ABSTRACT
In this chapter we begin a proper study of special curves lying on surfaces in ℝ3. An asymptotic curve on a surface M ⊂ ℝ3 is a curve whose velocity always points in a direction in which the normal curvature of M vanishes. In some sense, M bends less along an asymptotic curve than it does along a general curve. As a simple example, illustrated below, the straight lines on the cylinder (u,v) ↔ (cos u, sin u, v) formed by setting u constant are asymptotic curves. If p is a hyperbolic point of M (meaning that the Gaussian curvature is negative at p), there will be exactly two asymptotic curves passing through p.
