Curves

1-1. Introduction

The differential geometry of curves and surfaces has two aspects. One, which may be called classical differential geometry, started with the beginnings of calculus. Roughly speaking, classical differential geometry is the study of local properties of curves and surfaces. By local properties we mean those properties which depend only on the behavior of the curve or surface in the neighborhood of a point. The methods which have shown themselves to be adequate in the study of such properties are the methods of differential calculus. Because of this, the curves and surfaces considered in differential geometry will be defined by functions which can be differentiated a certain number of times.

The other aspect is the so-called global differential geometry. Here one studies the influence of the local properties on the behavior of the entire curve or surface. We shall come back to this aspect of differential geometry later in the book.

Perhaps the most interesting and representative part of classical differential geometry is the study of surfaces. However, some local properties of curves appear naturally while studying surfaces. We shall therefore use this first chapter for a brief treatment of curves.

The chapter has been organized in such a way that a reader interested mostly in surfaces can read only Secs. 1-2 through 1-5. Sections 1-2 through 1-4 contain essentially introductory material (parametrized curves, arc length, vector product), which will probably be known from other courses and is included here for completeness. Section 1-5 is the heart of the chapter and contains the material of curves needed for the study of surfaces. For those wishing to go a bit further on the subject of curves, we have included Secs. 1-6 and 1-7.

1-2. Parametrized Curves

We denote by R3 the set of triples (x, y, z) of real numbers. Our goal is to characterize certain subsets of R3 (to be called curves) that are, in a certain sense, one- dimensional and to which the methods of differential calculus can be applied. A natural way of defining such subsets is through differentiable functions. We say that a real function of a real variable is differentiable (or smooth) if it has, at all points, derivatives of all orders (which are automatically continuous). A first definition of curve, not entirely satisfactory but sufficient for the purposes of this chapter, is the following.

DEFINITION.A parametrized differentiable curve is a differentiable map a: I -> R3of an open interval I = (a, b) of the real line R into R3.

The word differentiable in this definition means that a is a correspondence which maps each t [member of] I into a point a (t) = (x t), y (t), z(t)) [member of] R3 in such a way that the functions x (t), y (t), z (t) are differentiable. The variable t is called the parameter of the curve. The word interval is taken in a generalized sense, so that we do not exclude the cases a = -8, b = +8.

If we denote by x'(t) the first derivative of x at the point t and use similar notations for the functions y and z, the vector (x'(t), y'(t), z'(t)) = a'(t) [member of] R3 is called the tangent vector (or velocity vector) of the curve a at t. The image set a(I) [subset] R3 is called the trace of a. As illustrated by Example 5 below, one should carefully distinguish a parametrized curve, which is a map, from its trace, which is a subset of R3

A warning about terminology. Many people use the term "infinitely differentiable" for functions which have derivatives of all orders and reserve the word "differentiable" to mean that only the existence of the first derivative is required. We shall not follow this usage.

Example 1. The parametrized differentiable curve given by

a(t) = (a cos t, a sin t, bt), t [member of] R,

has as its trace in R3 a helix of pitch 2pb on the cylinder x2 + y2 = a2 The parameter t here measures the angle which the x axis makes with the line joining the origin 0 to the projection of the point a(t) over the xy plane (see Fig. 1-1).

Example 2. The map a: R -> Rsup>2 given by a(t) = (t3t3, t2 [member of] R, is a parametrized differentiable curve which has Fig. 1-2 as its trace. Notice that a'(0) = (0,0); that is, the velocity vector is zero for t = 0.

Example 3. The map a: R ->R2 given by a (t) = (t3 - 4t, t2] - 4), t [member of] R, is a parametrized differentiable curve (see Fig. 1-3). Notice that a(2) = a(-2) = (0, 0); that is, the map a is not one-to-one.

Example 4. The map a: R ->R2 given by a(t)) = (t, |t]|), t [member of] R, is not a parametrized differentiable curve, since |t]| is not differentiable at t = 0 (Fig. 1-4).

Example 5. The two distinct parametrized curves

a(t) = (cos t, sin t),

ß(t) = (cos 2t, sin 2t),

where t [member of] (0 - [??], 2p + [??]), [??] > 0, have the same trace, namely, the circle x2 + y2 = 1. Notice that the velocity vector of the second curve is the double of the first one (Fig. 1-5).

We shall now recall briefly some properties of the inner (or dot) product of vectors in R3. Let u = (u1, u2, u3) [member of] R3 and define its norm (or length) by

|u| = [square root of u21 + u22 + u23.

Geometrically, [absolute value of u] is the distance from the point (u1, u2, u3 to the origin 0 = (0,0,0). Now, let u = (u1, u2, u3) and v = (v1, v2, v3) belong to R3, and let p 0 = f = p, be the angle formed by the segments 0 u and 0 v. The inner product u x v is defined by (Fig. 1-6)

u x v = |u||v| cos?.

The following properties hold:

1. Assume that u and v are nonzero vectors. Then u x v = 0 if and only if u is orthogonal to v.

2. u x v = v x u.

3. ?(u x v) = ?u x v = u x ?v.

4. u x (v + w) = u x v + u x w.

A useful expression for the inner product can be obtained as follows. Let [e1] = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1). It is easily checked that ei x ej = 1 if i = j and that ei x ej = 0 if i ? j, where i, j = 1, 2, 3. Thus, by writing

u = u1 e1 + u2 e2 + u3 e3, v = v1 e1 + v2 e2 + v3e3,

and using properties 2 to 4, we obtain

u x v = u1 v1 + u2 v2 + u3 v3.

From the above expression it follows that if u(t) and v(t), t [member of] I, are differentiable curves, then u(t) x v(t) is a differentiable function, and

d/dt (u(t) x v(t)) = u'(t) x v(t) + u(t) x v'(t).

EXERCISES

1. Find a parametrized curve a(t) whose...