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Unformatted text preview: 146 Chapter 6 • Additional Topics §6.4 6.4 Polar Coordinates
Suppose that from the point (1, 0) in the
x y-coordinate plane we draw a spiral around
the origin, such that the distance between
any two points separated by 360◦ along the
spiral is always 1, as in Figure 6.4.1. We
can not express this spiral as y = f ( x) for
some function f in Cartesian coordinates,
since its graph violates the vertical rule.
However, this spiral would be simple to
describe using the polar coordinate system.
Recall that any point P distinct from the
origin (denoted by O ) in the x y-coordinate
plane is a distance r > 0 from the origin,
and the ray OP makes an angle θ with the
positive x-axis, as in Figure 6.4.2. We call
the pair ( r , θ ) the polar coordinates of P ,
and the positive x-axis is called the polar
axis of this coordinate system. Note that ( r , θ ) = ( r , θ + 360◦ k) for k = 0, ± 1, ± 2, ..., so (unlike
for Cartesian coordinates) the polar coordinates of a point are not unique.
P (r, θ)
θ O θ x O x −r P (− r , θ )
Figure 6.4.2 Polar coordinates ( r , θ ) Figure 6.4.3 Negative r : (− r , θ ) In polar coordinates we adopt the convention that r can be negative, by deﬁning (− r , θ ) =
( r , θ + 180◦ ) for any angle θ . That is, the ray OP is drawn in the opposite direction from the
angle θ , as in Figure 6.4.3. When r = 0, the point ( r , θ ) = (0, θ ) is the origin O , regardless of
the value of θ .
You may be familiar with graphing paper, for plotting points or functions given in Cartesian coordinates (sometimes also called rectangular coordinates). Such paper consists of a
rectangular grid. Similar graphing paper exists for plotting points and functions in polar
coordinates, similar to Figure 6.4.4. ...
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