Ophiuride

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Additional Authors

Original notebook by Margherita Barile
June 21, 2002

Equation

FullSimplify[Eliminate[{x==(b sin[θ]-a cos[θ])sin[θ]/cos[θ]cos[θ],
y==(b sin[θ]-a cos[θ])sin[θ]/cos[θ]sin[θ],
sin[θ]^2+cos[θ]^2==1},{sin[θ],cos[θ]}]]

Simplify[Subtract@@{(x^3)+x y (a+y)-b (y^2),x(x^2+y^2)+(a x-b y)y}]

Plot

Implicit

<<Graphics`ImplicitPlot`
<<Graphics`Colors`

With[{a=5,b=2},ImplicitPlot[x(x^2+y^2)+(a x-b y)y==0,
{x,-2,5},PlotStyle->Red]
]

Polar

<<Graphics`Graphics`

PolarPlot[(2Sin[t]-5Cos[t])Tan[t], {t,-Pi/2,Pi/2},PlotStyle->Red,Ticks->None,AspectRatio->1.5]

Properties

ophiuride=(b Sin[t]-a Cos[t])Tan[t]{Cos[t],Sin[t]};

Arc Length

v4.2.1

ArcLength[ophiuride,t]//Timing

v5

ArcLength[ophiuride,t]//Timing

...ran overnight with no results...

v6

ArcLength[ophiuride,t]//Timing

...ran overnight with no results...

Curvature

FullSimplify[Curvature[ophiuride,t]]

Tangential Angle

v4.2.1

TangentialAngle[ophiuride,t]//Timing

v5

TangentialAngle[ophiuride,t]//Timing

...ran overnight with no results...

v6

TangentialAngle[ophiuride,t]//Timing

...ran overnight with no results...