The polar coordinate system consists of a pole and a polar axis.
The pole is a fixed point, and the polar axis is a directed ray whose endpoint
is the pole. Every point in the plane of the polar axis can be specified
according to two coordinates: r, the distance between the point and the pole,
and θ, the angle between the polar axis and the ray containing the point
whose endpoint is also the pole.
Figure %: The polar coordinate system
The distance r
and the angle θ
are both directed--meaning that they
represent the distance and angle in a given direction.
It is possible,
therefore to have negative values for both r
. However, we
typically avoid points with negative r
, since they could just as easily be
specified by adding Π
) to θ
. Similarly, we
typically ask that θ
be in the range 0≤θ < 2Π
, since there
is always some θ
in this range corresponding to our point. This doesn't
eliminate all ambiguity, however; the pole can still be specified by (0, θ)
for any angle θ
. But it is true that any other point can be
described uniquely with these conventions.
To convert equations between polar coordinates and rectangular coordinates,
consider the following diagram:
Figure %: The x and y coordinates in the polar coordinate system
See that sin(θ) =
, and cos(θ) =
To convert from rectangular to polar coordinates, use the following equations:
x = r cos(θ), y = r sin(θ). To convert from polar to rectangular
coordinates, use these equations: r = sqrtx2+y2, θ = arctan().