Freeth's Nephroid

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Plot

Polar

PolarPlot[1+2 Sin[t/2],{t,-2π,2π},Ticks->None,PlotStyle->Red,ImageSize->Small]

Plot[1+2 Sin[t/2],{t,-2π,2π}]

Show[{
PolarPlot[1+2 Sin[t/2],{t,0,π},Ticks->None,PlotStyle->Red],
PolarPlot[1+2 Sin[t/2],{t,π,2π},Ticks->None,PlotStyle->Orange],
PolarPlot[1+2 Sin[t/2],{t,-π,0},Ticks->None,PlotStyle->Blue],
PolarPlot[1+2 Sin[t/2],{t,-2π,-π},Ticks->None,PlotStyle->Green]
},ImageSize->Small,PlotRange->All]

Parametric

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

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

Implicit

(a^4) (-3 (x^2)+(y^2))+8 (a^3) x ((x^2)+(y^2))-6 (a^2) ((((x^2)+(y^2)))^2)+((((x^2)+(y^2)))^3)==0/.a->1

ContourPlot[-3 (x^2)+(y^2)+8 x ((x^2)+(y^2))-6 ((((x^2)+(y^2)))^2)+((((x^2)+(y^2)))^3)==0,{x,-3,1.5},{y,-2.5,2.5},AspectRatio->Automatic,ImageSize->Small]

Equations

Cartesian

Factor/@GroebnerBasis[Append[Thread[{x,y}==a(1+2 Sin[t/2]){Cos[t],Sin[t]}]//TrigExpand,Cos[t/2]^2+Sin[t/2]^2==1],{x,y},{Sin[t/2],Cos[t/2]},MonomialOrder->EliminationOrder]

FullSimplify[%]

eqn=(a^4) (-3 (x^2)+(y^2))+8 (a^3) x ((x^2)+(y^2))-6 (a^2) ((((x^2)+(y^2)))^2)+((((x^2)+(y^2)))^3);

Polar

PolarPlot[1+2 Sin[t/2],{t,0,4π}]

Properties

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

ArcLength

freeth/.a->1

V4.2

(Sqrt[2])Integrate[(Sqrt[7-3 Cos[t]+8 Sin[(t/2)]]),{t,-π,0}]

(Sqrt[2])Integrate[(Sqrt[7-3 Cos[t]+8 Sin[(t/2)]]),{t,0,π}]

V5.2

2Integrate[(Sqrt[2+4 Sin[t]+3 ((Sin[t])^2)]),{t,0,2π}]

i=2Integrate[(Sqrt[2+4 Sin[t]+3 ((Sin[t])^2)]),t]

i/.t->2π-0.000001

i/.t->0.000001

V6.1

ds=Sqrt[#.#&[D[freeth/.a->1,t]]]

FullSimplify[ds]

NIntegrate[((Sqrt[7-3 Cos[t]+8 Sin[(t/2)]])/(Sqrt[2])),{t,-2π,2π}]

(Sqrt[2])NIntegrate[(Sqrt[7-3 Cos[2t]+8 Sin[t]]),{t,0,2π}]

(Sqrt[2])NIntegrate[(Sqrt[7-3 (1-2Sin[t]^2)+8 Sin[t]]),{t,0,2π}]

FullSimplify[7-3 (1-2Sin[t]^2)+8 Sin[t]]

2NIntegrate[(Sqrt[2+4 Sin[t]+3 ((Sin[t])^2)]),{t,0,2π}]

2Integrate[(Sqrt[2+4 Sin[t]+3 ((Sin[t])^2)]),{t,0,2π}]

External

FullSimplify[2(1/3*3^(1/2)*(2/5*3^(1/2)*2^(1/2)+2*ArcTan@(1/2*3^(1/2)*2^(1/2)))-1/3*3^(1/2)*(-2/5*3^(1/2)*2^(1/2)-2*ArcTan@(1/2*3^(1/2)*2^(1/2)))+1/3*3^(1/2)*(1/2*3^(1/2)+2/3*Pi-2/5*3^(1/2)*2^(1/2)-2*ArcTan@(1/2*3^(1/2)*2^(1/2)))-4*3^(1/2)*ellipticK@(1/3*6^(1/2))+16/3*3^(1/2)*ellipticPi@@{-1/3,1/3*6^(1/2)}+4*3^(1/2)*ellipticE@(1/3*6^(1/2))+1/3*3^(1/2)*(-1/2*3^(1/2)-2/3*Pi)-1/3*3^(1/2)*(-1/2*3^(1/2)-2/3*Pi+2/5*3^(1/2)*2^(1/2)+2*ArcTan@(1/2*3^(1/2)*2^(1/2)))-1/3*3^(1/2)*(1/2*3^(1/2)+2/3*Pi)/.{ellipticK[k_]:>EllipticK[k^2],
ellipticE[k_]:>EllipticE[k^2],ellipticPi[x_,k_]:>EllipticPi[x,k^2]})]

N[%]

Arc Length Function

v5

ArcLength[freeth,t]

ArcLength[freeth,{t,-2π,2π}]//Timing

ArcLength[freeth,{t,-2π,2π}]//N

V6.0

ArcLength[freeth,t]

...crash...

ArcLength[freeth,{t,-2π,2π}]//Timing

...crash...

V6.1

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

Area

Area[freeth,{t,-2π,2π}]

AreaInertiaTensor

AreaInertiaTensor[freeth,{t,-2π,2π}]//Timing

Centroid

Centroid[freeth,{t,-2π,2π}]

Curvature

Parametric

FullSimplify[Curvature[freeth,t],a>0&&-2π<t<2π]

Polar

FullSimplify[Curvature[a(1+2 Sin[q/2]),q],a>0&&-2π<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 Sin[t/2]) {Cos[t],Sin[t]},t]//FullSimplify

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

Slope[a^4 (-3 x^2+y^2)+8 a^3 x (x^2+y^2)-6 a^2 (x^2+y^2)^2+(x^2+y^2)^3==0,{x,y}]//FullSimplify

SpecialAffineCurvature

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

Tangential Angle

FullSimplify[TangentialAngle[freeth,t],a>0&&-2π<t<2π]

Plot[-(π/4)+t+ArcTan[Sec[(t/2)]+2 Tan[(t/2)]],{t,-2π,2π}]

dt=Curvature[freeth,t] Sqrt[#.#&[D[freeth,t]]]/.a->1

Integrate[dt/.a->1,{t,0,t}]

Plot[

FullSimplify[dt]

Integrate[(3/8) (3+((3+Cos[t])/(7-3 Cos[t]+8 Sin[(t/2)]))),{t,0,t},Assumptions->-2π<t<2π]

Integrate[(3/8) (3+((3+Cos[t])/(7-3 Cos[t]+8 Sin[(t/2)]))),{t,0,t}]

FullSimplify[%,-2π<t<2π]

Plot[-(π/2)+t+I ArcTanh[((3-6 I (Sqrt[2])+(-2-5 I (Sqrt[2])) Tan[(t/4)])/(-7 I+4 (Sqrt[2])))]-I ArcTanh[((3+6 I (Sqrt[2])+(-2+5 I (Sqrt[2])) Tan[(t/4)])/(7 I+4 (Sqrt[2])))],{t,-2π,2π}]

int1=-(π/4)+t+ArcTan[Sec[(t/2)]+2 Tan[(t/2)]];
int2=(1/9) (9 t-(Sqrt[-7+4 I (Sqrt[2])]) ArcTan[(3/(Sqrt[-7-4 I (Sqrt[2])]))]+2 I (Sqrt[-14+8 I (Sqrt[2])]) ArcTan[(3/(Sqrt[-7-4 I (Sqrt[2])]))]-(Sqrt[-7-4 I (Sqrt[2])]) ArcTan[(3/(Sqrt[-7+4 I (Sqrt[2])]))]-2 I (Sqrt[-14-8 I (Sqrt[2])]) ArcTan[(3/(Sqrt[-7+4 I (Sqrt[2])]))]+(Sqrt[-7+4 I (Sqrt[2])]) (I+2 (Sqrt[2])) ArcTanh[(((Sqrt[-7-4 I (Sqrt[2])]) Sec[(t/4)] (3 Cos[(t/4)]+(2-I (Sqrt[2])) Sin[(t/4)]))/(-7 I+4 (Sqrt[2])))]+(Sqrt[-7-4 I (Sqrt[2])]) (-I+2 (Sqrt[2])) ArcTanh[(((Sqrt[-7+4 I (Sqrt[2])]) Sec[(t/4)] (3 Cos[(t/4)]+(2+I (Sqrt[2])) Sin[(t/4)]))/(7 I+4 (Sqrt[2])))]);

Plot[int1-int2,{t,-2π,2π}]

-(π/4)+t+ArcTan[Sec[(t/2)]+2 Tan[(t/2)]]

Plot[Floor[t/(2π)-1/2]π,{t,-2π,2π}]

Plot[-Floor[t/(2π)+1/2]π,{t,-2π,2π}]

int3=-(π/4)+t+ArcTan[Sec[(t/2)]+2 Tan[(t/2)]]+Floor[t/(2π)+1/2]π;

FullSimplify[int3]

Plot[int3,{t,-2π,2π}]

Plot[int3-int3,{t,-2π,2π}]

Vector Properties

parm=a (1+2 Sin[t/2]) {Cos[t],Sin[t]};
assum=a>0&&-2π<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]]