Epitrochoid

Author

Eric W. Weisstein
June 27, 2013

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

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

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

Curve

Epitrochoid[{a,b},h,t]//TraditionalForm

Figure

h<b

Epitrochoid[{16,2},1,{0,2Pi},PlotStyle->Red,ImageSize->200]

movie1=Show/@Most[Epitrochoid[{16,2},1,{0,2Pi,2Pi/30}]];

<<Utilities`GIF`;

Export[gifs<>"epitroc1.gif",movie1]//Timing

h>b

Epitrochoid[{16,2},5,{0,2Pi},PlotStyle->Red,ImageSize->200]

movie2=Show/@Most[Epitrochoid[{16,2},5,{0,2Pi,2Pi/30}]];

<<Utilities`GIF`;

Export[gifs<>"epitroc2.gif",movie2]//Timing

Properties

curve=Epitrochoid[{a,b},h,t]

ArcLengthFunction

(s=Assuming[a>0&&b>0&&h>0,ArcLength[curve,t]])//Timing

FullSimplify[s,a>0&&b>0&&h>0&&t∈Reals]

FullSimplify[Sqrt[#.#&[D[curve,t]]],a>0&&b>0&&h>0]

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

With[{a=1,b=2,h=.5},Plot[((2 (a+b) Abs[b-h] EllipticE[((a t)/(2 b)),-((4 b h)/(((b-h))^2))])/a),{t,0,50}]]

Curvature

FullSimplify[Curvature[curve,t],a>0&&b>0&&h>0]

Slope

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

SpecialAffineCurvature

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

TangentialAngle

FullSimplify[TangentialAngle[curve,t],a>0&&b>0&&h>0]

FullSimplify[%,a>0&&b>0&&h>0&&t∈Reals]

Limit[%,h->b]

Vector Properties

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