Coordinate systems are tools that let us use algebraic methods to understand geometry. While the rectangular (also called Cartesian) coordinates that we have been discussing are the most common, some problems are easier to analyze in alternate coordinate systems.

A coordinate system is a scheme that allows us to identify any point in the plane or in three-dimensional space by a set of numbers. In rectangular coordinates these numbers are interpreted, roughly speaking, as the lengths of the sides of a rectangular "box.''

In two dimensions you may already be familiar with an alternative, called polar coordinates. In this system, each point in the plane is identified by a pair of numbers $(r,\theta)$. The number $\theta$ measures the angle between the positive $x$-axis and a vector with tail at the origin and head at the point, as shown in figure 12.6.1; the number $r$ measures the distance from the origin to the point. Either of these may be negative; a negative $\theta$ indicates the angle is measured clockwise from the positive $x$-axis instead of counter-clockwise, and a negative $r$ indicates the point at distance $|r|$ in the opposite of the direction given by $\theta$. Figure 12.6.1 also shows the point with rectangular coordinates $\ds (1,\sqrt3)$ and polar coordinates $(2,\pi/3)$, 2 units from the origin and $\pi/3$ radians from the positive $x$-axis.

Figure 12.6.1. Polar coordinates: the general case and the point with rectangular coordinates $\ds (1,\sqrt3)$.

We can extend polar coordinates to three dimensions simply by adding a $z$ coordinate; this is called cylindrical coordinates. Each point in three-dimensional space is represented by three coordinates $(r,\theta,z)$ in the obvious way: this point is $z$ units above or below the point $(r,\theta)$ in the $x$-$y$ plane, as shown in figure 12.6.2. The point with rectangular coordinates $\ds (1,\sqrt3, 3)$ and cylindrical coordinates $(2,\pi/3,3)$ is also indicated in figure 12.6.2.

Figure 12.6.2. Cylindrical coordinates: the general case and the point with rectangular coordinates $\ds (1,\sqrt3, 3)$.

Some figures with relatively complicated equations in rectangular coordinates will be represented by simpler equations in cylindrical coordinates. For example, the cylinder in figure 12.6.3 has equation $\ds x^2+y^2=4$ in rectangular coordinates, but equation $r=2$ in cylindrical coordinates.