A Gallery of Algebraic Curves

Abstract

A great way to learn new mathematics is to work with examples. That's how we start. This chapter consists mostly of examples of algebraic curves in the real plane. A plane algebraic curve is defined to be the locus, or set of zeros, of a polynomial in two Cartesian variables with real coefficients. This may sound pretty special, but a surprisingly large number of familiar curves are exactly of this type. For example, many polar coordinate curves—lemniscates, limacons, all sorts of roses, folia, conchoids—are algebraic, as are many curves defined parametrically, such as Lissajous figures and the large assortment of curves obtained by rolling a circle of rational radius around a unit circle. Nearly all the curves the ancient Greeks knew are algebraic. So are many curves mechanically traced out by linkages. We begin this chapter with very simple algebraic curves, those defined by first and second degree polynomials. We then turn to curves of higher degree. CURVES OF DEGREE ONE AND TWO Definition 1.1 The degree of a monomial x m y n is m + n . The degree of a polynomial p ( x, y ) is the largest degree of its terms. The degree of a plane algebraic curve C is the degree of the lowest-degree polynomial defining C . Notation . In this book, we denote the set of all solutions of p ( x, y ) = 0 by C ( p ( x, y )) or by just C ( p ). DEGREE ONE The general form of a polynomial of degree one is Ax + By + C , where not both A and B are zero. Its zero set is a line, and conversely any line in ℝ 2 is the zero set of a polynomial of degree one.