Cissoid of Diocles

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Definition

eqn[a_][t_]:=a{2 Sin[t]^2,2 Sin[t]^2 Tan[t]}

Equation

Together[x^3/(2a-x)-y^2]

Simplify[(x^3)-2 a (y^2)+x (y^2)-(x(x^2+y^2)-2a y^2)]

Append[Thread[{x,y}==a{2 Sin[t]^2,2 Sin[t]^2 Tan[t]}],Sin[t]^2+Cos[t]^2==1]/.{_Cos->cos,_Sin->sin,_Tan->tan}

GroebnerBasis[{x==2 a (sin^2),y==2 a tan (sin^2),(cos^2)+(sin^2)==1,sin==cos tan},{x,y},{sin,cos,tan}]

eqn=(x^3)-2 a (y^2)+x (y^2);

AlgebraicDegree[(x^3)-2 a (y^2)+x (y^2),{x,y}]

Plot

V5

ParametricPlot[{
{2 Sin[t]^2,2 Sin[t]^2 Tan[t]},
{2,2t},
{1+Cos[Pi/1.5t],Sin[Pi/1.5t]}},
{t,-1.5,1.5},
PlotRange->{-3,3},
AspectRatio->Automatic,
Ticks->None,PlotStyle->{Red,{Dashing[{0.05,0.05}]},{Dashing[{0.05,0.05}]}}]

With[{t0=10},
ParametricPlot[{
{2t^2,2t^3}/(1+t^2),
{2,2t},
{1+Cos[Pi t/t0],Sin[Pi t/t0]}},
{t,-t0,t0},
PlotRange->{-3,3},
AspectRatio->Automatic,
Ticks->None,PlotStyle->{Red,{Dashing[{0.05}]},{Dashing[{0.05}]}}]
]

V6

With[{a=1},ContourPlot[(x^3)-2 a (y^2)+x (y^2)==0,{x,-2,2},{y,-2,2},AspectRatio->Automatic,ContourStyle->Red]]

ParametricPlot[{2 Sin[t]^2,2  Sin[t]^3/Cos[t]},{t,-1.5,1.5},PlotRange->{-3,3},Ticks->None,PlotStyle->Red]

ParametricPlot[{2 Sin[t]^2,2  Sin[t]^3/Cos[t]},{t,0,1.5},PlotRange->{-3,3},Ticks->None,PlotStyle->Red]

Properties a {2 (sin^2)(t),2 (sin^2)(t) tan(t)}

cissoid=a{2 Sin[t]^2,2  Sin[t]^3/Cos[t]};

Arc Length Function

v4.2

ArcLength[cissoid,t]

v5

FullSimplify[ArcLength[cissoid,t],a>0]//Timing

FullSimplify[ArcLength[cissoid,t,Assumptions->a>0&&-Pi/2<t<Pi/2],a>0]//Timing

v6

FullSimplify[ArcLength[eqn[a][t],t,Assumptions->0<t<π/2],a>0]

Plot[(-4+(Sqrt[3]) Log[14+8 (Sqrt[3])]-2 (Sqrt[3]) Log[(Sqrt[6]) Cos[t]+(Sqrt[5+3 Cos[2 t]])]+(Sqrt[10+6 Cos[2 t]]) Sec[t]),{t,0,π/2}]

ArcLength[cissoid,t,Assumptions->a>0&&-(π/2)≤t≤(π/2)]//Timing

FullSimplify[ArcLength[cissoid,t,Assumptions->a>0&&0<t<Pi/2]]//Timing

V8

ArcLength[cissoid,t,Assumptions->a>0&&-(π/2)≤t≤(π/2)]//Timing

FullSimplify[%,0<t<π/2]

Plot[(-4+(Sqrt[3]) Log[4]+2 (Sqrt[3]) Log[12+7 (Sqrt[3])]+4 (Sqrt[3]) Log[Sec[(t/2)]]-2 (Sqrt[3]) Log[(2+(Sqrt[3])) (6 Cos[t]+(Sqrt[6]) (Sqrt[5+3 Cos[2 t]])) ((Sec[(t/2)])^2)]+(Sqrt[10+6 Cos[2 t]]) Sec[t]),{t,0,π/2}]

Area Between Curve and Asymptote

2Integrate[(Sqrt[((x^3)/(2a-x))]),{x,0,2a},Assumptions->a>0]

Block[{a=1},RegionPlot[0<(x^3)-2 a (y^2)+x (y^2)&&x<2a,{x,-.5,2},{y,-10,10}]]

Integrate[Boole[0<(x^3)-2 a (y^2)+x (y^2)&&x<2a],{x,-∞,∞},{y,-∞,∞},Assumptions->a>0]

Asymptote

Limit[cissoid,t-># π/2,Assumptions->a>0,Direction->-#]&/@{-1,1}

Curvtaure

v4.2.1

FullSimplify[Curvature[cissoid,t],a>0]

v5

FullSimplify[Curvature[cissoid,t],a>0]

v6

FullSimplify[Curvature[cissoid,t],a>0]

Polar

FullSimplify[Curvature[2 a Sin[q] Tan[q],q],a>0&&0<q<2π]

Implicit Curvature

FullSimplify[ImplicitCurvature[x^3-2 a y^2+x y^2,{x,y}],a>0]

Singular Points

SingularPoints[eqn,{x,y}]

Slope

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

Slope[2 a Sin[q] Tan[q],q]//FullSimplify

Slope[y^2==x^3/(2 a-x),{x,y}]//FullSimplify

SpecialAffineCurvature

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

Tangential Angle

v4.2.1

FullSimplify[TangentialAngle[cissoid,t]]

FullSimplify[TangentialAngle[cissoid,t,Assumptions->a>0&&0<t<Pi/2]]//Timing

FullSimplify[TangentialAngle[cissoid,t,Assumptions->a>0&&-Pi/2<t<Pi/2]]//Timing

v5

TangentialAngle[cissoid,t]//Timing

FullSimplify[TangentialAngle[cissoid,t,Assumptions->a>0&&0<t<Pi/2]]//Timing

FullSimplify[TangentialAngle[cissoid,t,Assumptions->a>0&&-Pi/2<t<Pi/2]]//Timing

v6

TangentialAngle[cissoid,t]//Timing

FullSimplify[TangentialAngle[cissoid,t,Assumptions->a>0&&0<t<Pi/2]]//Timing

FullSimplify[TangentialAngle[cissoid,t,Assumptions->a>0&&-Pi/2<t<Pi/2]]//Timing

Limit[((1/6)-((ArcCot[2 Cot[t]])/(12 t))),t->0]

Integrate[(((Cos[(t^2)])^2)/(5+3 Cos[2 (t^2)])),{t,0,1},Assumptions->t==0]

Properties (Parameterization 2)

cissoid2=a{2t^2/(1+t^2),2t^3/(1+t^2)};

Arc Length

v5

FullSimplify[ArcLength[cissoid2,t],t∈Reals&&a>0]//Timing

v6

FullSimplify[ArcLength[cissoid2,t],t∈Reals&&a>0]//Timing

FullSimplify[ArcLength[cissoid2,t],t<0&&a>0]//Timing

FullSimplify[ArcLength[cissoid2,t],t>0&&a>0]//Timing

Curvature

v4.2

FullSimplify[Curvature[cissoid2,t],t∈Reals&&a>0]

v5

FullSimplify[Curvature[cissoid2,t],t∈Reals&&a>0]

v6

FullSimplify[Curvature[cissoid2,t],t∈Reals&&a>0]

Tangential Angle

v4.2

TangentialAngle[cissoid2,t]

FullSimplify[TangentialAngle[cissoid2,t],t∈Reals]

TangentialAngle[cissoid2,t,Assumptions->t∈Reals]

v5

TangentialAngle[cissoid2,t]

FullSimplify[TangentialAngle[cissoid2,t],t∈Reals&&a>0]

TangentialAngle[cissoid2,t,Assumptions->t∈Reals]

v6

TangentialAngle[cissoid2,t]

FullSimplify[TangentialAngle[cissoid2,t],t∈Reals]

TangentialAngle[cissoid2,t,Assumptions->t∈Reals]//InputForm

...doesn't evaluate because of Abs'...

Vector Properties

parm=a{2 Sin[t]^2,2 Sin[t]^2 Tan[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]]