One of the several systems for addressing points in the plane is the polar-coordinate system. In this system a point P is identified with an ordered pair (r,) where r is a distance and an angle. The angle is measured counter-clockwise from a fixed ray OA called the "polar axis."

The distance to P is measured from the end point O of the ray. This point is called the "pole." Thus each pair determines the location of a point precisely.

When a point P is given coordinates by this scheme, both r and will be positive. In working with polar coordinates, however, it occasionally happens that r, , or both take on negative values. To handle this one can either convert the negative values to positive ones by appropriate rules, or one can broaden the system to allow such possibilities. To do the latter, instead of a...