7.3 Polar coordinates  (Page 4/16)

We have now seen several examples of drawing graphs of curves defined by polar equations . A summary of some common curves is given in the tables below. In each equation, a and b are arbitrary constants.

A cardioid    is a special case of a limaçon    (pronounced “lee-mah-son”), in which $a=b$ or $a=\text{−}b.$ The rose    is a very interesting curve. Notice that the graph of $r=3\text{sin}2\theta$ has four petals. However, the graph of $r=3\text{sin}3\theta$ has three petals as shown.

If the coefficient of $\theta$ is even, the graph has twice as many petals as the coefficient. If the coefficient of $\theta$ is odd, then the number of petals equals the coefficient. You are encouraged to explore why this happens. Even more interesting graphs emerge when the coefficient of $\theta$ is not an integer. For example, if it is rational, then the curve is closed; that is, it eventually ends where it started ( [link] (a)). However, if the coefficient is irrational, then the curve never closes ( [link] (b)). Although it may appear that the curve is closed, a closer examination reveals that the petals just above the positive x axis are slightly thicker. This is because the petal does not quite match up with the starting point.

Since the curve defined by the graph of $r=3\text{sin}\left(\pi \theta \right)$ never closes, the curve depicted in [link] (b) is only a partial depiction. In fact, this is an example of a space-filling curve    . A space-filling curve is one that in fact occupies a two-dimensional subset of the real plane. In this case the curve occupies the circle of radius 3 centered at the origin.

Chapter opener: describing a spiral

Recall the chambered nautilus introduced in the chapter opener. This creature displays a spiral when half the outer shell is cut away. It is possible to describe a spiral using rectangular coordinates. [link] shows a spiral in rectangular coordinates. How can we describe this curve mathematically?

As the point P travels around the spiral in a counterclockwise direction, its distance d from the origin increases. Assume that the distance d is a constant multiple k of the angle $\theta$ that the line segment OP makes with the positive x -axis. Therefore $d\left(P,O\right)=k\theta ,$ where $O$ is the origin. Now use the distance formula and some trigonometry:

$\begin{array}{ccc}\hfill d\left(P,O\right)& =\hfill & k\theta \hfill \\ \hfill \sqrt{{\left(x-0\right)}^2+{\left(y-0\right)}^2}& =\hfill & k\text{arctan}\left(\frac{y}{x}\right)\hfill \\ \hfill \sqrt{x^2+y^2}& =\hfill & k\text{arctan}\left(\frac{y}{x}\right)\hfill \\ \hfill \text{arctan}\left(\frac{y}{x}\right)& =\hfill & \frac{\sqrt{x^2+y^2}}{k}\hfill \\ \hfill y& =\hfill & x\text{tan}\left(\frac{\sqrt{x^2+y^2}}{k}\right).\hfill \end{array}$

Although this equation describes the spiral, it is not possible to solve it directly for either x or y . However, if we use polar coordinates, the equation becomes much simpler. In particular, $d\left(P,O\right)=r,$ and $\theta$ is the second coordinate. Therefore the equation for the spiral becomes $r=k\theta .$ Note that when $\theta =0$ we also have $r=0,$ so the spiral emanates from the origin. We can remove this restriction by adding a constant to the equation. Then the equation for the spiral becomes $r=a+k\theta$ for arbitrary constants $a$ and $k.$ This is referred to as an Archimedean spiral , after the Greek mathematician Archimedes.

Another type of spiral is the logarithmic spiral, described by the function $r=a·b^{\theta }.$ A graph of the function $r=1.2\left({1.25}^{\theta }\right)$ is given in [link] . This spiral describes the shell shape of the chambered nautilus.

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