Trochoid

Author

Eric W. Weisstein
June 28, 2013

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

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

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

Equations

eqn=Trochoid[{a,b},t]

Eliminate[Thread[{x,y}==eqn],t]

Try2

eqn/.{x->a ArcCos[1-y/a]-Sqrt[2 a y-y^2]}

Solve[a t-b Sin[t]==x,t]

Try1

Solve[y==a-b Cos[t],t]

FullSimplify[x==a t-b Sin[t]/.t->ArcCos[((a-y)/b)],b>0]

FullSimplify[x+ (Sqrt[(b^2)-(((a-y))^2)])-a ArcCos[((a-y)/b)]/.Thread[{x,y}->eqn],b>0&&t>0]

FullSimplify[x+ (Sqrt[(b^2)-(((a-y))^2)])-a ArcCos[((a-y)/b)]/.Thread[{x,y}->eqn],b>0&&0<t<π]

Polar

Plot

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},List/@(Graphics[#,AspectRatio->Automatic,PlotRange->{{-1.1,4π+1.1},{-.5,2.5}}]&/@TrochoidFrames[{1,#},Range[0,4π,π]]&/@{.6,1,1.4})]]
]

Properties

curve={a t-b Sin[t],a-b Cos[t]};

Arc Length Function

ArcLength[eqn,t,Assumptions->a>0&&b>0&&t>0]//Timing

s[a_,b_]:=2 Abs[a-b] EllipticE[(t/2),-((4 a b)/(((a-b))^2))]

With[{a=1,b=.7},Plot[s[a,b],{t,0,6π}]]

Curvature

FullSimplify[Curvature[eqn,t],a>0&&b>0]

Slope

Slope

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

SpecialAffineCurvature

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

Tangential Angle

TangentialAngle[eqn,t,Assumptions->a>0&&b>0&&t>0]//Timing

With[{a=1,b=#},Plot[-(t/2)+((π Abs[a-b])/(2 (a-b)))-ArcTan[(((a-b) Cot[(t/2)])/(a+b))]+Floor[t/(2π)]π,{t,0,6π}]]&/@{.2,.5,.7,.99}

Limit[expr,t->2π,Direction->1,Assumptions->a>0&&b>0]

Limit[expr,t->2π,Direction->-1,Assumptions->a>0&&b>0]

Vector Properties

parm={a t-b Sin[t],a-b Cos[t]};
assum=a>0&&b>0&&t∈Reals;
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