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Rewrite the equation $r=\text{sec}\phantom{\rule{0.2em}{0ex}}\theta \phantom{\rule{0.2em}{0ex}}\text{tan}\phantom{\rule{0.2em}{0ex}}\theta $ in rectangular coordinates and identify its graph.
$y={x}^{2},$ which is the equation of a parabola opening upward.
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 $a=\text{\u2212}b.$ The rose is a very interesting curve. Notice that the graph of $r=3\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}2\theta $ has four petals. However, the graph of $r=3\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}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 $r=3\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\pi \theta \right)$ 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.
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:
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 $r=a\xb7{b}^{\theta}.$ 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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