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Polar Plot


PolarPlot

A plot of a function expressed in polar coordinates, with radius r as a function of angle theta. Polar plots can be drawn in the Wolfram Language using PolarPlot[r, {t, tmin, tmax}]. The plot above is a polar plot of the polar equation r(theta)=1+costheta, giving a cardioid.

Polar plots of r=sin(ntheta) give curves known as roses, while polar plots of r=a+btheta produce what's known as Archimedes' spiral, a special case of the Archimedean spiral r=a+btheta^(1/n) corresponding to n=1. Other specially-named Archimedean spirals include the lituus when n=-2, the hyperbolic spiral when n=-1, and Fermat's spiral when n=2. Note that lines and circles are easily-expressed in polar coordinates as

 theta=c
(1)

and

 r^2-2rr_0cos(theta-theta_0)+r_0^2=a^2
(2)

for the circle with center (r_0,theta_0) and radius a, respectively. Note that equation () is merely a particular instance of the equation

 r=L/(1+ecostheta)
(3)

defining a conic section of eccentricity e and semilatus rectum L. In particular, the circle is the conic of eccentricity e=0, while e<1 yields a general ellipse, e=1 a parabola, and e>1 a hyperbola.

The plotting of a complex number z=x+iy in terms of its complex modulus |z| and its complex argument theta is closely related to polar coordinates due, e.g., to the Euler formula. As such, the plotting of complex numbers in the Cartesian plane by way of an Argand diagram can be viewed as a specialized polar plot.


See also

Archimedean Spiral, Archimedes' Spiral, Argand Diagram, Cardioid, Circle, Complex Argument, Complex Modulus, Complex Number, Conic Section, Eccentricity, Ellipse, Fermat's Spiral, Hyperbola, Hyperbolic Spiral, Line, Lituus, Parabola, Polar Coordinates, Polar Curve, Polar Equation, Rose Curve, Spherical Plot

Portions of this entry contributed by Christopher Stover

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Cite this as:

Stover, Christopher and Weisstein, Eric W. "Polar Plot." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PolarPlot.html

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