

ArcLength[Sec[t]^2,t,Assumptions->-Pi/2<t<Pi/2]//Timing

ArcLength[Sec[t]^2,{t,-∞,∞}]//Timing

v6

ArcLength[Sec[t]^2,t,Assumptions->t>0]//Timing

(s=ArcLength[Sec[t]^2,t,Assumptions->-Pi/2<t<Pi/2])//Timing

FullSimplify[Expand[s],-Pi/2<t<Pi/2]

ArcLength[Sec[t]^2,{t,-∞,∞}]//Timing

V6.1

ArcLength[curve,t,Assumptions->a>0&&0<t<2π]

ArcLength[curve,t,Assumptions->a>0&&0<t<π/2]

Plot[(ArcSinh[2 Tan[t]]+(Sqrt[10-6 Cos[2 t]]) Sec[t] Tan[t]),{t,0,π/2}]

Curvature

Parametric

FullSimplify[Curvature[curve,t],0<t<2π]

Polar

FullSimplify[Curvature[a Sec[q]^2,q],a>0&&0<q<2π]

Implicit

FullSimplify[ImplicitCurvature[(a^2) (x^2)-(x^4)+(a^2) (y^2),{x,y}],0<t<2π&&a>0]

Inflection Points

pts=InflectionPoints[curve,t]//Union

InflectionPoints[eqn,{x,y}]

Block[{a=1},ContourPlot[eqn==0,{x,-2,2},{y,-2,2},Epilog->{PointSize[.02],Point[pts]}]]

Singular Points

eqn/.{x->0,y->0}

SingularPoints[eqn,{x,y}]

Slope

Slope[a Sec[t]^2 {Cos[t],Sin[t]},t]//FullSimplify

Slope[a Sec[q]^2,q]//FullSimplify

Slope[x^4==a^2(x^2+y^2),{x,y}]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&0<t<2π]

Tangential Angle

v5

TangentialAngle[Sec[t]^2,t]//FullSimplify

v6

TangentialAngle[curve,t,Assumptions->a>0&&0<t<2π]

Plot[t-ArcTan[2 Tan[t]],{t,0,2π}]

..make continuous...

Vector Properties

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

VectorLength

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

Plot[(Sqrt[((Sec[t])^4)])-Sec[t]^2,{t,0,2π}]

NormalVector

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

{-((Cos[t] (((Sec[t])^2)+((Tan[t])^2)))/(Sqrt[ -3+4 ((Sec[t])^2)])),((Cos[t] Tan[t])/(Sqrt[ -3+4 ((Sec[t])^2)]))}

TangentVector

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

{((Cos[t] (((Sec[t])^2)+((Tan[t])^2)))/(Sqrt[ -3+4 ((Sec[t])^2)])),((Cos[t](((Sec[t])^2)+((Tan[t])^2)))/(Sqrt[ -3+4 ((Sec[t])^2)]))}