Chapter 10 - 10 PARAMETRIC EQUATIONS AND POLAR COORDINATES...

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So far we have described plane curves by giving as a function of or
as a function of or by giving a relation between and that defines
implicitly as a function of . In this chapter we discuss two new methods
for describing curves.
Some curves, such as the cycloid, are best handled when both and are given in
terms of a third variable called a parameter . Other curves, such as
the cardioid, have their most convenient description when we use a new coordinate
system, called the polar coordinate system.
x f t , y t t t
y x
f x , y 0 x
y y x x t y y
x y f x x y
Parametric equations and polar coordinates enable us to
describe a great variety of new curves—some practical,
some beautiful, some fanciful, some strange.
PARAMETRIC EQUATIONS
AND POLAR COORDINATES
10

CURVES DEFINED BY PARAMETRIC EQUATIONS
Imagine that a particle moves along the curve C shown in Figure 1. It is impossible to
describe C by an equation of the form because C fails the Vertical Line Test. But
the x - and y -coordinates of the particle are functions of time and so we can write
and . Such a pair of equations is often a convenient way of describing a curve and
gives rise to the following definition.
Suppose that and are both given as functions of a third variable (called a param-
eter ) by the equations
(called parametric equations ). Each value of determines a point , which we can
plot in a coordinate plane. As varies, the point varies and traces out
a curve , which we call a parametric curve . The parameter t does not necessarily repre-
sent time and, in fact, we could use a letter other than t for the parameter. But in many
applications of parametric curves, t does denote time and therefore we can interpret
as the position of a particle at time t .
EXAMPLE 1
Sketch and identify the curve defined by the parametric equations
SOLUTION
Each value of gives a point on the curve, as shown in the table. For instance, if
, then , and so the corresponding point is . In Figure 2 we plot the
points determined by several values of the parameter and we join them to produce
a curve.
A particle whose position is given by the parametric equations moves along the curve
in the direction of the arrows as increases. Notice that the consecutive points marked on
the curve appear at equal time intervals but not at equal distances. That is because the
particle slows down and then speeds up as increases.
It appears from Figure 2 that the curve traced out by the particle may be a parabola.
This can be confirmed by eliminating the parameter as follows. We obtain
from the second equation and substitute into the first equation. This gives
and so the curve represented by the given parametric equations is the parabola
.
M
x y 2 4 y 3
x t 2 2 t y 1 2 2 y 1 y 2 4 y 3
t y 1 t
t
t
FIGURE 2
0
t=0
t=1
t=2
t=3
t=4
t=_1
t=_2
(0, 1)
y
x
8
x , y
0, 1 y 1 x 0 t 0
t
y t 1 x t 2 2 t
x , y f t , t t
C
x , y f t , t t t
x , y t
y t t x f t
t y x
y t t
x f t
y f x
10.1
t x y
2 8 1
1
3 0
0 0 1
1 1 2
2 0 3
3 3 4
4 8 5
N
This equation in and describes where the
particle has been, but it doesn’t tell us when the