Ranunculoid

Author

Eric W. Weisstein
June 28, 2013

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

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Figure

curve=a Epicycloid[{5,1},t]

Show[Graphics[{
ParametricPlot[{6 Cos[t]-Cos[6 t],6 Sin[t]-Sin[6 t]},{t,0,2π},
PlotStyle->Red,DisplayFunction->Identity][[1]],
Circle[{0,0},5]
}],AspectRatio->Automatic,ImageSize->200,Ticks->None]

Show[EpicycloidFrames[{5,1},Range[0,2Pi-.01,2Pi/7]],ImageSize->200]

Animation

movie=Show/@Epicycloid[{5,1},{0,2Pi-.001,2Pi/40.}]

<<Utilities`GIF`;

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

Equations

Polar

curve={a (6 Cos[t]-Cos[6 t]),a (6 Sin[t]-Sin[6 t])};

Solve[Tan[θ]==1/Divide@@curve,t]//FullSimplify

Parametric

a Epicycloid[{5,1},t]

FullSimplify[Epicycloid[{5,1},t],0<t<2π]//Column//TraditionalForm

Implicit

Factor/@GroebnerBasis[Append[Thread[{x,y}== a TrigExpand[ Epicycloid[{5,1},t]]],Cos[t]^2+Sin[t]^2==1]/.{Cos[t]->cos,Sin[t]->sin},{x,y},{sin,cos}]

FullSimplify[%]

eqn=-52521875 (a^12)-1286250 (a^10) ((x^2)+(y^2))-32025 (a^8) ((((x^2)+(y^2)))^2)-812 (a^6) ((((x^2)+(y^2)))^3)-21 (a^4) ((((x^2)+(y^2)))^4)-42 (a^2) ((((x^2)+(y^2)))^5)+((((x^2)+(y^2)))^6)+93312 (a^7) ((x^5)-10 (x^3) (y^2)+5 x (y^4));

Properties

curve=a{(6 Cos[t]-Cos[6 t]),(6 Sin[t]-Sin[6 t])};

Arc Length

ds=Sqrt[#.#&[D[curve,t]]]

Simplify[ds,a>0&&0<t<2π]

V6 .0

ArcLength[curve,{t,0,2π}]

Integrate[ds,{t,0,2π}]

Integrate[Simplify[ds],{t,0,2π}]

V7

curve

Sqrt[Total[D[#,t]^2&/@curve]]

expr=(Sqrt[(((6 Cos[t]-6 Cos[6 t]))^2)+(((-6 Sin[t]+6 Sin[6 t]))^2)]);

Integrate[expr,{t,0,2π}]

ArcLength[curve,{t,0,2π}]

Integrate[ds,{t,0,2π}]

Arc Length Function

V6.0

ArcLength[curve,{t,0,t}]

ArcLength[curve,t]

Simplify[%,a>0&&t>0]

...wrong...

Plot[(24/5) (1- Abs[Sin[((5 t)/2)]] Cot[((5 t)/2)]),{t,0,2π}]

h[t_]:=(24/5) (1- Abs[Sin[((5 t)/2)]] Cot[((5 t)/2)])+(48/5)Floor[t/(2π/5)]

FullSimplify[h[t],0<t<2π]

Limit[h[t],t->2π]

FullSimplify[h[t],0<t<2π/5]

PiecewiseExpand[h[t],0<t<2π]

Plot[Sign[Sin[((5 t)/2)]],{t,0,2π}]

Plot[(24/5) (1- Sign[Sin[((5 t)/2)]] Cos[((5 t)/2)])+(48/5)Floor[t/(2π/5)],{t,0,2π}]

Plot[h[t],{t,0,2π}]

V7

s=ArcLength[curve,{t,0,t}]

Plot[s/.a->1,{t,0,2π}]

...wrong...

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

...wrong...

ds=Sqrt[#.#&[D[curve,t]]]

ds/.a->1

Integrate[(Sqrt[(((6 Cos[t]-6 Cos[6 t]))^2)+(((-6 Sin[t]+6 Sin[6 t]))^2)]),{t,0,t}]

Plot[%,{t,0,2π}]

h[t_]:=12  ((2/5)+(Piecewise[{{(2/5) Cos[((5 t)/2)], Sin[((5 t)/2)]<0}, {-(2/5) Cos[((5 t)/2)], True}}]))+(48/5)Floor[t/(2π/5)]

FullSimplify[h[t],t>0]

h[2π]

Plot[h[t],{t,0,2π}]

Simplify[Sqrt[#.#&[D[{6 Cos[t]-Cos[6 t],6 Sin[t]-Sin[6 t]},t]]]]

Plot[12 Abs[Sin[((5 t)/2)]],{t,0,2π}]

NIntegrate[12 Abs[Sin[((5 t)/2)]],{t,0,2π}]

Integrate[12 Abs[Sin[((5 t)/2)]],{t,0,2π}]

g[x_?NumericQ]:=NIntegrate[12 Abs[Sin[((5 t)/2)]],{t,0,x}]

g[2π]

Plot[g[t],{t,0,2π},PlotPoints->10,MaxRecursion->2]

Table[-(24/5) (-1+Abs[Sin[((5 x)/2)]] Cot[((5 x)/2)])-g[x],{x,.01,2π,.2}]//Chop

V8

Integrate[Sqrt[(6 Cos[t]-6 Cos[6 t])^2+(-6 Sin[t]+6 Sin[6 t])^2],{t,0,t},Assumptions->t>0]

$Version

Plot[-(12/5) (-2+(Sqrt[((Csc[((5 t)/2)])^2)]) Sin[5 t]),{t,-10,10}]

s=ArcLength[curve,{t,0,t}]

FullSimplify[s,a>0&&t>0]

Plot[(48/5) ((Sin[((5 t)/4)])^2),{t,0,2π/5}]

Area

Parametric

Area[curve,{t,0,2π}]

Inequality

Area[curve,{t,0,2π}]

Area[{a (6 Cos[t]-Cos[6 t]),a (6 Sin[t]-Sin[6 t])},{t,0,2π}]

Area[-52521875 (a^12)-1286250 (a^10) ((x^2)+(y^2))-32025 (a^8) ((((x^2)+(y^2)))^2)-812 (a^6) ((((x^2)+(y^2)))^3)-21 (a^4) ((((x^2)+(y^2)))^4)-42 (a^2) ((((x^2)+(y^2)))^5)+((((x^2)+(y^2)))^6)+93312 (a^7) ((x^5)-10 (x^3) (y^2)+5 x (y^4))<0,{x,y}]

N[%4/.a->1]//Chop

42.π

AreaInertiaTensor

Parametric

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

Inequality

AreaInertiaTensor[-52521875 (a^12)-1286250 (a^10) ((x^2)+(y^2))-32025 (a^8) ((((x^2)+(y^2)))^2)-812 (a^6) ((((x^2)+(y^2)))^3)-21 (a^4) ((((x^2)+(y^2)))^4)-42 (a^2) ((((x^2)+(y^2)))^5)+((((x^2)+(y^2)))^6)+93312 (a^7) ((x^5)-10 (x^3) (y^2)+5 x (y^4))<0,{x,y}]

Centroid

Parametric

Centroid[curve,{t,0,2π}]

Inequality

Assuming[a>0,Centroid[-52521875 (a^12)-1286250 (a^10) ((x^2)+(y^2))-32025 (a^8) ((((x^2)+(y^2)))^2)-812 (a^6) ((((x^2)+(y^2)))^3)-21 (a^4) ((((x^2)+(y^2)))^4)-42 (a^2) ((((x^2)+(y^2)))^5)+((((x^2)+(y^2)))^6)+93312 (a^7) ((x^5)-10 (x^3) (y^2)+5 x (y^4))<0,{x,y}]]

N[%10/.a->1]//Chop

Simplify[%10/.a->1]

Curvature

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

SortBy[Simplify[Curvature[eqn,{x,y}],a>0],ByteCount]

ByteCount/@%

SortBy[FullSimplify[%4,a>0&&(x|y)∈Reals],ByteCount]

ByteCount/@%

SingularPoints

all=Union[FullSimplify[SingularPoints[eqn,{x,y}],a>0]];//Timing

pts=Block[{a=1},Select[all,Im[#]==0.{1,1}&]]

eqn

Block[{a=1},ContourPlot[eqn==0,{x,-8,8},{y,-8,8},Epilog->{PointSize[.02],Point[pts]}]]

Slope

Slope[a{(6 Cos[t]-Cos[6 t]),(6 Sin[t]-Sin[6 t])},t]//FullSimplify

Slope[-52521875 a^12-1286250 a^10 (x^2+y^2)-32025 a^8 (x^2+y^2)^2-812 a^6 (x^2+y^2)^3-21 a^4 (x^2+y^2)^4-42 a^2 (x^2+y^2)^5+(x^2+y^2)^6+93312 a^7 (x^5-10 x^3 y^2+5 x y^4)==0,{x,y}]//FullSimplify

SpecialAffineCurvature

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

Tangential Angle

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

Vector Properties

parm=a{(6 Cos[t]-Cos[6 t]),(6 Sin[t]-Sin[6 t])};
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]]/.xform

NormalVector

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

TangentVector

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