Cycloid of Ceva

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Plot

PolarPlot[1+2Cos[2θ],{θ,0,2π},PlotStyle->Red,Ticks->None]

Manipulate[ParametricPlot[(1+2Cos[2t]){Cos[t],Sin[t]},{t,0,tmax},PlotRange->{{-1,1}3,{-1,1}}],{{tmax,.5},0,2π}]

ParametricPlot[(1+2Cos[2t]){Cos[t],Sin[t]},{t,0,2Pi},Ticks->None,
PlotStyle->Red]

ParametricPlot[(1+2Cos[2t]){Cos[t],Sin[t]},{t,-Pi/3,Pi/3},AspectRatio->Automatic,Ticks->None,
PlotStyle->Red]

ParametricPlot[(1+2Cos[2t]){Cos[t],Sin[t]},{t,Pi/3,2Pi/3},AspectRatio->Automatic,Ticks->None,
PlotStyle->Red]

ParametricPlot[(1+2Cos[2t]){Cos[t],Sin[t]},{t,2Pi/3,4Pi/3},AspectRatio->Automatic,Ticks->None,
PlotStyle->Red]

ParametricPlot[(1+2Cos[2t]){Cos[t],Sin[t]},{t,4Pi/3,5π/3},AspectRatio->Automatic,Ticks->None,
PlotStyle->Red]

Equations

Algebraic (a==1)

FullSimplify[Eliminate[Append[Thread[{x,y}==TrigExpand[(1+2Cos[2t]){Cos[t],Sin[t]}]],
Cos[t]^2+Sin[t]^2==1],{Cos[t],Sin[t]}]]

Factor[%]

Expand[Subtract@@{(x^2+y^2)^3-(3x^2-y^2)^2,(x^6)+(y^6)+3 (x^4) (-3+(y^2))+3 (x^2) (y^2) (2+(y^2))-(y^4)}]

Expand[eqn=(x^6)+(y^6)+3 (x^4) (-3+(y^2))+3 (x^2) (y^2) (2+(y^2))-(y^4)]

Algebraic

FullSimplify[Eliminate[Append[Thread[{x,y}==TrigExpand[a(1+2Cos[2t]){Cos[t],Sin[t]}]],
Cos[t]^2+Sin[t]^2==1],{Cos[t],Sin[t]}]]

Subtract@@%

Expand[%]

Properties

curve=a(1+2Cos[2t]){Cos[t],Sin[t]};

Arc Length Function

By Hand

Sasha Pavlyk
Sun, 16 Mar 2008 23:58:27 -0500

s0[t_]:=-8 EllipticE[ArcSin[2-3/(2+Cos[2 t])],13/16]+3/2 EllipticF[ArcSin[2-3/(2+Cos[2 t])],13/16]+3/2 EllipticPi[1/4,-ArcSin[2-3/(2+Cos[2 t])],13/16]+Log[(5+4 Cos[2 t])/(51+12 Cos[2 t]-36 Cos[4 t]-12 Sqrt[3] Sqrt[(11+4 Cos[2 t]-6 Cos[4 t]) Sin[2 t]^2])]/Sqrt[3]+(2 Sqrt[(11+4 Cos[2 t]-6 Cos[4 t]) Sin[2 t]^2])/(2+Cos[2 t])

FullSimplify[s0[t]-s0[0],0<t<π/2]

s[t_]:=(1/6) (48 EllipticE[(13/16)]-48 EllipticE[ArcSin[2-(3/(2+Cos[2 t]))],(13/16)]+9 EllipticF[ArcSin[2-(3/(2+Cos[2 t]))],(13/16)]-9 EllipticK[(13/16)]+9 EllipticPi[(1/4),(13/16)]+9 EllipticPi[(1/4),-ArcSin[2-(3/(2+Cos[2 t]))],(13/16)]+(Sqrt[3]) Log[9]+2 (Sqrt[3]) Log[(5+4 Cos[2 t])/(51+12 Cos[2 t]-36 Cos[4 t]-12 (Sqrt[3]) (Sqrt[11+4 Cos[2 t]-6 Cos[4 t]]) Sin[2 t])]+((12 (Sqrt[11+4 Cos[2 t]-6 Cos[4 t]]) Sin[2 t])/(2+Cos[2 t])))

s[π/2]//FullSimplify

N[%]

Plot[s[t],{t,0,π/2}]

f[x_?NumericQ]:=4NIntegrate[Sqrt[1+56 Cos[t]^2-48 Cos[t]^4],{t,0,x}]

Plot[f[t],{t,0,π/2}]

Arc Length

By hand

s[π/2]//FullSimplify

N[%]

V6.1

NIntegrate[(Sqrt[(((1+2 Cos[2 θ]))^2)+16 ((Sin[2 θ])^2)]),{θ,0,2π}]

FullSimplify[(Sqrt[(((1+2 Cos[2 θ]))^2)+16 ((Sin[2 θ])^2)]),0<θ<2π]

4NIntegrate[(Sqrt[11+4 Cos[2 θ]-6 Cos[4 θ]]),{θ,0,π/2}]

4NIntegrate[Sqrt[9+40*Sin[t]^2-48*Sin[t]^4],{t,0,π/2}]

4NIntegrate[Sqrt[1+56 ((Cos[t])^2)-48 ((Cos[t])^4)],{t,0,π/2}]

v5

ArcLength[1+2Cos[2θ],{θ,0,2π}]//Timing

2ArcLength[1+2Cos[2θ],{θ,0,π}]//Timing

Integrate[(Sqrt[11+4 Cos[2 θ]-6 Cos[4 θ]]),{θ,0,2π}]//Timing

2Integrate[(Sqrt[11+4 Cos[2 θ]-6 Cos[4 θ]]),{θ,0,π}]//Timing

v6

ArcLength[1+2Cos[2θ],{θ,0,2π}]//Timing

2ArcLength[1+2Cos[2θ],{θ,0,π}]//Timing

Integrate[(Sqrt[11+4 Cos[2 θ]-6 Cos[4 θ]]),{θ,0,2π}]//Timing

4Integrate[(Sqrt[11+4 Cos[2 θ]-6 Cos[4 θ]]),{θ,0,π/2}]//Timing

V6.1

4NIntegrate[Sqrt[1+56 ((Cos[t])^2)-48 ((Cos[t])^4)],{t,0,π/2}]

4Integrate[Sqrt[1+56 ((Cos[t])^2)-48 ((Cos[t])^4)],{t,0,π/2}]

Integrate[(Sqrt[11+4 Cos[2 θ]-6 Cos[4 θ]]),{θ,0,2π}]

External

lprint(4*int(sqrt(1+56*cos(t)^2-48*cos(t)^4),t=0..Pi/2));

lprint(int(sqrt(11+4*cos(2*t)-6*cos(4*t)),t=0..2*Pi));

Indefinite

Simplify[Integrate[(Sqrt[11+4 Cos[2 t]-6 Cos[4 t]]),t],0<t<π/2]

FullSimplify[%33,0<t<π/2]

s[t_]:=((29+8 (Sqrt[13])) (16 Sin[2 t]-6 Sin[4 t]+Tan[t])+I ((Cos[t])^2) ((220+61 (Sqrt[13])) EllipticE[-I ArcCsch[(-4+(Sqrt[13])) Cot[t]],(9/(((29+8 (Sqrt[13])))^2))]-8 (4+(Sqrt[13])) EllipticPi[29-8 (Sqrt[13]),-I ArcCsch[(-4+(Sqrt[13])) Cot[t]],(9/(((29+8 (Sqrt[13])))^2))]) (Sqrt[9+58 ((Tan[t])^2)+((Tan[t])^4)])-I EllipticF[-I ArcCsch[(-4+(Sqrt[13])) Cot[t]],(9/(((29+8 (Sqrt[13])))^2))] (104 (Sqrt[11+4 Cos[2 t]-6 Cos[4 t]])+32 (Sqrt[13]) (Sqrt[11+4 Cos[2 t]-6 Cos[4 t]])+60 (Sqrt[9+58 ((Tan[t])^2)+((Tan[t])^4)])+15 (Sqrt[13]) (Sqrt[9+58 ((Tan[t])^2)+((Tan[t])^4)])+15 (4+(Sqrt[13])) Cos[2 t] (Sqrt[9+58 ((Tan[t])^2)+((Tan[t])^4)])))/(2 (29+8 (Sqrt[13])) (Sqrt[11+4 Cos[2 t]-6 Cos[4 t]]))

Plot[s[t],{t,0,π/2},PlotStyle->Red]

4s[t]/.t->π/2-.001

ss=s[t]/.t->π/2-u

FullSimplify[(-I (104 (Sqrt[11+4 Cos[2 ((π/2)-u)]-6 Cos[4 ((π/2)-u)]])+32 (Sqrt[13]) (Sqrt[11+4 Cos[2 ((π/2)-u)]-6 Cos[4 ((π/2)-u)]])+60 (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)])+15 (Sqrt[13]) (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)])+15 (4+(Sqrt[13])) Cos[2 ((π/2)-u)] (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)])) EllipticF[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]+(29+8 (Sqrt[13])) (Cot[u]+16 Sin[2 ((π/2)-u)]-6 Sin[4 ((π/2)-u)])+I (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)]) ((220+61 (Sqrt[13])) EllipticE[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]-8 (4+(Sqrt[13])) EllipticPi[29-8 (Sqrt[13]),-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]) ((Sin[u])^2))/(2 (29+8 (Sqrt[13])) (Sqrt[11+4 Cos[2 ((π/2)-u)]-6 Cos[4 ((π/2)-u)]])),0<u<.1]

Limit[((29+8 (Sqrt[13])) (9 Cos[u]-5 Cos[3 u]-3 Cos[5 u]) Csc[u]-I (104 (Sqrt[11-4 Cos[2 u]-6 Cos[4 u]])+32 (Sqrt[13]) (Sqrt[11-4 Cos[2 u]-6 Cos[4 u]])+60 (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)])+15 (Sqrt[13]) (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)])-15 (4+(Sqrt[13])) Cos[2 u] (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)])) EllipticF[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]+I (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)]) ((220+61 (Sqrt[13])) EllipticE[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]-8 (4+(Sqrt[13])) EllipticPi[29-8 (Sqrt[13]),-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]) ((Sin[u])^2))/(2 (29+8 (Sqrt[13])) (Sqrt[11-4 Cos[2 u]-6 Cos[4 u]])),u->0,Direction->-1]

Limit[(9 Cos[u]-5 Cos[3 u]-3 Cos[5 u]),u->0]

4((29+8 (Sqrt[13])) Csc[u]-I (104 (Sqrt[11-4 Cos[2 u]-6 Cos[4 u]])+32 (Sqrt[13]) (Sqrt[11-4 Cos[2 u]-6 Cos[4 u]])+60 (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)])+15 (Sqrt[13]) (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)])-15 (4+(Sqrt[13])) Cos[2 u] (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)])) EllipticF[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]+I (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)]) ((220+61 (Sqrt[13])) EllipticE[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]-8 (4+(Sqrt[13])) EllipticPi[29-8 (Sqrt[13]),-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]) ((Sin[u])^2))/(2 (29+8 (Sqrt[13])) (Sqrt[11-4 Cos[2 u]-6 Cos[4 u]]))/.u->.0001

Limit[ (104 (Sqrt[11-4 Cos[2 u]-6 Cos[4 u]])+32 (Sqrt[13]) (Sqrt[11-4 Cos[2 u]-6 Cos[4 u]])+60 (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)])+15 (Sqrt[13]) (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)])-15 (4+(Sqrt[13])) Cos[2 u] (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)])) ,u->0]

4((29+8 (Sqrt[13])) Csc[u]-I (224+62 (Sqrt[13])) EllipticF[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]+I (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)]) ((220+61 (Sqrt[13])) EllipticE[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]-8 (4+(Sqrt[13])) EllipticPi[29-8 (Sqrt[13]),-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]) ((Sin[u])^2))/(2 (29+8 (Sqrt[13])) (Sqrt[11-4 Cos[2 u]-6 Cos[4 u]]))/.u->.00001

Limit[ (Sqrt[11-4 Cos[2 u]-6 Cos[4 u]]),u->0]

4((29+8 (Sqrt[13])) Csc[u]-I (224+62 (Sqrt[13])) EllipticF[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]+I (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)]) ((220+61 (Sqrt[13])) EllipticE[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]-8 (4+(Sqrt[13])) EllipticPi[29-8 (Sqrt[13]),-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]) ((Sin[u])^2))/(2 (29+8 (Sqrt[13])) )/.u->.00001

FullSimplify[4((29+8 (Sqrt[13])) Csc[u]-I (224+62 (Sqrt[13])) EllipticF[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]+I (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)]) ((220+61 (Sqrt[13])) EllipticE[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]-8 (4+(Sqrt[13])) EllipticPi[29-8 (Sqrt[13]),-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]) ((Sin[u])^2))/(2 (29+8 (Sqrt[13])) )]

expr=(1/(29+8 (Sqrt[13])))(2 ((29+8 (Sqrt[13])) Csc[u]-2 I (112+31 (Sqrt[13])) EllipticF[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]+I (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)]) ((220+61 (Sqrt[13])) EllipticE[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]-8 (4+(Sqrt[13])) EllipticPi[29-8 (Sqrt[13]),-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]) ((Sin[u])^2)));

Limit[expr,u->0,Direction->-1]

EllipticF[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]/. u->.001

Limit[EllipticF[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))],u->0]

FullSimplify[%30]

EllipticK[(9/(((29+8 (Sqrt[13])))^2))]-(1/3) (29+8 (Sqrt[13])) EllipticK[(1/9) (((29+8 (Sqrt[13])))^2)]//N//Chop

Limit[EllipticE[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))],u->0]

EllipticE[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]/.u->.001

Series[EllipticE[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))],{u,0,0}]

Normal[EllipticE[(((-1))^(Floor[((π+2 Arg[u]-Arg[(u^2) ((Cot[u])^2)])/(2 π))])) (((I (-4+(Sqrt[13])))/(2 (4-(Sqrt[13]))))+((O[u])^2)) (-2 I π Floor[((π+2 Arg[u]-Arg[(u^2) ((Cot[u])^2)])/(2 π))]-Floor[((Arg[-1+(u^2) ((Cot[u])^2)])/(2 π))] (Log[(4/(((-4+(Sqrt[13])))^2))]+Log[(1/4) (((-4+(Sqrt[13])))^2)])+((-Log[(4/(((-4+(Sqrt[13])))^2))]+2 Log[u])+((O[u])^2))),(9/(((29+8 (Sqrt[13])))^2))]]

EllipticE[(1/(2 (4-(Sqrt[13]))))(I  (-4+(Sqrt[13])) (-Log[(4/(((-4+(Sqrt[13])))^2))]+2 Log[u])),(9/(((29+8 (Sqrt[13])))^2))]/.u->.001

FullSimplify[EllipticE[(1/(2 (4-(Sqrt[13]))))(I  (-4+(Sqrt[13])) (-Log[(4/(((-4+(Sqrt[13])))^2))]+2 Log[u])),(9/(((29+8 (Sqrt[13])))^2))]]

Limit[%,u->0]

(1/(29+8 (Sqrt[13])))(2 ((29+8 (Sqrt[13])) Csc[u]-2 I (112+31 (Sqrt[13])) (EllipticK[(9/(((29+8 (Sqrt[13])))^2))]-(1/3) (29+8 (Sqrt[13])) EllipticK[(1/9) (((29+8 (Sqrt[13])))^2)])+I (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)]) ((220+61 (Sqrt[13])) EllipticE[-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]-8 (4+(Sqrt[13])) EllipticPi[29-8 (Sqrt[13]),-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]) ((Sin[u])^2)))/.u->.00001

EllipticPi[29-8 (Sqrt[13]),-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))]/.u->.0001

Series[EllipticPi[29-8 (Sqrt[13]),-I ArcCsch[(-4+(Sqrt[13])) Tan[u]],(9/(((29+8 (Sqrt[13])))^2))],{u,0,0}]

Limit[%,u->0]

Normal[EllipticPi[29-8 (Sqrt[13]),(((-1))^(Floor[((π+2 Arg[u]-Arg[(u^2) ((Cot[u])^2)])/(2 π))])) (((I (-4+(Sqrt[13])))/(2 (4-(Sqrt[13]))))+((O[u])^2)) (-2 I π Floor[((π+2 Arg[u]-Arg[(u^2) ((Cot[u])^2)])/(2 π))]-Floor[((Arg[-1+(u^2) ((Cot[u])^2)])/(2 π))] (Log[(4/(((-4+(Sqrt[13])))^2))]+Log[(1/4) (((-4+(Sqrt[13])))^2)])+((-Log[(4/(((-4+(Sqrt[13])))^2))]+2 Log[u])+((O[u])^2))),(9/(((29+8 (Sqrt[13])))^2))]]/.u->.001

Normal[EllipticPi[29-8 (Sqrt[13]),(((-1))^(Floor[((π+2 Arg[u]-Arg[(u^2) ((Cot[u])^2)])/(2 π))])) (((I (-4+(Sqrt[13])))/(2 (4-(Sqrt[13]))))+((O[u])^2)) (-2 I π Floor[((π+2 Arg[u]-Arg[(u^2) ((Cot[u])^2)])/(2 π))]-Floor[((Arg[-1+(u^2) ((Cot[u])^2)])/(2 π))] (Log[(4/(((-4+(Sqrt[13])))^2))]+Log[(1/4) (((-4+(Sqrt[13])))^2)])+((-Log[(4/(((-4+(Sqrt[13])))^2))]+2 Log[u])+((O[u])^2))),(9/(((29+8 (Sqrt[13])))^2))]]

EllipticPi[29-8 (Sqrt[13]),(1/(2 (4-(Sqrt[13]))))(I (((-1))^(Floor[((π+2 Arg[u]-Arg[(u^2) ((Cot[u])^2)])/(2 π))])) (-4+(Sqrt[13])) (-2 I π Floor[((π+2 Arg[u]-Arg[(u^2) ((Cot[u])^2)])/(2 π))]-Log[(4/(((-4+(Sqrt[13])))^2))]-Floor[((Arg[-1+(u^2) ((Cot[u])^2)])/(2 π))] (Log[(4/(((-4+(Sqrt[13])))^2))]+Log[(1/4) (((-4+(Sqrt[13])))^2)])+2 Log[u])),(9/(((29+8 (Sqrt[13])))^2))]/.u->.001

Limit[Floor[((π+2 Arg[u]-Arg[(u^2) ((Cot[u])^2)])/(2 π))],u->0]

EllipticPi[29-8 (Sqrt[13]),(1/(2 (4-(Sqrt[13]))))(I  (-4+(Sqrt[13])) (-Log[(4/(((-4+(Sqrt[13])))^2))]-Floor[((Arg[-1+(u^2) ((Cot[u])^2)])/(2 π))] (Log[(4/(((-4+(Sqrt[13])))^2))]+Log[(1/4) (((-4+(Sqrt[13])))^2)])+2 Log[u])),(9/(((29+8 (Sqrt[13])))^2))]/.u->.001

Limit[Floor[((Arg[-1+(u^2) ((Cot[u])^2)])/(2 π))],u->0]

EllipticPi[29-8 (Sqrt[13]),(1/(2 (4-(Sqrt[13]))))(I  (-4+(Sqrt[13])) (-Log[(4/(((-4+(Sqrt[13])))^2))]-0(Log[(4/(((-4+(Sqrt[13])))^2))]+Log[(1/4) (((-4+(Sqrt[13])))^2)])+2 Log[u])),(9/(((29+8 (Sqrt[13])))^2))]/.u->.001

EllipticPi[29-8 (Sqrt[13]),(1/(2 (4-(Sqrt[13]))))(I  (-4+(Sqrt[13])) (-Log[(4/(((-4+(Sqrt[13])))^2))]-0(Log[(4/(((-4+(Sqrt[13])))^2))]+Log[(1/4) (((-4+(Sqrt[13])))^2)])+2 Log[u])),(9/(((29+8 (Sqrt[13])))^2))]

FullSimplify[%]

Limit[%,u->0]

EllipticPi[29-8 (Sqrt[13]),(1/2) I (Log[(4/(((-4+(Sqrt[13])))^2))]-2 Log[u]),(9/(((29+8 (Sqrt[13])))^2))]/.u->.001

Series[EllipticPi[29-8 (Sqrt[13]),(1/2) I (Log[(4/(((-4+(Sqrt[13])))^2))]-2 Log[u]),(9/(((29+8 (Sqrt[13])))^2))],{u,0,4}]

Log[.001]

Limit[%,u->0]

Limit[EllipticPi[29-8 (Sqrt[13]),(1/2) I Log[(4/(((-4+(Sqrt[13])))^2))]-I x,(9/(((29+8 (Sqrt[13])))^2))],x->-∞]

EllipticPi[29-8 (Sqrt[13]),(1/2) I Log[(4/(((-4+(Sqrt[13])))^2))]+I Log[x],(9/(((29+8 (Sqrt[13])))^2))]/.x->9999999.

EllipticPi[29-8 (Sqrt[13]), I Log[x(Sqrt[(4/(((-4+(Sqrt[13])))^2))])],(9/(((29+8 (Sqrt[13])))^2))]/.x->9999999.

Limit[EllipticPi[29-8 (Sqrt[13]), I Log[x(Sqrt[(4/(((-4+(Sqrt[13])))^2))])],(9/(((29+8 (Sqrt[13])))^2))],x->∞]

Series[EllipticPi[29-8 (Sqrt[13]),I Log[((2 x)/(4-(Sqrt[13])))],(9/(((29+8 (Sqrt[13])))^2))],{x,∞,1}]

(1/(29+8 (Sqrt[13])))(2 ((29+8 (Sqrt[13])) Csc[u]-2 I (112+31 (Sqrt[13]))( EllipticK[(9/(((29+8 (Sqrt[13])))^2))]-(1/3) (29+8 (Sqrt[13])) EllipticK[(1/9) (((29+8 (Sqrt[13])))^2)])+I (Sqrt[9+58 ((Cot[u])^2)+((Cot[u])^4)]) ((220+61 (Sqrt[13])) EllipticE[(1/2) I (Log[(4/(((-4+(Sqrt[13])))^2))]-2 Log[u]),(9/(((29+8 (Sqrt[13])))^2))]-8 (4+(Sqrt[13])) EllipticPi[29-8 (Sqrt[13]),(1/2) I Log[(4/(((-4+(Sqrt[13])))^2))]-I Log[u],(9/(((29+8 (Sqrt[13])))^2))]) ((Sin[u])^2)))/.u->.0001

Area

Polar

(1/2)Integrate[(1+2Cos[2θ])^2,{θ,0,2π}]

Integrate[(1+2Cos[2θ])^2,{θ,0,π}]

Area[a(1+2Cos[2t]),{t,0,2π}]

Parametric

Area[curve,{t,0,2π}]

N[3π]

Inequality

<<Graphics`InequalityGraphics`

RegionPlot[(x^6)+(y^6)+3 (x^4) (-3+(y^2))+3 (x^2) (y^2) (2+(y^2))<(y^4),{x,-3,3},{y,-1.1,1.1},AspectRatio->Automatic,MaxRecursion->5]

NIntegrate[Boole[(x^6)+(y^6)+3 (x^4) (-3+(y^2))+3 (x^2) (y^2) (2+(y^2))<(y^4)],{x,-∞,∞},{y,-∞,∞}]

Integrate[Boole[(x^6)+(y^6)+3 (x^4) (-3+(y^2))+3 (x^2) (y^2) (2+(y^2))<(y^4)],{y,-∞,∞},{x,-∞,∞}]//Timing

N[%]

Integrate[Boole[(x^6)+(y^6)+3 (x^4) (-3+(y^2))+3 (x^2) (y^2) (2+(y^2))<(y^4)],{x,-∞,∞},{y,-∞,∞}]//Timing

AreaMomentOfInertia

Parametric

AreaInertiaTensor[curve,{t,0,2π}]//Timing

Polar

AreaInertiaTensor[a(1+2Cos[2t]),{t,0,2π}]

Inequality

AreaInertiaTensor[(x^6)+(y^6)+3 (x^4) (-3+(y^2))+3 (x^2) (y^2) (2+(y^2))<(y^4),{x,y}]//Timing

AreaInertiaTensor[(x^6)+(y^6)+3 (x^4) (-3+(y^2))+3 (x^2) (y^2) (2+(y^2))<(y^4),{y,x}]//Timing

Centroid

Numeric

NIntegrate[{x,y}Boole[(x^6)+(y^6)+3 (x^4) (-3+(y^2))+3 (x^2) (y^2) (2+(y^2))<(y^4)],{y,-1,3},{x,-3,3}]//Timing

NIntegrate[{x,y}Boole[(x^6)+(y^6)+3 (x^4) (-3+(y^2))+3 (x^2) (y^2) (2+(y^2))<(y^4)],{y,-1,3},{x,-3,3},MaxPoints->10^7,MaxRecursion->20]//Timing

Polar

Centroid[a(1+2Cos[2t]),{t,0,2π}]//Timing

Inequality

Integrate[{x,y}Boole[(x^6)+(y^6)+3 (x^4) (-3+(y^2))+3 (x^2) (y^2) (2+(y^2))<(y^4)],{y,-∞,∞},{x,-∞,∞}]//Timing

Integrate[{x,y}Boole[(x^6)+(y^6)+3 (x^4) (-3+(y^2))+3 (x^2) (y^2) (2+(y^2))<(y^4)],{x,-∞,∞},{y,-∞,∞}]//Timing

Curvature

Parametric

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

Plot[((3 (9+4 Cos[2 t]-2 Cos[4 t]))/(((11+4 Cos[2 t]-6 Cos[4 t]))^(3/2))),{t,0,2π}]

Polar

FullSimplify[Curvature[a(1+2 Cos[2 q]),q],a>0&&0<q<2π]

Implicit

SortBy[FullSimplify[Curvature[eqn,{x,y}],a>0&&(x|y)∈Reals],ByteCount]

ByteCount/@%

Inflection Points

InflectionPoints[eqn,{x,y}]//Union

Singular Points

SingularPoints[eqn,{x,y}]//Union

Slope

Slope[a(1+2 Cos[2 t]) {Cos[t],Sin[t]},t]//FullSimplify

Slope[a(1+2 Cos[2 q]),q]//FullSimplify

Slope[x^6+y^6+3 x^4 (-3+y^2)+3 x^2 y^2 (2+y^2)==y^4,{x,y}]//FullSimplify

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

ByteCount/@%

SpecialAffineCurvature

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

Tangential Angle

FullSimplify[TangentialAngle[curve,t],a>0]

Plot[t+ArcTan[((4 Sin[2 t])/(1+2 Cos[2 t]))],{t,0,2π}]

Vector Properties

parm=a(1+2 Cos[2 t]) {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

NormalVector

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

TangentVector

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