Abel's theorem in problems and solutions - School of Mathematics

66 Chapter 2
The plane on which the values of are represented will be called simply
the plane’. If the functions and are continuous for
then as varies from 0 to 1, the point describes a continuous curve
in the plane. We provide this curve with an orientation, assuming that
and are the initial and the final points respectively.
The function is called the parametric equation of this curve.
EXAMPLE 16. Let Thus and
for every i.e., the point lies on the parabola
for every As varies from 0 to 1, also varies from 0 to 1 and the
point runs along the parabola from the point to the
point (Figure 16).
FIGURE 16
244. Trace on the plane the curves given by the following parametric
equations: a) b) c) d) e)
f) g)
h)
i)
245. Write a parametric equation for the segment joining the points
and
REMARK. In the following problems the parametric equations have
some indices. These numbers have to be viewed only as labels, but all
curves lie in the same plane.
246. By means of which geometrical transformations of the curve
with equation can we obtain the curve with equation if: