PARAMETRIC EQUATIONS
If the variables x and y of the Cartesian coordinate
system are expressed in terms of a third variable, say t (or 9), then the
variable
t (or
) is
called a parameter.
The two equations x = x(t) and y = y(t) [or x = x(
) and y =
y(
)] are
called parametric equations.
MOTION IN A STRAIGHT LINE
To illustrate the application of a parameter, we will
assume that an aircraft takes off from azoo
field, which we will call the origin. Figure 36 shows
the diagram we will use. The aircraft is flying on a compass heading of due
north. There is a wind blowing from the west at 20 miles per hour, and the
airspeed of the aircraft is 400 miles per hour. Let the direction of the
positive Y axis be due north and the positive X axis be due east, as shown in
figure 36. Use the scales as shown.
Figure 36.Aircraft position.
One hour after takeoff the position of the aircraft,
represented by point P, is 400 miles north and 20 miles east of the origin. If
we use t as the parameter, then at any time, t, the aircraft's position (x,y) will be given by x
equals 20t and y equals 400t. The equations are
x = 20t
and
y = 400t
and are called parametric equations. Notice that time is
not plotted on the graph of figure 36. The parameter t is used only to plot
the position (x,y) of the aircraft.
We may eliminate the parameter t to obtain a direct
relationship between x and y as follows:
if
^{
}
then
and we find the graph to be a straight line. When we
eliminated the parameter, the result was the rectangular coordinate equation of
the line.
