Polar coordinates are extremely useful, especially when thinking about complex numbers. Note, however, that the representation of a point is very non-unique.

First, is not determined by the point. You could add to it and get the same point:

Also that can be negative introduces further non-uniqueness:

We can convert back and forth between cartesian and polar coordinates using that

(4.1) | ||

(4.2) |

and in the other direction

(4.3) | ||

(4.4) |

(Thus and )

We plug in points for one period of the function we are graphing--in this case :

0 | |

To more accurately draw the graph, let's try converting the equation to one involving polar coordinates. This is easier if we multiply both sides by :

The graph of (4.1.5) is the same as that of . To confirm this we complete the square:

Thus the graph of (4.1.5) is a circle of radius centered at .

Actually *any* polar graph of the form
is a circle, as you will see in homework problem 67
by generalizing what we just did.

William Stein 2006-03-15