10
Polar Coordinates,
Parametric Equations
Coordinate systems are tools that let us use algebraic methods to understand geometry.
While the
rectangular
(also called
Cartesian
) coordinates that we have been using are
the most common, some problems are easier to analyze in alternate coordinate systems.
A coordinate system is a scheme that allows us to identify any point in the plane or
in threedimensional space by a set of numbers. In rectangular coordinates these numbers
are interpreted, roughly speaking, as the lengths of the sides of a rectangle.
In
polar
coordinates
a point in the plane is identified by a pair of numbers (
r, θ
). The number
θ
measures the angle between the positive
x
axis and a ray that goes through the point,
as shown in figure 10.1; the number
r
measures the distance from the origin to the point.
Figure 10.1 shows the point with rectangular coordinates (1
,
√
3) and polar coordinates
(2
, π/
3), 2 units from the origin and
π/
3 radians from the positive
x
axis.
√
3
1
.
.
(2
, π/
3)
•
.
Figure 10.1
Polar coordinates of the point (1
,
√
3).
217
218
Chapter 10 Polar Coordinates, Parametric Equations
Just as we describe curves in the plane using equations involving
x
and
y
, so can we
describe curves using equations involving
r
and
θ
. Most common are equations of the form
r
=
f
(
θ
).
EXAMPLE 10.1
Graph the curve given by
r
= 2. All points with
r
= 2 are at distance
2 from the origin, so
r
= 2 describes the circle of radius 2 with center at the origin.
EXAMPLE 10.2
Graph the curve given by
r
= 1 + cos
θ
.
We first consider
y
=
1 + cos
x
, as in figure 10.2. As
θ
goes through the values in [0
,
2
π
], the value of
r
tracks
the value of
y
, forming the “cardioid” shape of figure 10.2. For example, when
θ
=
π/
2,
r
= 1 + cos(
π/
2) = 1, so we graph the point at distance 1 from the origin along the
positive
y
axis, which is at an angle of
π/
2 from the positive
x
axis.
When
θ
= 7
π/
4,
r
= 1 + cos(7
π/
4) = 1 +
√
2
/
2
≈
1
.
71, and the corresponding point appears in the fourth
quadrant.
This illustrates one of the potential benefits of using polar coordinates:
the
equation for this curve in rectangular coordinates would be quite complicated.
0
1
2
π/
2
π
3
π/
2
2
π
•
•
.
•
•
..
.
.
.
.
.
.
.
.
Figure 10.2
A cardioid:
y
= 1 + cos
x
on the left,
r
= 1 + cos
θ
on the right.
Each point in the plane is associated with exactly one pair of numbers in the rect
angular coordinate system; each point is associated with an infinite number of pairs in
polar coordinates. In the cardioid example, we considered only the range 0
≤
θ
≤
2
π
, and
already there was a duplicate: (2
,
0) and (2
,
2
π
) are the same point. Indeed, every value of
θ
outside the interval [0
,
2
π
) duplicates a point on the curve
r
= 1+cos
θ
when 0
≤
θ <
2
π
.
We can even make sense of polar coordintes like (

2
, π/
4): go to the direction
π/
4 and
then move a distance 2 in the opposite direction; see figure 10.3. As usual, a negative angle
θ
means an angle measured clockwise from the positive
x
axis. The point in figure 10.3
also has coordinates (2
,
5
π/
4) and (2
,

3
π/
4).