1 Polar Coordinates

In this Section we consider the application of polar coordinates to the description of curves; in particular, to conics.

If the Cartesian coordinates of a point $P$ are $\left(x,y\right)$ then $P$ can be located on a Cartesian plane as indicated in Figure 10.

Figure 10

However, the same point $P$ can be located by using polar coordinates $r,\theta$ where $r$ is the distance of $P$ from the origin and $\theta$ is the angle, measured anti-clockwise, that the line $OP$ makes when measured from the positive $x$ -direction. See Figure 10(b). In this Section we shall denote the polar coordinates of a point by using square brackets.

From Figure 10 it is clear that Cartesian and polar coordinates are directly related. The relations are noted in the following Key Point.

From these relations we see that it is a straightforward matter to calculate $\left(x,y\right)$ given $\left[r,\theta \right]$ . However, some care is needed (particularly with the determination of $\theta$ ) if we want to calculate $\left[r,\theta \right]$ from $\left(x,y\right)$ .

Example 4

On a Cartesian plane locate points $P,Q,R,S$ which have their locations specified by polar coordinates $\left[2,\frac{\pi }{2}\right],\left[2,3\frac{\pi }{2}\right]$ , $\left[3,\frac{\pi }{6}\right]$ , $\left[\sqrt{2},\pi \right]$ respectively.

Solution

Figure 11

Task!

Two points $P,Q$ have polar coordinates $\left[3,\frac{\pi }{3}\right]$ and $\left[2,\frac{5\pi }{6}\right]$ respectively. By locating these points on a Cartesian plane find their equivalent Cartesian coordinates.

Answer

The polar coordinates of a point are not unique. So, the polar coordinates $\left[a,\theta \right]$ and $\left[a,\varphi \right]$ represent the same point in the Cartesian plane provided $\theta$ and $\varphi$ differ by an integer multiple of $2\pi$ . See Figure 12.

Figure 12

For example, the polar coordinates $\left[2,\frac{\pi }{3}\right]$ , $\left[2,\frac{7\pi }{3}\right]$ , $\left[2,-\frac{5\pi }{3}\right]$ all represent the same point in the Cartesian plane.

The angle $-\varphi$ is interpreted as the angle $\varphi$ measured in a clockwise direction.

Figure 13

Exercises

1. The Cartesian coordinates of $P,Q$ are $\left(1,-1\right)$ and $\left(-1,\sqrt{3}\right)$ . What are their equivalent polar coordinates?
2. Locate the points $P,Q,R$ with polar coordinates $\left[1,\frac{\pi }{3}\right]$ , $\left[2,\frac{7\pi }{3}\right]$ , $\left[2,\frac{10\pi }{3}\right]$ . What do you notice?

Answer