Epicycloid

Author

Eric W. Weisstein
October 3, 2018

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

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

©2018 Wolfram Research, Inc. except for portions noted otherwise

Parametric Equation

Hypocycloid[{a,b},t]

FullSimplify[Hypocycloid[{a,b},t]/.b->a/n]

xy=PlaneCurveData["Epicycloid","ParametricEquations"][a,b][t]

Function[{a,b},ParametricPlot[{(a+b) Cos[t]-b Cos[(((a+b) t)/b)],(a+b) Sin[t]-b Sin[(((a+b) t)/b)]},{t,0,2π LCM[a,b]/a},ColorFunction->(Hue[#3]&)]]@@@{{5,1},{5,2},{3,1},{6,2},{6,1}}

Array of n-Cusped Epicycloids

Show[GraphicsArray[Block[{$DisplayFunction=Identity},
Table[Epicycloid[{1,1/n},{0,2π},Prolog->Red,
Epilog->{Black,Circle[{0,0},1]}],{n,4}]]
]]

Denominator[a/b] turns around the wheel, Numerator[a/b] "cusps"

Function[{a,b,th},ParametricPlot[{(a+b) Cos[t]-b Cos[(((a+b) t)/b)],(a+b) Sin[t]-b Sin[(((a+b) t)/b)]},{t,0,th},PlotLabel->{Symbol["a"]==a,Symbol["b"]==b,Symbol["θ"]==th,Numerator[a/b],2π Denominator[a/b]}]]@@@{{1,2,4π},{1,3,6π},{2,1,2π},{3,2,4π},{2,3,6π},{(1/2),(1/3),4π},{(1/6),5,60π},{(π/7),2(π/9),28π},{5,(1/2),2π},{(5/3),(7/6),14π}}

Animation of n-Cusped Epicycloids

Show[GraphicsArray[Block[{$DisplayFunction=Identity},
Table[EpicycloidFrames[{1,1/n},Range[0,1,1/(n+3+Mod[n,2])]2Pi,Prolog->Red,
Epilog->{Black,Circle[{0,0},1]}],{n,4}]]
]]

Movie

<<Utilities`GIF`

movie=Show[GraphicsArray[#,GraphicsSpacing->-.1],ImageSize->500]&/@Transpose[Table[Epicycloid[{1,1/n},{0,2π-π/20,π/20},Prolog->Red,
Epilog->{Black,Circle[{0,0},1]}],{n,4}]];

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

Equation

Parametric

TrigExpand[{(a+b) Cos[t]-b Cos[(((a+b) t)/b)],(a+b) Sin[t]-b Sin[(((a+b) t)/b)]}]

Polar

Curve Properties

Epicycloid[{a,b},t]

curve={(a+b) Cos[t]-b Cos[(((a+b) t)/b)],(a+b) Sin[t]-b Sin[(((a+b) t)/b)]};

Arc Length

xy=PlaneCurveData["Epicycloid","ParametricEquations"][a,b][t]

FullSimplify[(Sqrt[Total[D[xy,t]^2]]),0<b<a]

FullSimplify[%/.b->a/n,n≥1&&a>0&&n∈Integers]

ArcLengthFunction

V4.2

ArcLength[Epicycloid[{a,b},t],t]

V5

FullSimplify[ArcLength[Epicycloid[{a,b},t],t],0<t<2π&&a>0&&b>0]

V6

FullSimplify[ArcLength[Epicycloid[{a,b},t],t],0<t<2π&&a>0&&b>0]

V8

Assuming[a>0&&b>0,ArcLength[Epicycloid[{a,b},t],t]]

...wrong...

With[{a=1,b=2},Plot[-((4 b (a+b) (-1+Abs[Sin[((a t)/(2 b))]] Cot[((a t)/(2 b))]))/a),{t,0,20}]]

Sqrt[#.#&[D[Epicycloid[{a,b},t],t]]]

Integrate[√(((((a+b) Cos[t]-(a+b) Cos[(((a+b) t)/b)]))^2)+(((-(a+b) Sin[t]+(a+b) Sin[(((a+b) t)/b)]))^2)),{t,0,t},Assumptions->a>0&&b>0]

FullSimplify[Sqrt[#.#&[D[Epicycloid[{a,b},t],t]]],a>0&&b>0]

Integrate[2 (a+b) (Sqrt[((Sin[((a t)/(2 b))])^2)]),{t,0,t},Assumptions->a>0&&b>0]

V9

Assuming[a>0&&b>0,ArcLength[Epicycloid[{a,b},t],t]]

Integrate[2 (a+b) (Sqrt[((Sin[((a t)/(2 b))])^2)]),{t,0,t},Assumptions->a>0&&b>0]

V12

$Version

For n == a/b and b == 1:

Integrate[((2 a (n+1) )/n)Abs[Sin[((n t)/2)]],{t,0,2π }]//Timing

Assuming[n∈Integers&&n>0,Integrate[((2 a (n+1) )/n)Abs[Sin[((n t)/2)]],{t,0,2π }]]//Timing

Simplify[-((4 a (1+n) (-1+Abs[Sin[n π]] Cot[n π]))/(n^2)),n∈Integers]

bug(358864): Refine[IntegerPart[LCM[a, b]/b], (a | b) ∈ Integers && b > 0] could return LCM[a, b]/b (etc.)

LCM[a, b]/a == b/GCD[a, b]

Do the integral assuming (a|b)∈Integers:

(res1=Assuming[a>0&&b>0&&(a|b)∈Integers,Integrate[2(a+b) Abs[Sin[((a t)/(2 b))]],{t,0,2π b/GCD[a,b]}]])//Timing

Assuming[a>0&&b>0&&(a|b)∈Integers,FullSimplify[res1]]===res1//Timing

(res2=Assuming[a>0&&b>0&&(a|b)∈Integers,Integrate[2(a+b)Abs[Sin[((a t)/(2 b))]],{t,0,2π ((LCM[a,b])/a)}]])//Timing

Assuming[a>0&&b>0&&(a|b)∈Integers,FullSimplify[res2]]===res2//Timing

Unforunately, this gives a result equal to 0/0:

res2/.{IntegerPart[x_]:>x,_FractionalPart->0}

Omit the integer condition in the integral:

(res3=Assuming[a>0&&b>0,Integrate[2(a+b)Abs[Sin[((a t)/(2 b))]],{t,0,2π ((LCM[a,b])/a)}]])//Timing

Assuming[a>0&&b>0&&(a|b)∈Integers,FullSimplify[res3]]//Timing

simp3=Assuming[a>0&&b>0&&(a|b)∈Integers,Simplify[res3]]

Simplify[simp3/.{IntegerPart[x_]:>x,_FractionalPart->0}]

...wrong...

Apparently correct:

((8 (a+b) b IntegerPart[((LCM[a,b])/b)])/a)-((2b (-1+Cos[π 0]))/a)

%/.IntegerPart[x_]:>x

Area

With[{b=a},Area[{(a+b) Cos[t]-b Cos[(((a+b) t)/b)],(a+b) Sin[t]-b Sin[(((a+b) t)/b)]},{t,0,2π}]]

With[{a=1,b=1/2},ParametricPlot[{(a+b) Cos[t]-b Cos[(((a+b) t)/b)],(a+b) Sin[t]-b Sin[(((a+b) t)/b)]},{t,0,2π},ImageSize->Small]]

With[{a=2b},Area[{(a+b) Cos[t]-b Cos[(((a+b) t)/b)],(a+b) Sin[t]-b Sin[(((a+b) t)/b)]},{t,0,2π}]]

With[{a=1,b=1/5},ParametricPlot[{(a+b) Cos[t]-b Cos[(((a+b) t)/b)],(a+b) Sin[t]-b Sin[(((a+b) t)/b)]},{t,0,2π},ImageSize->Small]]

With[{a=π,b=π/5},ParametricPlot[{(a+b) Cos[t]-b Cos[(((a+b) t)/b)],(a+b) Sin[t]-b Sin[(((a+b) t)/b)]},{t,0,2π},ImageSize->Small]]

With[{a=5b},Area[{(a+b) Cos[t]-b Cos[(((a+b) t)/b)],(a+b) Sin[t]-b Sin[(((a+b) t)/b)]},{t,0,2π}]]

FullSimplify[With[{b=a/n},Area[{(a+b) Cos[t]-b Cos[(((a+b) t)/b)],(a+b) Sin[t]-b Sin[(((a+b) t)/b)]},{t,0,2π}]],n∈Integers&&n>1]

Table[(((a^2) (1+n) (2+n) π)/(n^2)),{n,5}]

FullSimplify[With[{a=n b},Area[{(a+b) Cos[t]-b Cos[(((a+b) t)/b)],(a+b) Sin[t]-b Sin[(((a+b) t)/b)]},{t,0,2π}]],n∈Integers&&n>1]

Table[(b^2) (1+n) (2+n) π,{n,10}]

FullSimplify[b^2 (1+a/b) (2+a/b) Pi]

Curvature

FullSimplify[Curvature[Epicycloid[{a,b},t],t],0<t<2π&&a>0&&b>0]

V8

FullSimplify[Assuming[a>0&&b>0,Curvature[Epicycloid[{a,b},t],t]],a>0&&b>0]

Slope

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

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&b>0&&t>0]

Tangential Angle

v4.2

FullSimplify[TangentialAngle[Epicycloid[{a,b},t],t]]

v5.0

FullSimplify[TangentialAngle[Epicycloid[{a,b},t],t]]

v6

FullSimplify[TangentialAngle[Epicycloid[{a,b},t],t]]

Construction by Diameter Rotation

EpicycloidByDiameters[n_,pts_]:=Module[{i},
{
Table[Line[Transpose[Through[{Cos,Sin}[2i Pi/pts{1,n+1}+{0,Pi}]]]],{i,0,pts}],
ParametricPlot[Evaluate[Epicycloid[{n,1},t]/(n+2)],{t,0,2π},
DisplayFunction->Identity,
AspectRatio->Automatic,
PlotStyle->{Thickness[.02],Red},
Axes->None][[1]],
{Blue,Thickness[.02],Circle[{0,0},1]}
}
]

Show[GraphicsArray[Partition[
Graphics[#,AspectRatio->Automatic]&/@Table[EpicycloidByDiameters[i,20(i+1)],{i,6}],3]]]

Vector Properties

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