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In **polar coordinates** a point *P* is also characterized by two
numbers: the distance *r*0 to a fixed **pole** or **origin** *O*,
and the angle the ray *OP* makes with a fixed ray originating at
*O*, which is generally drawn pointing to the right (this is called
the **initial ray**). The angle
is only defined up to a multiple of 360° or 2.
In addition, it is sometimes convenient to relax the condition *r*0
and allow *r* to be a signed distance, so (*r*,) and
(-*r*, +180°) represent the same point
(Figure 1).

**Figure 1:** Among the possible sets of polar coordinates for *P* are:
(10, 30°),
(10, 390°)
and (10, -330°).
Among the sets of polar coordinates for *Q* are:
(2.5, 210°)
and (-2.5, 30°).

Consider a system of polar coordinates and a system of cartesian
coordinates with the same origin. Assume the initial ray of the polar
coordinate system coincides with the positive *x*-axis, and that the
ray =90° coincides with the positive *y*-axis. Then
the polar coordinates (*r*,) and the cartesian coordinates
(*x*,*y*) of the same point are related as follows:

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.