LECTURE 1
Rational and Polynomial Parametrizations
Let us begin with some simple examples of plane curves.
Conic Sections. These include the circle, ellipse, hyperbola, and parabola
defined by the equations I
2
+ F
2
= l ,
X2/a2
+
Y2/b2
= 1, XY = 1, and
Y = X respectively. We are quite familiar with their graphs as depicted in
Figure 1.1.
(a)
Circle. X
2
+ y
2
= l
Y2 Y2
Ellipse. ^ +r ? = 1
a2 b2
(c)
(d)
Hyperbola. XY = 1
Parabola. Y =X
FIGURE 1.1
Further, we can similarly consider
Surfaces in 3-space. Some examples are shown in Figure 1.2 on page 2.
Now the examples of the plane curves that we considered previously are
plane curves of degree 2. We can also consider curves of higher degree such
as those shown in Figures 1.3-1.5 on pages 2 and 3.
Notice that near the origin we may think of the tacnodal quartic as the
!
http://dx.doi.org/10.1090/surv/035/01
Previous Page Next Page