Do Carmo, Differential Geometry Of Curves And Surfaces.pdf 164io

2

Curves

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 \Ve denote by R 3 the set of triples (x, y, z) of real numbers. Our goal is to characterize certain subsets of R 3 (to be called curves) that are, in a certain sensc, one-dimcnsional and to which the mcthods of differential calculus can be applied. A natural way of defining such subsets is through differenti­ able functions. We say that a real function of a real variable is dijferentiable (or smooth) ifit has, at all points, derivatives ofall orders (which are automa­ tically continuous). A first de:finítion of curve, not entirely satisfactory but sufficient for the purposes of this chapter, is the following. DEFINITION. A parametrized differentiable curve is a diflerentiable map (J.: r- R 3 of an open interval I = (a, b) ofthe real line R into R 3 .t The word d(/ferentíab!e in this definition means that (1, is a correspondence which maps each t E I into a point (J.(t) = (x(t), y(t), z(t)) E: R 3 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= -oo, b = +oo. Ifwe denote by x'(t) the first derivative ofx at the point t and use similar notations for the functions y and z, the vector (x'(t), y'(t), z'(t)) = (1, (t) E R 3 is called the tangent vector (or velocity vector) of thc curve (1, at t. Thc image set (J.(/) e R 3 is called the trace of (J.. As illustrated by Example 5 below, one should carcfully distinguish a parametrized curve, which is a map, from its trace, which is a subset of R 3 • 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 deriva­ tive is rcquired. We shall not follow this usage. 1

Example l. The parametrized differentiable curve given by cx(t)

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

t E R,

has as its trace in R 3 a helix of pitch 2nb on the cylinder x2 + y 2 = a 2 • The parameter t here measures the angle which the x axis makes with the line ing the origin O to the projection of the point cx(t) over the xy plane (see Fig. 1-1). tin italic context, letter symbols will not be italicized so they will be clearly distin­ guished from the surrounding text.

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Curves

where t E (O - f, 2n + f), f > O, havc the same trace, namely, the circle x 2 + y 2 = L Notice that the velocity vector of the second curve is the double of the first one (Fig. 1-5).

y

/ o Figure 1-5

We shall now recall briefly sorne properties of the inner (or dot) product of vectors in R 3 . Let u = (u 1 , u 2 , u 3 ) E R 3 and define its norm (or length) by

Geometrically, 1 u j is the distance from the point (u 1 , u 2 , u 3 ) to the origin O = (O, O, O). Now, let u = (u 1 , u2 , u 3 ) and v = (v 1 , v 2 , v 3 ) belong to R 3 , and let (),O<()< n, be the angle formed by the segments Ou and Ov. The inner product u· v is defined by (Fig. 1-6) U• V �-=- j U 11

VI COS (},

z

7 u¡

... y

X

Figure 1-6 The following properties hold: l. Assume that u and v are nonzero vectors. Then u• v = O if and only if u is orthogonal to v.

Curves Parametrized by Are Length

19

Let us summarize our position. To each value of the parameter s, we have associated three orthogonal unit vectors t(s), n(s), b(s). The trihedron thus formed is referred to as the Frenet trihedron at s. The derivatives t'(s) = kn, b'(s) = rn of the vectors tls) and b(s), when expressed in the basis {t, n, b}, yield geometrical entities (curvature k and torsion r) which give us infor­ mation about the behavior of rx in a neighborhood of s. The search for other local geometrical entities would lead us to compute n'(s). However, since n = b /\ t, we have n'(s)

= b'(s)

¡\ t(s)

+ b(s) /\ t'(s) = -rb - kt,

and we obtain again the curvature and the torsion. For later use, we shall call the equations t' = kn' n' = -kt - rb,

b' = rn the Frenetformulas (we have ommited the s, for convenience). In this context, the following terminology is usual. The tb plane is called the rectifying plane, and the nb plane the normal plane. The lines which contain n(s) and b(s) and through rx(s) are called the principal normal and the binormal, respecti­ vely. The inverse R = l/k of the curvature is called the radius of curvature at s. Of course, a circle of radius r has radius of curvature equa1 to r, as one can easily . Physically, we can think of a curve in R 3 as being obtained from a straight line by bending (curvature) and twisting (torsion). After reflecting on this construction, we are led to conjecture the following statement, which, roughly speaking, shows that k and r describe completely the local behavior of the curve. FUNDAMENTAL THEOREM OF THE LOCAL THEORY OF CURVES. Given dif.ferentiable functions k(s) > O and r(s), s E I, there exists a regular parametrized curve a,: I - R 3 such that sis the are length, k(s) is the curvature, and r(s) is the torsion of a. Moreover, any other curve ii, satisfying the same conditions, dif.fers from a, by a rigid motion; that is, there exists an orthogonal linear map p of R 3, with positive determinant, anda vector e such that iX = porx + c. The above statement is true. A complete proof involves the theorem of existence and uniqueness of solutions of ordinary differential equations and will be given in the appendix to Chap. 4. A proof of the uniqueness, up to

Curves

22

s = s(t) =

f la'Ct)I dt, to

t, t0 E

[.

Since ds/dt = 1 !X'(t) 1 =,t:. O, the function s = s(t) has a differentiable inverse t = t(s),. s E s(I) = J7 where; by an abuse of notation_, t also denotes the inverse function s- 1 of s. Now set p = aot: J - , R 3 • Clearly, p(J) = a(J) and I P'(s) 1 = 1 a'(t) · (dt/ds) J = l. This shows that p has the same trace as a and is parametrized by are length. It is usual to say that pis a reparametriza­ tion of !X(I) by are length. This fact allows us to extend all local concepts previously defined to regular curves with an arbitrary parameter. Thus, we say that the curvature k(t) of a: I--;. R 3 at t E I is the curvature of a reparametrization p: J--;. .R... 3 of a(I) by are length at the corresponding point s = s(t). This is clear]y independent of the choice of p and shows that the restriction, made at the end of Sec. 1-3. of considering only curves parametrized by are length is not essential. In applications, it is often convenient to have explicit formulas for the geometrical entities in of an arbitrary parameter; we shall present sonie of them in Exercise 12. EXERCJSES

*

Unless explicity stated, !X: I --, R 3 is a curve parametrized by are length s, with O, Jor ali s E I. curvature k(s)

l. Given the paramet rized curve (helix) !X(s) = (a cos : , a sin : , b : ),

s

E

R,

where c 2 = a 2 + b2, a. Show that the parameter s is the are length. b. Determine the curvature and the torsion of rt.,. c. Determine the osculating plane of IX. d. Show that the lines containing n(s) and ing through !X(s) meet the z aJt:is under a comstant angle equal to n/2. e. Show that the tangent Iines to IX make a constant angle with the z axis. *2. Show that the torsioµ T of oc is given by /\ IX"(s)· IX"' (s) r(s) = - IX'(s) lk(s) ¡2 3. Assume that IX([) e R 2 (i.e., IX is aplane curve) and give ka sign as in the text. Transport the vectors t(s) parallel to themselves in such a way that the orígins of

Figure 1-17

a. Show that the signed curvature (cf. Remark 1) of the catenary is 1 • k(t) - cosh 2 t

b. Show that the evolute (cf. Exercise 7) of the catenary is PU) = (t - sinh t cosh t, 2 cosh t). 9. Given a differentiable function k(s), s E I, show that the parametrized planc curve having k(s) = k as curvature is gi ven by IX(s) =

(f cos 8(s) ds + a, f sin 8(s) ds + b),

where 8(s) -

f k(s) ds + rp,

and that the curve is determined up to a translation of the vector (a, b) and a rotation of the angle rp. 10. Consider the map

(t, O, e- 1 1r2), IX(t) = {(t, e- l/r•, 0), (O, O, O),

t>O t
a. Prove that IX is a differentiable curve. b. Prove that IX is regular for all t and that the curvature k(t) t =/= ± v2/3, and k(O) = O.

* O, for * O, t

Curves Parametrized by Are Length

25

c. Show that the Iimit of the osculating planes as t ---- O, t > O, is the plane y = O but that the limit of the osculating planes as t - -" O, t < O, is the plane z = O (this implies that the normal vector is discontinuous at t = O and shows why we excluded points where k = O). d. Show that r can be defined so that r ce O, cvcn though rx is not a plane curve. 11. Onc oftcn givcs aplane curve in polar coordinates by p = p(O), a < B

< b.

a. Show that the are lcngth is

f:,,;p

2

+ (p') 2 dO,

where the prime denotes thc dcrivative relative to O. b. Show that thc curvature is k(fJ) =

2(p') 2 - pp'' + p 2 {(p ')2 + p2}3/2 .

12. Let rx: I ---- R 3 be a regular parametrized curve (not necessarily by are Iength) and Jet P: J �, R 3 be a reparametrization of rx(I) by the are length s = s(t), measurcd from t 0 e I (scc Rcmark 2). Lct t = t(s) be thc inverse function of s and set da/dt = a', d 2rx/dt 2 = rx", etc. Prove that a. dt/ds = 1/[rx'I, d 2 t/ds 2 = · (ri'·rt."/Jrt.'[ 4 ). b. The curvature of a at t E I is k(t) =

c. The torsion of a at t

E::

1 a' /\ rx" 1 .

1 a' 13

I is

(a'/\ a")·rt." r(t) = - I rxl /\ rx'I ¡2 ' · d. If a: I----+ R 2 is a plane curve rt.(t) '--' (x(t), y(t)), the signed curvature (see Remark 1) of rx at t is

*

k(t) =

x'yn - x"y' ((x')2 + (y/)2)3/2

*

�13. Assume that r(s) O and k'(s) O for all s E J. Show that a necessary and suflicient condition for rx(I) to lie on a sphere is that R2

+ (R;2T 2 = const.,

where R = l/k, T = 1/r, and R' is the derivative of R relative to s. 14. Let rx: (a, b) - R 2 be a regular parametrized plane curve. Assume that there exists t 0, a < t O < b, such that the distance I rx(t) 1 from the origin to the trace of