12.3: Polar Coordinates

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Cartesian coordinates (\(x,y\)) are not the only way of labeling a point \(P\) on a flat plane by a pair of numbers. Other ways exist, and they can be more useful in special situations.

One system (“polar coordinates") uses the length \(r\) of the line \(OP\) from the origin to \(P\) (i. e. the distance of \(P\) distance to the origin) and the angle that line makes with the x-axis. Angles are often denoted by Greek letters, and here we follow conventions by marking it with \(\theta\). Note that while in the cartesian system \(x\) and \(y\) play very similar roles, here roles are divided: \(r\) gives distance and \(\theta\) direction.

The two representations are closely related. From the definitions of the sine and cosine:

\[x\,=\,r\,\times\, \cos \theta\]

\[y\,=\,r\,\times\, \sin \theta\]

This allows \((x,y\)) to be derived from polar coordinates. This relationship is illustrated below:

To go in the reverse direction, we can use the Pythagorean theorem to find \(r\):

\[r^2\,=\,x^2\,+\,y^2\]

Once \(r\) is known, the rest is easy:

\[\cos \theta\,=\frac{x}{r}\]

\[\sin \theta\,=\frac{y}{r}\]

These relations fail only at the origin, where \(x\,=\,y\,=\,r\,=\,0\). At that point, \(\theta\) is undefined and one can choose for it whatever one pleases.

In three dimensional space, the cartesian labeling (\(x,y,z\)) is nicely symmetric, but sometimes it is convenient to follow the style of polar coordinates and label distance and direction separately. Distance is easy: you take the line OP from the origin to the point and measure its length r. You can even show from the theorem of Pythagoras that in this case

\[r^2\,=\,x^2\,+\,y^2\,+\,z^2\]

All the points with the same value of \(r\) form a sphere of radius \(r\) around the origin \(O\). On a sphere we can label each point by latitude \(\lambda\) (lambda, small Greek L) and longitude \(\phi\) (phi, small Greek F), so that the position of any point in space is defined by the 3 numbers (\(r,\lambda ,\phi \)).