Kappa Curve

Author

Eric W. Weisstein
March 16, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Equation

FullSimplify[Eliminate[{x==a Cot[θ]Cos[θ],y==a Cot[θ]Sin[θ],Cos[θ]^2+Sin[θ]^2==1}/.{Cot[θ]:>cos[θ]/sin[θ],Sin->sin,Cos->cos},{sin[θ],cos[θ]}]]

Plot

V5

<<Graphics`Graphics`

PolarPlot[Cot[t],{t,0,3},PlotRange->{{-.1,2},{-1,1}},Ticks->None,PlotStyle->Red]

V6

PolarPlot[Cot[t],{t,0,π},PlotRange->{{-2,2},{-1,1}},Ticks->None,PlotStyle->Red,ImageSize->200]

PolarPlot[Cot[t],{t,π+.001,2π},PlotRange->{{-2,2},{-1,1}},Ticks->None,PlotStyle->Red,ImageSize->200]

ContourPlot[(x^2)==(y^2) ((x^2)+(y^2)),{x,-1,1},{y,-1,1}]

Properties

Arc Length

PolarPlot[Cot[t],{t,0,Pi/2},PlotRange->{{-.1,2},{-1,1}},Ticks->None,PlotStyle->Red]

v5

ArcLength[a Cot[θ],θ]

ArcLength[a Cot[θ],{θ,Pi/2,θ}]

v6

ArcLength[a Cot[θ],θ]

ArcLength[a Cot[θ],{θ,Pi/2,θ}]

Curvature

FullSimplify[Curvature[a Cot[θ],θ],a>0&&-Pi/2<θ<Pi/2]

Tangential Angle

FullSimplify[TangentialAngle[a Cot[θ],θ],a>0]