# 1.3 Polar Coordinates in the Plane

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°).

### Relations between Cartesian and Polar Coordinates

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: Next: 1.4 Homogeneous Coordinates in the Plane
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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.