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

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http://dx.doi.org/10.1090/surv/035/01