Curtate Cycloid

Author

Eric W. Weisstein
January 26, 2011

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

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

©2010 Wolfram Research, Inc. except for portions noted otherwise

Equation

eqns=Append[Thread[{x,y}==Cycloid[{a,b},t]],Sin[t]^2+Cos[t]^2==1]

Solve[eqns,{x,y},t]

Plots

V5

ParametricPlot[Cycloid[{1,.6},t],{t,-.01,4 Pi+.01},AspectRatio->Automatic,
PlotRange->{Automatic,{0,1.6}},Ticks->None,PlotStyle->Red]

TrochoidFrames[{1,.6},4π Range[0,1,1/5]]

V5 Animation

<<Utilities`GIF`;

Export[gifs<>"cycloidc.gif",Trochoid[{1,.6},{0,4Pi,4Pi/20},ImageSize->400]]//Timing

V6

GraphicsColumn[{
ParametricPlot[Cycloid[{1,.6},t],{t,-.01,4 Pi+.01},AspectRatio->Automatic,
PlotRange->{Automatic,{0,1.6}},Ticks->None,PlotStyle->Red],
TrochoidFrames[{1,.6},4π Range[0,1,1/5],PlotStyle->Red]
}]

Properties

CurtateCycloid[a_,b_]:={a t-b Sin[t],a-b Cos[t]}

Arc Length

v4.2

ArcLength[CurtateCycloid[a,b],t,Assumptions->0<b<a&&t>0]

v5.0.1

ArcLength[CurtateCycloid[a,b],t,Assumptions->0<b<a&&t>0]

Simplify[Limit[%,b->a],t>0&&a>0]

v5.1

ArcLength[CurtateCycloid[a,b],t,Assumptions->0<b<a&&t>0]

ArcLength[CurtateCycloid[a,b],t,GenerateConditions->False]

FullSimplify[%,0<b<a&&t>0]

v6

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

ArcLength[CurtateCycloid[a,b],t,GenerateConditions->False]//Timing

FullSimplify[Last[%],0<b<a&&t>0]

FullSimplify[Sqrt[-(1/(((a-b))^2))],0<b<a]

Check

ArcLength[CurtateCycloid[a,b],t,blah->1]//Timing

With[{t0=1.,a=.5,b=.3},
{
NIntegrate[(Sqrt[(((a-b Cos[t]))^2)+(b^2) ((Sin[t])^2)]),{t,0,t0}],
2 (a-b) EllipticE[(t0/2),-((4 a b)/(((a-b))^2))]
}
]//Chop

Curvature

Curvature[CurtateCycloid[a,b],t]//FullSimplify

Tangential Angle

v4.2

TangentialAngle[CurtateCycloid[a,b],t,Assumptions->0<b<a&&t>0]

v5.0.1

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

FullSimplify[Last[%],Assumptions->0<b<a&&t>0]

FullSimplify[%,0<t<Pi/2]

v5.1

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

FullSimplify[Last[%3],0<t<Pi/2]

TangentialAngle[CurtateCycloid[a,b],t,Assumptions->0<b<a&&0<t<Pi/2]//Timing