Graphing Polar Functions

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The first (and simplest) method to try for drawing a polar graph is to rewrite $r=f(\theta)$ as a relation between $x$ and $y$ , and then draw the graph of this relation. For example, when $r=2\cos(\theta)$ , then we have $r^2 - 2r\cos(\theta) = 0$ . But

$r^2 \ = \ x^2+y^2 \quad \hbox{and} \quad 2r \cos(\theta) \ = \ 2 x,$ so

$x^2-2x + y^2=0$ . Completing the square with

$x^2-2x = (x-1)^2 - 1$ gives

$(x-1)^2 + y^2=1.$ This is a circle of radius 1 centered at

$(1,0)$ .

In other cases, the curve doesn't have a nice description in rectangular coordinates -- that's usually why we're using polar coordinates! We then plot points as follows: start with the graph of $r=f(\theta)$ with $(r,\theta)$ as rectangular coordinates, and then plot the corresponding values of $(x,y)$ using

$x=r\cos(\theta),\qquad y=r\sin(\theta).$ In other words, we obtain a parametrization of our polar curve as

$\theta=t$ ,

$r=f(t)$ , and so:

$x(t) = f(t) \cos(t); \qquad y(t) = f(t) \sin(t).$ We can then apply everything we know about parametrized curves to polar curves.

The following video shows three ways to plot the polar curve $r = 2 \cos(\theta)$ .