Differential Geometry of Curves and Surfaces, 2nd Edition

CHAPTER 1

Plane Curves: Local Properties

Just as calculus courses introduce real functions of one variable before tackling multivariable calculus, so it is natural to study curves before addressing surfaces and higher-dimensional objects. This first chapter presents local properties of plane curves, where by local property we mean properties that are defined in a neighborhood of a point on the curve. For the sake of comparison with calculus, the derivative f′(a) of a function f at a point a is a local property of the function since we only need knowledge of f(x) for x in (a − ε, a + ε), where ε is any positive real number, to define f′(a). In contrast, the definite integral of a function over an interval is a global property since we need ...