Nephroid

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Code

NephroidEnvelope::usage="NephroidEnvelope[n] gives a Graphics object constructing the nephroid using n circles."

NephroidEnvelope[n_]:=Module[{x},Graphics[{Circle[{0,0},1],Line[{{0,-1},{0,1}}],Table[{Circle[{Sqrt[1-x^2],x},Sqrt[1-x^2]],Circle[{-Sqrt[1-x^2],x},Sqrt[1-x^2]]},{x,-1,1,2/(n-1)}]}]]

Equation

Cartesian

eqn=FullSimplify[Eliminate[{x==a(3Cos[t]-Cos[3t]),y==4a Sin[t]^3,Sin[t]^2+Cos[t]^2==1}/.c:Cos[3t]:>TrigExpand[c],{Sin[t],Cos[t]}]]

FullSimplify[ExpandAll[Subtract@@eqn]]

FullSimplify[Subtract@@{-Subtract@@eqn,(x^2+y^2-4a^2)^3-108a^4y^2}]

eqn=(x^2+y^2-4a^2)^3-108a^4y^2;

Polar

((3Sin[ϕ]-Sin[3ϕ])/(3Cos[ϕ]-Cos[3ϕ]))/.trig:(_Cos|_Sin):>TrigExpand[trig]

Sin[t]/Cos[t]

FullSimplify[Subtract@@FullSimplify[Eliminate[{r^2==(1/2)(a^2)(5+3Cos[2ϕ]),sin[θ]/cos[θ]==((3Sin[ϕ]-Sin[3ϕ])/(3Cos[ϕ]-Cos[3ϕ])),
Sin[ϕ]^2+Cos[ϕ]^2==1}/.trig:(_Cos|_Sin):>TrigExpand[trig],{Cos[ϕ],Sin[ϕ]}]/.{sin:>Sin,cos:>Cos},Cos[θ]≠0]]

FullSimplify[Subtract@@{((((r/(2a))))^(2/3)),((Sin[θ/2])^(2/3))+((Cos[θ/2])^(2/3))}]

Plot

Cartesian Implicit

<<Graphics`ImplicitPlot`

With[{a=1},ImplicitPlot[
(x^2+y^2-4a^2)^3==108a^4y^2,
{x,-3,3},PlotStyle->Red,Ticks->None,AspectRatio->Automatic]
]

Polar

Show[Block[{$DisplayFunction=Identity},PolarPlot[2#(((((Sin[θ/2])^(2/3))+((Cos[θ/2])^(2/3))))^(3/2)),{θ,0,π},PlotStyle->Red,Ticks->None]&/@{-1,1}
]]

FullSimplify[2((((((((Sin[θ/2])^2)))^(1/3))+(((((Cos[θ/2])^2)))^(1/3))))^(3/2)),0<θ<2π]

PolarPlot[(Sqrt[2]) ((((((1-Cos[θ]))^(1/3))+(((1+Cos[θ]))^(1/3))))^(3/2)),{θ,0,# π}]&/@{1,2}

Parametric

ParametricPlot[Evaluate[{Epicycloid[{1,1/2},t],{Cos[t],Sin[t]}}],{t,0,2π},PlotStyle->{Red,{}},PlotPoints->50,
AspectRatio->Automatic,Ticks->None]

Flat Visor Curve

Source

http://arxiv.org/pdf/1206.1312.pdf

Plot

ParametricPlot[{s(3-2s^2),2(1-s^2)^(3/2)},{s,-1,1}]

Equations

Factor/@GroebnerBasis[{x-s(3-2s^2),y^2-4(1-s^2)^3},{x,y},s]

Expand[4((((x^2+y^2-1))^3)-27/4(y^2))]

Animation

Show[EpicycloidFrames[{2,1},Table[x,{x,0,2Pi-.01,2Pi/5}]],
ImageSize->200]

Show/@(movie=Epicycloid[{2,1},{0,2Pi-.001,2Pi/40.}])

<<GIF`;

Export[gifs<>"nephroid.gif",movie]//Timing

Envelope

Show[NephroidEnvelope[20],AspectRatio->Automatic]

Properties

FullSimplify[Epicycloid[{2a,a},t]]

nephroid={-a (-3 Cos[t]+Cos[3 t]),4 a ((Sin[t])^3)};

ArcLength

v5

FullSimplify[ArcLength[nephroid,t,Assumptions->a>0]]

Plot[-6  (-1+Cot[t] (Sqrt[((Sin[t])^2)])),{t,0,2π}]

Simplify[ArcLength[nephroid,{t,0,2π}],a>0]

v6

ArcLength[nephroid,t]

Simplify[%,a>0&&0<t<π]

Simplify[ArcLength[nephroid,{t,0,2π}],a>0]

V6.1

ArcLength[nephroid,{t,0,2π}]

...wrong...

Area

Area[nephroid,{t,0,2π}]

Area[2((((((((Sin[θ/2])^2)))^(1/3))+(((((Cos[θ/2])^2)))^(1/3))))^(3/2)),{θ,0,2π}]//Timing

Area[(Sqrt[2]) ((((((1-Cos[θ]))^(1/3))+(((1+Cos[θ]))^(1/3))))^(3/2)),{θ,0,2π}]//Timing

Area[(x^2+y^2-4a^2)^3-108a^4y^2<0,{x,y}]//Timing

Area[(x^2+y^2-4a^2)^3-108a^4y^2<0,{y,x}]//Timing

<<Graphics`InequalityGraphics`

InequalityPlot[(x^2+y^2-4)^3<108y^2,{x},{y}]

<<Calculus`Integration`

Integrate[Boole[(x^2+y^2-4)^3<108y^2],{x},{y}]

Integrate[Boole[(x^2+y^2-4)^3<108y^2],{y},{x}]

N[%]

N[12π]

AreaInertiaTensor

AreaInertiaTensor[(Sqrt[2]) ((((((1-Cos[θ]))^(1/3))+(((1+Cos[θ]))^(1/3))))^(3/2)),{θ,0,2π}]//Timing

Centroid

Centroid[(Sqrt[2]) ((((((1-Cos[θ]))^(1/3))+(((1+Cos[θ]))^(1/3))))^(3/2)),{θ,0,2π}]//Timing

Curvature

Parametric

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

FullSimplify[%,0<t<π]

Polar

FullSimplify[Curvature[Sqrt[2] a ((1-Cos[q])^(1/3)+(1+Cos[q])^(1/3))^(3/2),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

InflectionPoints[nephroid,t]//Union

Singular Points

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

Slope

Slope[a{3Cos[t]-Cos[3t],3Sin[t]-Sin[3t]},t]//FullSimplify

FullSimplify[Slope[Sqrt[2] a ((1-Cos[q])^(1/3)+(1+Cos[q])^(1/3))^(3/2),q],0<q<2π]

Slope[(x^2+y^2-4a^2)^3==108a^4 y^2,{x,y}]//FullSimplify

SpecialAffineCurvature

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

Tangential Angle

v5

TangentialAngle[nephroid,t]

v6

TangentialAngle[nephroid,t]

Vector Properties

parm=a{3Cos[t]-Cos[3t],3Sin[t]-Sin[3t]};
assum=a>0&&0<t<2π;
xform={Abs[x_]^(n_/;Mod[n,2]==0):>x^n,Sqrt[x_^(n_/;Mod[n,2]==0)]:>Abs[x]^(n/2)};

VectorLength

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

NormalVector

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

TangentVector

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