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Chapter 6 Some Special Curves on Surfaces Chapter 6 Some Special Curves on Surfaces In this chapter we introduce three families of curves on surfaces, namely, principal, asymptotic and geodesic curves. Principal curves follow paths of maximal bending, asymptotic curves follow paths of zero bending and geodesicscurvesalwaysgo"striaghtahead"inthesurface. 6.1 Vector Fields Along Curves Our discussion begins with an introduction to vector fields along curves. Definition 135 Let α :(a, b) R3 be a curve given componentwise by α(t)= (x(t),y(t),z(t)).Avectorfield→V along α is a function defined on (a, b) such that V(t) R3 for all t (a, b). Each vector field V along α has the form ∈ α(t) ∈ V(t)=(V1(t),V2(t),V3(t))α(t) , where Vi :(a, b) R. Avectorfield V along α is differentiable if each Vi is differentiable; the→ derivative of a differentiable vector field V along α is the vector field V0 along α defined by V0(t)=(V10(t),V20(t),V30(t))α(t) . The velocity vector field along α is defined by α0(t)=(x0(t),y0(t),z0(t))α(t) ; the acceleration vector field along α is defined by 93 94 CHAPTER 6. SOME SPECIAL CURVES ON SURFACES α00(t)=(x00(t),y00(t),z00(t))α(t) . All vector fields henceforth are assumed to be infinitely differentiable, i.e., smooth. Definition 136 Let α :(a, b) R3 be a curve, let f :(a, b) R be a differentiable function and let V→, W be vector fields along α. For al→ l t (a, b), define ∈ a. (V + W)(t)=V(t)+W(t) b. (fV)(t)=f(t)V(t) c. (V W)(t)=V(t) W(t). • • The proof of the next Proposition is left for the reader. Proposition 137 Differentiation of vector fields has the following proper- ties: a. (V + W)0= V0 + W0 b. (fV)0=f 0V+fV0 c. (V W)0= V0 W + V W0. • • • 6.2 Principal, Asymptotic & Geodesic Curves Now suppose that the image of a curve α is contained in the image of some regular patch x. Define Definition 138 A tangent vector vp p is principal if k (vp)=S (vp) ∈ M • vp = k1 or k2; it is asymptotic if k (vp)=S (vp) vp =0. Directions in • p that contain principal tangent vectors are called principal directions;di- M rections in p that contain asymptotic tangent vectors are called asymptotic directions. M Note that in an asymptotic direction at p, does not bend away from M p (at least instantaneously at p). M 6.2. PRINCIPAL, ASYMPTOTIC & GEODESIC CURVES 95 Definition 139 Acurveα is a principal curve if α0 is a principal tan- ⊂ M gent vector field along α; α is an asymptotic curve if α0 (t) is an asymptotic tangent vector field along α. Principal curves travel in directions of extreme bending. If p is a nonum- bilic point of , there are exactly two principal curves through p intersecting each other orthogonallyM at p. Asymptotic curves, on the other hand, travel in directions of no bending. Since normal curvature k measures the com- ponent of acceleration normal to ,normalcurvaturek =0implies that M acceleration α00 along an asymptotic curve α is always tangent to . Principal curves on a cylinder are circular crossections and linesM parallel to the axis; these lines are also asymptotic curves. Principal curves on the hyperbolic paraboloid z = xy are the parabolas z = x2; the asymptotic curves are the xy-coordinate axes xy =0. ± Lemma 140 Let α be a regular curve in and let U be a surface normal. M a. α is a principal curve if and only if Dα U (α)=λα0. − 0 b. If α is principal curve, then the principal curvature α00 U (α) k (α0)= • . α 2 k 0k Proof. For part a, note that Dα U (α)=λα0 if and only if S (α0)=λα0 − 0 if and only if λ (t) is a principal curvature at α (t) for all t if and only if α0 is a principal tangent vector field along α. For part b, the vector field α0 consists of principal tangent vectors belonging to some principal curvature, say ki. Hence S (α0) α0 α00 U (α) ki = k (α0)= • = • , α 2 α 2 k 0k k 0k where the last equality is given by Lemma 101. In this lemma, statement (a) gives a simple criterion for a curve to be a principal curve, while statement (b) gives the principal curvature along a curve known to be principal. 96 CHAPTER 6. SOME SPECIAL CURVES ON SURFACES Definition 141 Let be the image of a regular patch. A geodesic on M M is a curve α :(a, b) whose acceleration vector field α00 is always in the direction normal to→ M, i.e., if U is a surface normal, then M α00 = λU (α) . Thusageodesicisacurvethatalwaysgoes“straightahead”inthe surface; its acceleration only serves to keep it in the surface. There is no component of the acceleration tangent to the surface. Let p⊥ denote the 3 M orthogonal complement of p in Rp. M Example 142 Suppose the image of a regular patch contains a straight M line segment α(t)=p+tv, where t (a, b). Then α00(t)=0 for all t and ∈ in particular, α00(t) vα(t) =0for all tangent vectors vα(t) α(t). Hence • ∈M α00(t) α⊥(t), for all t and it follows that straight line segments on a regular patch∈ areM geodesics. Example 143 Let be the cylinder x2 + y2 =1thought of as the level surface for g(x, y, z)=Mx2+y2 1 of height zero. The gradient is g(x, y, z)= (2x, 2y, 0), so a surface normal− U on is given by 5 M U(x, y, z)=(x, y, 0)p, where p =(x, y, z) . For each a, b, c, d, the curve ∈ M α(t)=(cos(at + b), sin(at + b),ct+ d) is a geodesic in the cylinder because 2 2 2 α00(t)= a cos(at + b), a sin(at + b), 0 = a U(α(t)), − − α(t) − for all t. Furthermore,¡ the geodesic is a ¢ 1. (a) Helix if a, c =0: 6 α(t)=(cos(at + b), sin(at + b),ct+ d) (b) Line if a =0and c =0: 6 α(t)=(cos(b), sin(b),ct+ d) 6.2. PRINCIPAL, ASYMPTOTIC & GEODESIC CURVES 97 (c) Circle if a =0and c =0: 6 α(t)=(cos(at + b), sin(at + b),d) (d) Point if a, c =0: α(t)=(cos(b), sin(b),d) . Example 144 Let be the sphere x2 + y2 + z2 = r2.Slice with a plane Γ through theM origin and obtain a great circle C = ΓM.Choose M ∩ 3 any orthonormal basis e1, e2 for Γ (thought of as a subspace of R ) and parametrize C by { } α(t)=(r cos at) e1 +(r sin at) e2, with t R and a =0. As with the cylinder, there is a surface normal U on given∈ by 6 M x y z U(x, y, z)= , , , r r r (x,y,z) where (x, y, z) . Then for all t, ³ ´ ∈M 2 2 2 2 α00(t)= a r cos(at)e1 a r sin(at)e2 = a α(t)α(t) = a (U α)(t) , − − α(t) − − ◦ so the great¡ circle C is a geodesic. Since¢ Γ and e1 were arbitrarily chosen, every great circle is a geodesic. Furthermore, if α(t)=re1 is a constant curve, then α00(t)=0 r⊥e1 for all t, and the point re1 is a geodesic. We conclude that points and∈ M great circles on a sphere are geodesics. Proposition 145 Geodesic on the image of a regular patch have constant speed. Proof. Let be the image of a regular patch and let α :(a, b) be M → M a geodesic. Then α0(t) α(t) and α00(t) α⊥(t) so that α0(t) α00(t)=0. Therefore ∈M ∈M • d 2 d α0(t) = [α0(t) α0(t)] = 2α0(t) α00(t)=0, dt k k dt • • so the speed is constant. Here is a simple geometric way to find examples of geodesics. Let U be a surface normal on .WesaythataplaneΠ is orthogonal to at p if U (p) Π. M M ⊂ 98 CHAPTER 6. SOME SPECIAL CURVES ON SURFACES Proposition 146 If Π is a plane everywhere orthogonal to and α is a unit speed curve in Π, then α is a geodesic of . M M∩ M Proof. Since α has constant speed, α0 α0 = c so that α0 α00 =0. • • Furthermore, α0, α00 Π since α Π. By assumption, U (α (t)) Π for all t. ⊂ ⊂ ⊂ Therefore α00 (t)=λ (t) U (α (t)) and is α is a geodesic. We could have used this proposition to find the geodesics in all of the examples above except the helices on the cylinder. Note that meridians on a surface of revolution are geodesics since they are cut from by a plane passing through the axis. M Exercise 6.2.1 Let α(t)=(x(t),y(t)) be a constant speed curve in the upper half plane y>0 with no self-intersections and let be the surface of revolution obtained by revolving the trace of α about theMx-axis. For each t (a, b) define βt : R by ∈ →M βt(θ)=(x(t),y(t)cosθ, y(t)sinθ). The circles βt are called the parallels on . Show that a parallel βt is a y t M geodesic if and only if the slope 0( ) of the line tangent to α at α (t) is zero.
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