Topic 18 Polar coordinates
Figure 18.1: Polar coordinates chart
Given a point \(P\) in the plane as in Figure 18.1:
- \(r\) and \(\theta\) are called the polar coordinates of \(P\), also written as \(r\angle\theta\) or \([r\angle\theta]\).
- The principal angle convention is that \(-\pi<\theta\leq\pi\). This makes \(\theta\) uniquely determined by \(P\).
- The convention \(0\leq\theta<2\pi\) is often used as well.
18.1 Converting polar to Cartesian coordinates
Figure 18.2: Conversion triangle
Given a point \(P=[r\angle\theta]\) as in Figure 18.2, its Cartesian coordinates are given by the following formulas:
\[\boxed{x=r\cos\theta, \quad y=r\sin\theta}\]
Example 18.1 \([2\sqrt2\angle \frac\pi4] = (2\sqrt{2}\cdot\frac{1}{\sqrt2},2\sqrt{2}\cdot\frac{1}{\sqrt2})=(2,2)\).
Example 18.2 \([6.8\angle 55^\circ] = (6.8\cos 55^\circ,6.8\sin 55^\circ) \approx(3.9,5.6)\).
18.2 Converting Cartesian to polar coordinates
Given a point \(P=(x,y)\) as in Figure 18.2, its polar coordinates are given by the following formulas:
\[\boxed{r=\sqrt{x^2+y^2}}\]
\[\boxed{\theta=\begin{cases} \arctan(\frac{y}{x}) & \text{if }x>0\\ \arctan(\frac{y}{x})+\pi &\text{if }x<0\text{ and }y\geq0\\ \arctan(\frac{y}{x})-\pi &\text{if }x<0\text{ and }y<0\\ \frac{\pi}2&\text{if }x=0\text{ and }y>0\\ -\frac{\pi}2&\text{if }x=0\text{ and }y<0\\ \text{undefined} &\text{if }x=0\text{ and }y=0 \end{cases}}\]
Note: The value of “\(\arctan(\dots)\)” is a quantity that describes size of an angle. The formula above assumes that it is in radians (hence the appearance of \(\pi\)). Some care is required when using a calculator which can be set to work in degrees instead, in which case the above formulas need to be adjusted by replacing every \(\pi\) by \(180^\circ\).
Example 18.3 \[\begin{align*} (-67.4,20.31) &= \left[\sqrt{(-67.4)^2+(20.31)^2}\angle \left(\arctan\left(\frac{20.31}{-67.4}\right)+\pi\right)\cdot\frac{180^\circ}{\pi}\right]\\ &\approx\left[70.39\angle 163.23^\circ\right] \end{align*}\]
Example 18.4 Six holes in the plane are given, each in polar coordinates relative to the preceding one: see Table 18.1. Sketch the system and determine the coordinates of hole 6 relative to hole 1 in rectangular and polar coordinates.
| Hole | Relative polar coordinates |
|---|---|
| Hole 1 | \(90\angle 120^\circ\) |
| Hole 2 | \(40\angle 90^\circ\) |
| Hole 3 | \(55\angle 60^\circ\) |
| Hole 4 | \(25\angle 45^\circ\) |
| Hole 5 | \(20\angle-30^\circ\) |
| Hole 6 | \(150\angle -150^\circ\) |
Solution. For a sketch, see Figure 18.3.
Use the above formulas to convert relative polar coordinates to relative Cartesian coordinates: see Table 18.2. Cartesian coordinates of hole 6 relative to hole 1 are the sum of the rectangular coordinates of holes 2 to 6. Executing this we arrive at: \((-67.4,20.31)\).
We convert back to polar coordinates (see Example 18.3) and obtain the polar coordinates of hole 6 relative to hole 1 as: \(70.39\angle 163.23^\circ\).
Figure 18.3: A diagram for the example of holes in the plane
| Hole | Relative polar coords | Relative Cartesian coords |
|---|---|---|
| Hole 1 | \(90\angle 120^\circ\) | \((-45,77.94)\) |
| Hole 2 | \(40\angle 90^\circ\) | \((0,40)\) |
| Hole 3 | \(55\angle 60^\circ\) | \((27.5,47.63)\) |
| Hole 4 | \(25\angle 45^\circ\) | \((17.68,17.68)\) |
| Hole 5 | \(20\angle-30^\circ\) | \((17.32,-10)\) |
| Hole 6 | \(150\angle -150^\circ\) | \((-129.9,-75)\) |
