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[parent] polar curve (Example)

Polar curves are plane curves in $ \mathbb{R}^2$ that are expressed in polar coordinates $ (r,\theta)$. The two simplest polar curves are obtained when one of the two coordinates is set to be a constant. If the first coordinate is set to a constant $ r$, we have a circle with radius $ \lvert r \rvert$, or a point when $ r=0$. When the second coordinate is the constant instead, say $ c$, we have a straight line through the (polar) origin, with slope = $ \tan c$.

\includegraphics{polar_curve_1.eps} \includegraphics{polar_curve_2.eps}

Using polar coordinates, one can generate many visually pleasing curves. Below are some of the most popular ones.

\includegraphics{polar_curve_3.eps}


\includegraphics{polar_curve_4.eps} \includegraphics{polar_curve_5.eps}


\includegraphics{polar_curve_6.eps}


\includegraphics{polar_curve_7.eps}


\includegraphics{polar_curve_8.eps} \includegraphics{polar_curve_9.eps}



"polar curve" is owned by CWoo.
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See Also: area of plane region, Cassini oval

Other names:  limaçon
Also defines:  lemniscate, rhodonea, cardioid, limacon

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Cross-references: curves, generate, slope, origin, polar, line, straight, point, radius, circle, coordinates, polar coordinates, plane curves
There are 2 references to this entry.

This is version 3 of polar curve, born on 2005-05-18, modified 2005-05-20.
Object id is 7076, canonical name is PolarCurve.
Accessed 8105 times total.

Classification:
AMS MSC: 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)
 51-01 (Geometry :: Instructional exposition )
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