Garfield Curve

Author

Eric W. Weisstein
January 20, 2015

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

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

©2015 Wolfram Research, Inc. except for portions noted otherwise

Plots

PolarPlot

PolarPlot[q Cos[q],{q,-2π,2π},PlotStyle->Red,Ticks->None]

ParametricPlot

ParametricPlot[t Cos[t]{Cos[t],Sin[t]},{t,-2π,2π},PlotStyle->Red,Ticks->None]

ParametricPlot[t Cos[t]{Cos[t],Sin[t]},{t,-5π-π/2,5π+π/2},PlotStyle->Red,Ticks->None]

Equations

Cartesian

r==a θ Cos[θ]==a θ x/r

x^2+y^2==a θ x

x^2+y^2==a ArcTan[y/x] x

ContourPlot[x^2+y^2== ArcTan[x,y] x,{x,-4,3},{y,-3,3}]

Properties

curve=t Cos[t]{Cos[t],Sin[t]};

ArcLength

MyArcLength[curve,{t,-2π,2π}]

ArcLengthFunctions

MyArcLength[curve,t]

Curvature

Parametric

FullSimplify[Curvature[curve,t],-2π<t<2π]

Polar

FullSimplify[Curvature[a q Cos[q],q],a>0&&-2π<q<2π]

Slope

Parametric

Slope[curve,t]//FullSimplify

Polar

Slope[a q Cos[q],q]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&-2π<t<2π]

Tangential Angle

TangentialAngle[curve,t,Assumptions->a>0&&t>0]

Vector Properties

parm=a t Cos[t]{Cos[t],Sin[t]};
assum=a>0&&-2π<t<2π;
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]]