Atzema Spiral

Author

Eric W. Weisstein
April 8, 2021

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

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

©2021 Wolfram Research, Inc. except for portions noted otherwise

Parametric equations

Plot[{Sinc[t]-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},{t,-5,5},PlotLegends->"Expressions"]

Maximize[{-((Cos[t])/t)-2Sin[t]+t Cos[t],0<t<2},t]

ParametricPlot[{Sinc[t]-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},{t,.4,2},ColorFunction->Function[{x,y,t},If[t<1,Red,Blue]],ColorFunctionScaling->False]

ParametricPlot[{Sinc[t]-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},{t,1,30},PlotRange->20]

Catacaustic for a point at the origin is a circle

General radiant point

({cx,cy}=FullSimplify[Catacaustic[{((Sin[t])/t)-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},{x,y},t],t>1])//Timing

GraphicsRow[Table[CatacausticPlot[{Sinc[t]-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},{x,0},{t,1.01,7,1}],{x,0,2,.5}]]

Radiant point at origin

FullSimplify[Catacaustic[{((Sin[t])/t)-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},{0,0},t],t>1]

Manipulate[CatacausticPlot[{Sinc[t]-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},{0,0},{t,1.01,tmax,.1}],{tmax,1.02,15}]

CatacausticPlot[{Sinc[t]-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},{0,0},{t,1.01,2Pi+1,.1},PlotRange->All]

GraphicsGrid[Partition[Table[CatacausticPlot[{Sinc[t]-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},{0,0},{t,1.01,tmax,.1},PlotRange->5],{tmax,2,9.,1}],UpTo[4]]]

Properties

ArcLengthFunction

FullSimplify[ArcLength[{Sinc[u]-2Cos[u]-u Sin[u],-((Cos[u])/u)-2Sin[u]+u Cos[u]},{u,1,t}],t>1]

FullSimplify[(Sqrt[Total[D[{Sinc[t]-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},t]^2]]),t>0]

Factor[-1+(1/(t^4))-(1/(t^2))+(t^2)]

Integrate[(Sqrt[-1+(1/(t^4))-(1/(t^2))+(t^2)]),{t,1,u},Assumptions->u>1]

Limit[(1/2) (-3 (Sqrt[2])+2 (Sqrt[1+(1/(u^2))])+u (Sqrt[1+(u^2)])+ArcSinh[1]-ArcSinh[u]),u->1]

Limit[(1/2) (-3 (Sqrt[2])+2 (Sqrt[1+(1/(u^2))])+u (Sqrt[1+(u^2)])+ArcSinh[1]-ArcSinh[u]),u->∞]

Plot[(1/2) (-3 (Sqrt[2])+2 (Sqrt[1+(1/(u^2))])+u (Sqrt[1+(u^2)])+ArcSinh[1]-ArcSinh[u]),{u,1,10}]

Curvature

FullSimplify[Curvature[a{((Sin[t])/t)-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},t],t>1&&a>0]

NormalVector

FullSimplify[NormalVector[{((Sin[t])/t)-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},t],t>1]

PolarEquation

FullSimplify[a#.#&@{((Sin[t])/t)-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]}]

ArcTan[((-((Cos[t])/t)-2Sin[t]+t Cos[t])/(((Sin[t])/t)-2Cos[t]-t Sin[t]))]//FullSimplify

Solve[ArcTan[((Cos[t]-(t^2) Cos[t]+2 t Sin[t])/(2 t Cos[t]+(-1+(t^2)) Sin[t]))]==θ,t]

Solve[{((Sin[t])/t)-2Cos[t]-t Sin[t]==r Cos[θ],((Cos[t])/t)-2Sin[t]+t Cos[t]==r Sin[θ]},{r,θ}]//FullSimplify

Slope

FullSimplify[Slope[a{((Sin[t])/t)-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},t],t>1]

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[a{((Sin[t])/t)-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},t],t>1&&a>0]

TangentialAngle

FullSimplify[TangentialAngle[a{((Sin[t])/t)-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},t],t>1&&a>0]

TangentVector

FullSimplify[TangentVector[{((Sin[t])/t)-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]},t],t>1]

VectorLength

(Sqrt[FullSimplify[#.#&@{((Sin[t])/t)-2Cos[t]-t Sin[t],-((Cos[t])/t)-2Sin[t]+t Cos[t]}]])