Kilroy Curve

Author

Eric W. Weisstein
June 28, 2013

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

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

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

Definition

curve[x_]:={x,Log[Sqrt[Sinc[x]^2]]}

Plot

V5

Plot[Log[Abs[Sin[x]/x]],{x,-15,15},PlotDivision->70,PlotPoints->200,PlotRange->{-7,.2},PlotStyle->Red,Ticks->None]

V6

ParametricPlot[curve[x],{x,-15,15}]

<<Utilities`Plot`

ComplexContourPlot[Log[Sqrt[Sinc[z]^2]],{z,-3,3}]

ComplexPlot3D[Log[Sqrt[Sinc[z]^2]],{z,-3,3}]

Properties

curve=a {t,Log[Abs[Sinc[t]]]};

Arc Length

v5

ArcLength[{x,Log[Abs[Sin[x]/x]]},x,Assumptions->x>0]//Timing

v6

c=curve[t]/.Abs[x_]:>Sqrt[x^2]

ArcLength[c,t]//Timing

FullSimplify[Sqrt[FunctionExpand[1+((((((Cos[t])/t)-((Sin[t])/(t^2))))^2)/((Sinc[t])^2))]],t>0]

Integrate[(Sqrt[(1/(t^2))-((2 Cot[t])/t)+((Csc[t])^2)]),t]

Curvature

v5

Curvature[{x,Log[Abs[Sin[x]/x]]},x]//FullSimplify

v6

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

Slope

Slope[{t,Log[Sqrt[Sinc[t]^2]]},t]//FullSimplify

SpecialAffineCurvature

SpecialAffineCurvature[curve,t]

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&t∈Reals]

Tangential Angle

v5

TangentialAngle[{x,Log[Abs[Sin[x]/x]]},x]//Timing

v6

TangentialAngle[c,t]//Timing

Vector Properties

parm=a {t,Log[Abs[Sinc[t]]]};
assum=a>0&&t∈Reals;
xform={Abs[x_]^(n_/;Mod[n,2]==0):>x^n,((Sign[Sinc[_]])^2):>1};

VectorLength

Assuming[assum,FullSimplify[Norm[parm]/.xform]]

NormalVector

Assuming[assum,FullSimplify[FullSimplify[NormalVector[parm,t]/.xform]]/.xform]

TangentVector

Assuming[assum,FullSimplify[FullSimplify[TangentVector[parm,t]/.xform]]/.xform]