We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
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, limaçons, 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 xm yn 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 ℝ2 is the zero set of a polynomial of degree one.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
