Plotting polar equations requires the use of polar coordinates, in which points have the form $\left(r\,\mathrm{theta;}\right)$, where r measures the radial distance from the pole O to a point P and $\mathrm{\θ}$ measures the counterclockwise angle from the positive polar axis to the line segment OP.
When plotting a polar function, $r\left(\mathrm{\θ}\right)$, it is often helpful to first plot the function on a rectilinear grid, treating $\left(\mathrm{\θ}\,r\right)$ as Cartesian coordinates, with $\mathrm{\θ}$ being plotted along the horizontal axis and r being plotted along the vertical axis. From this Cartesian plot, you can transfer critical points, such as minima, maxima, roots, and endpoints to a polar grid using the coordinates $\left(r\,\mathrm{theta;}\right)$, and then fill in the behavior of r between the critical values of $\mathrm{\θ}$ to create the final polar graph.

Common Figures in Polar Plots


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Circle: The equation $requals;a$ creates a circle of radius a centered at the pole. The equation $r\=a\mathrm{cos}\left(\mathrm{\theta}\right)$ creates a circle of diameter a centered at the point $\left(r\,\mathrm{\θ}\right)equals;\left(\frac{a}{2}comma;0\right)$. The equation $r\=a\mathrm{sin}\left(\mathrm{\theta}\right)$creates a circle of diameter a centered at the point $\left(r\,\mathrm{\theta}\right)equals;\left(\frac{a}{2}comma;\frac{\mathrm{pi;}}{2}\right)$.

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Cardioid: The equations $requals;aplus;a\mathrm{cos}\left(\mathrm{\theta}\right)$ and $requals;aplus;a\mathrm{sin}\left(\mathrm{theta;}\right)$ create horizontal and vertical heartlike shapes called cardioids.

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Archimedean Spiral: any equation of the form $r\=a{\mathrm{\θ}}^{\frac{1}{n}}$ creates a spiral, with the constant n determining how tightly the spiral winds around the pole. Special cases of Archimedean spirals include: Archimedes' Spiral when $nequals;1$, Fermat's Spiral when $n\=2$, a hyperbolic spiral when $nequals;1$, and a lituus when $nequals;2$.

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Polar Rose: Equations of the form $r\=a\mathrm{cos}\left(k\mathrm{\θ}\right)$create curves which look like petaled flowers, where a represents the length of each petal and k determines the number of petals. If k is an odd integer, the rose will have k petals; if k is an even integer, the rose will have $2k$ petals; and if k is rational, but not an integer, a roselike shape may form with overlapping petals.

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Ellipse: Equations of the form $r\=\frac{l}{1\+e\mathrm{cos}\left(\mathrm{\theta}\right)}$, where $1e1$ is the eccentricity of the curve (a measure of how much a conic section deviates from being circular) and l is the semilatus rectum (half the chord parallel to the directrix passing through a focus), create ellipses for which one focus is the pole and the other lies somewhere along the line $\mathrm{\θ}equals;0$. The special case where $eequals;0$ creates a circle of radius l.


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