Definition 5 of Book I of Euclid's The Elements tells us that “a surface is that which has length and breadth only.” Analytic geometry turns this definition into functions on a surface that behave like length and breadth, namely, a pair of independent coordinates that determine uniquely each point on the surface. For example, spherical coordinates, longitude and colatitude (chapter 1), apply to the surface of a sphere, and they permit new arguments via the calculus, revealing the geometry of a sphere.

Our goal in developing the classical topics of differential geometry is to discover the surfaces on which non-Euclidean geometry holds. On the way to this goal the major themes of differential geometry emerge, which include the notions of intrinsic properties, curvature, geodesics, and abstract surfaces. In this chapter we develop the analogues of notions for curves such as parameters and their transformations, tangent directions, and lengths.