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Differential Geometry of Curves & Surfaces
Revised & Updated
By Manfredo P. Do Carmo Dover Publications, Inc.
Copyright © 2016 Manfredo P. do Carmo
All rights reserved.
ISBN: 978-0-486-80699-0
CHAPTER 1
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 α: I -> R3of an open interval I = (a, b) of the real line R into R3.
The word differentiable in this definition means that α is a correspondence which maps each t [member of] I into a point α (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 α = -∞, b = +∞.
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)) = α'(t) [member of] R3 is called the tangent vector (or velocity vector) of the curve α at t. The image set α(I) [subset] R3 is called the trace of α. 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
α(t) = (a cos t, a sin t, bt), t [member of] R,
has as its trace in R3 a helix of pitch 2πb 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 α(t) over the xy plane (see Fig. 1-1).
Example 2. The map α: R -> Rsup>2 given by α(t) = (t3t3, t2 [member of] R, is a parametrized differentiable curve which has Fig. 1-2 as its trace. Notice that α'(0) = (0,0); that is, the velocity vector is zero for t = 0.
Example 3. The map α: R ->R2 given by α (t) = (t3 - 4t, t2] - 4), t [member of] R, is a parametrized differentiable curve (see Fig. 1-3). Notice that α(2) = α(-2) = (0, 0); that is, the map α is not one-to-one.
Example 4. The map α: R ->R2 given by α(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
α(t) = (cos t, sin t),
β(t) = (cos 2t, sin 2t),
where t [member of] (0 - [??], 2π + [??]), [??] > 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 π 0 ≤ φ ≤ π, 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| cosθ.
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. λ(u x v) = λu x v = u x λv.
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 trace is the circle x2 + y2 = 1 such that α(t) runs clockwise around the circle with α(0) = (0, 1).
2. Let α(t) be a parametrized curve which does not pass through the origin. If α (t)0 is a point of the trace of α closest to the origin and α'(t)0 ≠ 0, show that the position vector α(t)0 is orthogonal to α'(t)0.
3. A parametrized curve [α(t) has the property that its second derivative [α"(t) is identically zero. What can be said about α?
4. Let α: I ->R3 be a parametrized curve and let v [member of] R3 be a fixed vector. Assume that a]'(t) is orthogonal to v for all t [member of] I and that α(0) is also orthogonal to v. Prove that α(t) is orthogonal to v for all t [member of] I.
5. Let α: I ->R3 be a parametrized curve, with α'(t) ≠ 0 for all t [member of] I. Show that |α(t)| is a nonzero constant if and only if α(t)) is orthogonal to α'(t) for all t [member of] I.
1-3. Regular Curves; Arc Length
Let α: I ->R3 be a parametrized differentiable curve. For each t [member of] I where α'(t) [not member of] 0, there is a well-defined straight line, which contains the point α: I and the vector α' (t). This line is called the tangent line to α at t. For the study of the differential geometry of a curve it is essential that there exists such a tangent line at every point. Therefore, we call any point t where α'(t) = 0 a singular point of α and restrict our attention to curves without singular points. Notice that the point t = 0 in Example 2 of Sec. 1-2 is a singular point.
DEFINITION.A parametrized differentiable curve α: I -> R3is said to be regular if α'(t) ≠ 0 for all t [member of] I.
From now on we shall consider only regular parametrized differentiable curves (and, for convenience, shall usually omit the word differentiable).
Given [t.sub.0] [member of] I, the arc length of a regular parametrized curve α: I ->R3 from the point t0, is by definition
[MATHEMATICAL EXPRESSION OMITTED]
where?
[MATHEMATICAL EXPRESSION OMITTED]
is the length of the vector α'(t)ITL. Since α'(t) ≠ 0, the arc length s is a differentiable function of t and ds/dt = |α'(t)]|.
In Exercise 8 we shall present a geometric justification for the above definition of arc length.
It can happen that the parameter t is already the arc length measured from some point. In this case, ds/dt = 1 = |α'(t)|; that is, the velocity vector has constant length equal to 1. Conversely, if |α'(t)| [equivalent] 1, then
[MATHEMATICAL EXPRESSION OMITTED]
i.e., t is the arc length of α measured from some point.
To simplify our exposition, we shall restrict ourselves to curves parametrized by arc length; we shall see later (see Sec. 1-5) that this restriction is not essential. In general, it is not necessary to mention the origin of the arc length s, since most concepts are defined only in terms of the derivatives of α(s).
It is convenient to set still another convention. Given the curve α parametrized by arc length s [member of] (a, b), we may consider the curve β defined in (-b, -a) by β(-s) = α(s), which has the same trace as the first one but is described in the opposite direction. We say, then, that these two curves differ by a change of orientation.
EXERCISES
1. Show that the tangent lines to the regular parametrized curve α(t) = (3t, 3t2, 2t3) make a constant angle with the line y = 0, z = x.
2. A circular disk of radius 1 in the plane xy rolls without slipping along the x axis. The figure described by a point of the circumference of the disk is called a cycloid (Fig. 1-7).
*a. Obtain a parametrized curve α: R -> R2 the trace of which is the cycloid, and determine its singular points.
b. Compute the arc length of the cycloid corresponding to a complete rotation of the disk.
3. Let 0A = 2a be the diameter of a circle S1 and 0y and AV be the tangents to S1 at 0 and A, respectively. A half-line r is drawn from 0 which meets the circle S1 at C and the line AV at B. On 0B mark off the segment 0p = CB. If we rotate r about 0, the point p will describe a curve called the cissoid of Diocles. By taking 0A as the x axis and 0Y as the y axis, prove that
a. The trace of
[MATHEMATICAL EXPRESSION OMITTED]
is the cissoid of Diocles (t = tan θ; see Fig. 1-8).
b. The origin (0, 0) is a singular point of the cissoid.
c. As t -> ∞, α(t)) approaches the line x = 2a, and α'(t) -> 0, 2a. Thus, as t [right arrow] ∞, the curve and its tangent approach the line x = 2a; we say that x = 2a is an asymptote to the cissoid.
4. Let α: (0, π) ->R2 be given by
α(t) = (sin , cost t + log tan t/2),
where t is the angle that the y axis makes with the vector α'(t). The trace of α is called the tractrix (Fig. 1-9). Show that
a. α is a differentiable parametrized curve, regular except at t = π/2.
b. The length of the segment of the tangent of the tractrix between the point of tangency and the y axis is constantly equal to 1.
5. Let α: (-1, +∞) ->Rsup>2 be given by
α(t) = (3at/ 1 + t3, 3at2/ 1 + t3).
Prove that:
a. For t = 0, α is tangent to the x axis.
b. As t -> + ∞, a(t) -> (0, 0) and α'(t) -> (0, 0).
c. Take the curve with the opposite orientation. Now, as t -> -1, the curve and its tangent approach the line x + y + a = 0.
The figure obtained by completing the trace of α in such a way that it becomes symmetric relative to the line y = x is called the folium of Descartes (see Fig. 1-10.
6. Let α(t) = (aebt cos t], aebt] sin t), t [member of] R, a and b constants, a > 0, b< 0, be a parametrized curve.
a. Show that as t -> +∞, a(t) approaches the origin 0, spiraling around it (because of this, the trace of α is called the logarithmic spiral; see Fig. 1-11).
b. Show that α'(t)K -> (0, 0) as t -> +∞ and that
[MATHEMATICAL EXPRESSION OMITTED]
is finite; that is, α has finite arc length in t0, ∞).
Figure 1-11. Logarithmic spiral.
7. A map α I ->R3 is called a curve of class Ck] if each of the coordinate functions in the expression α(t) = (x(t)], y(t), z(t)) has continuous derivatives up to order k. If α is merely continuous, we say that α is of class C0. A curve α is called simple if the map α is one-to-one. Thus, the curve in Example 3 of Sec. 1-2 is not simple.
(Continues...)
Excerpted from Differential Geometry of Curves & Surfaces by Manfredo P. Do Carmo. Copyright © 2016 Manfredo P. do Carmo. Excerpted by permission of Dover Publications, Inc..
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