Visualizing Polar Coordinates

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Section 3.9 Visualizing Polar Coordinates

Polar coordinates are useful for situations with circular symmetry in the plane. The polar coordinates (\(r\text{,}\)\(\phi\)) of a point \(P\) are given by the distance \(r\) of \(P\) from the origin and the angle \(\phi\) from the positive \(x\)-axis to \(P\text{,}\) as shown in Figure 3.9.1.

Figure 3.9.1. The construction of the polar coordinates (\(r\text{,}\)\(\phi\)) at an arbitrary point.

It is important to remember that the angle \(\phi\) does not measure distance; it has the wrong dimensions. Angles (in radians) are defined as the ratio of arclength \(\ell\) to radius on the circle, i.e.

\begin{equation} \phi=\frac{\ell}{r}\text{.}\tag{3.9.1} \end{equation}

so the arclength from the positive \(x\)-axis to \(P\) along the circle shown in the figure is \(\ell=r\phi\text{.}\)

See [cross-reference to target(s) "coordsnotation" missing or not unique] for a discussion of the relationship between polar coordinates and both cylindrical and spherical coordinates in three-dimensions.