配极曲线

When working with rectangular coordinates, we can take a function $f$, and graph it by drawing the points $(x,y)$ where $y = f(x)$. You can also think of graphs as starting with a curve, and describing it by saying what the $y$-value is for each $x$-value.

Now that we have polar coordinates, we can do something different. We could take a function $f$ and get a completely different curve by drawing the points with polar coordinates $(r, \theta)$ where $r = f(\theta)$. Or, in the other paradigm, we now have a different way of describing a curve: we can try to say what the $r$-value is for each $\theta$-value.

The spiral below is a graph of the curve $r = \theta$. We start with $\theta = 0$ (the positive $x$-axis) at the origin. As $\theta$ increases, we go counterclockwise, and $r$ increases as well: we start getting further away from the origin.

It's not entirely clear whether this is a particularly interesting way of looking at the function $f(\theta) = \theta$. On the other hand, this is a very simple description of the spiral we get: trying to write down the relationship between $x$-coordinates and $y$-coordinates on the curve would be much more painful.

This is generally the reason why we'd resort to polar curves: to describe curves for which polar coordinates are more natural to use. For example, a circle of radius $R$ is the graph of the polar curve $r = R$. In rectangular coordinates, we'd have to write $x^2 + y^2 = R^2$, and if we wanted to make $y$ a function of $x$, it would be even more work.