Local theory of curves
Elementary geometry gives a fairly accurate and well-established notion of what is a straight line, whereas is somewhat vague about curves in general. Intuitively, the difference between a straight line and a curve is that the former is, well, straight while the latter is curved. But is it possible to measure how curved a curve is, that is, how far it is from being straight? And what, exactly, is a curve? The main goal of this chapter is to answer these questions. After comparing in the first two sections advantages and disadvantages of several ways of giving a formal definition of a curve, in the third section we shall show how Differential Calculus enables us to accurately measure the curvature of a curve. For curves in space, we shall also measure the torsion of a curve, that is, how far a curve is from being contained in a plane, and we shall show how curvature and torsion completely describe a curve in space. Finally, in the supplementary material, we shall present (in Section 1.4) the local canonical shape of a curve; we shall prove a result (Whitney’s Theorem 1.1.7, in Section 1.5) useful to understand what cannot be the precise definition of a curve; we shall study (in Section 1.6) a particularly well-behaved type of curves, foreshadowing the definition of regular surface we shall see in Chapter 3; and we shall discuss (in Section 1.7) how to deal with curves in ℝ n when n ≥ 4.