Cochleoid

Author

Eric W. Weisstein
June 6, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Definition

cochleoid=a{Sinc[t] Cos[t],Sinc[t]Sin[t]};

Equation

FullSimplify[Sqrt[#.#]&[cochleoid],a>0&&t>0]

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

Numerator[Factor[FullSimplify[(x^2+y^2)-((a^2)/(θ^2))Sin[θ]^2/.θ->ArcTan[y/x]]]]//FullSimplify

Plot

V5

<<Graphics`Graphics`

PolarPlot[Sin[t]/t,{t,0,20},Ticks->None,PlotRange->{{-.3,1},{-.1,.8}},PlotStyle->Red]

ParametricPlot[cochleoid,{t,0,20},PlotStyle->Red,AspectRatio->Automatic,Ticks->None,PlotRange->{{-.3,1},{-.1,.8}}]

V6

PolarPlot[Sinc[t],{t,0,8π},PlotRange->All]

ParametricPlot[cochleoid,{t,0,20},PlotStyle->Red,AspectRatio->Automatic,Ticks->None,PlotRange->All]

Properties

cochleoid=a{Sinc[t] Cos[t],Sinc[t]Sin[t]};

Arc Length

v4.2

ArcLength[cochleoid,t]

v5.0

ArcLength[cochleoid,t]

FullSimplify[√((((((2 Cos[t] Sin[t])/t)-(((Sin[t])^2)/(t^2))))^2)+((((((Cos[t])^2)/t)-((Cos[t] Sin[t])/(t^2))-(((Sin[t])^2)/t)))^2)),t>0]

Integrate[((Sqrt[(1/2)+(t^2)-(1/2) Cos[2 t]-t Sin[2 t]])/(t^2)),{t,0,t}]

v6 TrigExpand

Integrate[(1/(t^2))(Sqrt[(1/2)+(t^2)-(((Cos[t])^2)/2)-2 t Cos[t] Sin[t]+(((Sin[t])^2)/2)]),{t,0,t}]//Timing

Integrate[(1/(t^2))(Sqrt[(1/2)+(t^2)-(((Cos[t])^2)/2)-2 t Cos[t] Sin[t]+(((Sin[t])^2)/2)]),t]//Timing

v6

ArcLength[cochleoid,t]

FullSimplify[√((((((2 Cos[t] Sin[t])/t)-(((Sin[t])^2)/(t^2))))^2)+((((((Cos[t])^2)/t)-((Cos[t] Sin[t])/(t^2))-(((Sin[t])^2)/t)))^2)),t>0]

Integrate[((Sqrt[(1/2)+(t^2)-(1/2) Cos[2 t]-t Sin[2 t]])/(t^2)),{t,0,t}]//Timing

Integrate[((Sqrt[(1/2)+(t^2)-(1/2) Cos[2 t]-t Sin[2 t]])/(t^2)),t]//Timing

Curvature

Plot[((4 (Sqrt[2]) (t^3) (t-Cos[t] Sin[t]))/(((1+2 (t^2)-Cos[2 t]-2 t Sin[2 t]))^(3/2))),{t,0,10},PlotStyle->Red]

v5.2

FullSimplify[Curvature[cochleoid,t],t>0]

v6

FullSimplify[Curvature[cochleoid,t],t>0&&a>0]

Polar

FullSimplify[Curvature[a Sinc[q],q],a>0&&q>0]

Slope

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

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[cochleoid,t],a>0&&0<t<8π]

Tangential Angle

Plot[(1/2) (3 t-ArcCot[Cot[t]-t ((Csc[t])^2)]+ArcTan[(t/(-1+t Cot[t]))]),{t,0,10},PlotStyle->Red]

v4.2

TangentialAngle[cochleoid,t]

v5.0

TangentialAngle[cochleoid,t]

FullSimplify[%4,t>0]

...contains a jump...

v5.2

TangentialAngle[cochleoid,t]//Timing

v6

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

Vector Properties

parm=a {Sinc[t] Cos[t],Sinc[t] Sin[t]};
assum=a>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