Plane Curves in Rectangular Coordinates

Abstract

There are many ways to classify curves. One of them is to think of curves as either algebraic or transcendental. An algebraic (plane) curve is given by a polynomial equation P(x, y) = 0. Its degree n = deg P is called the order of the curve. Curves of order n = 2 are studied in analytic geometry. The first classification of curves of order n = 3 was obtained by Newton. The case n > 3 is more difficult. But among easily obtained curves, there are many that are nonalgebraic, for example, the cycloid and spiral of Archimedes; we study them using parametrized or implicit equations (Chapter 5) or polar coordinates (see Chapter 6).