Curves on a Surface. (Com S 477/577 Notes) Yan-Bin Jia. Oct 24, 2017

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1 Curves on a Surface (Com S 477/577 Notes) Yan-Bin Jia Oct 24, Normal and Geodesic Curvatures One way to examine how much a surface bends is to look at the curvature of curves on the surface. Let γ(t) = σ(u(t),v(t)) be a unit-speed curve in a surface patch σ. Thus, γ is a unit tangent vector to σ, and it is perpendicular to the surface normal ˆn at the same point. The three vectors γ, ˆn γ, and ˆn form a local coordinate frame by the right-hand rule. Differentiating γ γ yields that γ is orthogonal to γ. Hence γ is a linear combination of ˆn and ˆn γ: γ = κ nˆn+κ gˆn γ (1) Here κ n is called the normal curvature and κ g is the geodesic curvature of γ. ˆn ˆn γ γ φ γ κ n γ κ g σ Since ˆn and ˆn γ are orthogonal to each other, (1) implies that Since γ is unit-speed, its curvature is κ n = γ ˆn and κ g = γ (ˆn γ). κ = γ = κ nˆn+κ gˆn γ = κ 2 n +κ2 g. (2) The material is adapted from the book Elementary Differential Geometry by Andrew Pressley, Springer-Verlag, 1

2 Let ψ be the angle between the principal normal ˆn γ of the curve, in the direction of γ, and the surface normal ˆn. We have κ n = κˆn γ ˆn = κcosψ. (3) Therefore, equation (2) implies κ g = ±κsinψ. If γ is regular but arbitrary-speed, the normal and geodesic curvatures of γ are defined to be those of a unit-speed reparametrization of the same curve. A unit-speed curve γ is a geodesic if κ g = 0. By (1), its acceleration γ is always normal to the surface. Geodesics have many applications that we will devote one lecture to the topic later on. 2 Darboux Frame on a Curve On the unit-speed surface curve γ, the frame formed by the unit vectors γ, ˆn γ, and ˆn is called the Darboux frame on the curve. This frame is different from the Frenet frame on the curve defined by γ, the principal normal γ/ γ, and the binormal γ γ/ γ. Let us rename the three unit vectors γ, ˆn γ, ˆn as T, V, U, respectively. Their derivatives must be respectively orthogonal to themselves. Differentiation of U T = 0 yields U T = U T = U (κ n U +κ g V) (by (1)) = κ n. Thevector U has a second component along V. Thegeodesic torsion, definedto beτ g = U V, describes the negative rate of change of U in the direction of V. With τ g we characterize this change rate completely as below: U = κ g T τ g V. Similarly, we express V in terms of T,U,V, and combine it with the above equation and (1) into the following compact form (where the vectors are viewed as scalars ): T V U 0 κ g κ n = κ g 0 τ g κ n τ g 0 T V U (4) The formulas (4) describe the geometry of a curve at a point in a local frame suiting the curve as well as a surface on which it lies, whereas the Frenet formulas describe its geometry at the point in a local frame best suiting the curve alone. The curve γ is asymptotic provided its tangent γ always points in a direction in which κ n = 0. In some sense, the surface bends less along γ than it does along a general curve. In (4), T = κ g V. Thus, κ = κ g and V is aligned with the principal normal T / T (assuming κ g 0). Consequently, the Darboux frame T-V-U coincides with the Frenet frame everywhere. 2

3 3 The Second Fundamental Form Let σ be a surface patch in R 3 with standard unit normal ˆn = σ u σ v σ u σ v = σ u σ v EG F 2. (5) With an increase ( u, v) in the parameter values, the movement of the point is described by Taylor s series below: σ(u+ u,v + v) σ(u,v) = σ u u+σ v v + 1 ( ( σ uu ( u) 2 +2σ uv u v +σ vv ( v) 2 )+O ( u+ v) 3). 2 The first order terms are tangent to the surface, hence perpendicular to ˆn. The terms of order higher than two tend to zero as ( u) 2 +( v) 2 does so. The deviation of σ from the tangent plane is determined by the dot product of the second order term with the surface normal ˆn, namely, it is where 1 ( L( u) 2 +2M u v +N( v) 2), (6) 2 L = σ uu ˆn = σ uu (σ u σ v ) σ u σ v = det(σ uu σ u σ v ), EG F 2 by (5) M = σ uv ˆn = det(σ uv σ u σ v ), (7) EG F 2 N = σ vv ˆn = det(σ vv σ u σ v ). EG F 2 Recall that a unit-speed space curve α(s) can be approximated around s = 0 up to the second order as α(0) + sˆt(0) κ(0)s2ˆn(0), where ˆt(0) and ˆn(0) are the tangent and principal normal, respectively, and κ(0) the curvature. The expression(6) for the surface is analogous to the curvature term 1 2 κ(0)s2ˆn(0) for a curve. In particular, the expression Ldu 2 +2Mdudv +Ndv 2 is the second fundamental form of σ. While the first fundamental form permits the calculation of metric properties such as length and area on a surface patch, the second fundamental form captures how curved a surface patch is. The roles of the two fundamental forms on describing the local geometry are analogous to those of speed and acceleration for a parametric curve. Just as a unit-speed space curve is determined up to a rigid motion by its curvature and torsion, a surface patch is determined up to a rigid motion by its first and second fundamental forms. 3

4 Example 1. Consider a surface of revolution σ(u,v) = (f(u)cosv,f(u)sinv,g(u)), where f(u) > 0 always holds. The figure below plots a catenoid where ( u f(u) = 2cosh and g(u) = u 2) with (u,v) [ 2,2] [0,2π]. Assume the profile curve u (f(u), 0, g(u)) is unit-speed, i.e., f 2 + ġ 2 = 1, where a dot denotes differentiation with respect to u. We perform the following calculations: σ u = ( f cosv, f sinv,ġ), σ v = ( f sinv,f cosv,0), E = σ u σ u = f 2 +ġ 2 = 1, F = σ u σ v = 0, G = σ v σ v = f 2, σ u σ v = ( fġcosv, fġsinv,ff), σ u σ v = f, σ u σ v ˆn = σ u σ v = ( ġcosv, ġsinv, f), σ uu = ( f cosv, f sinv, g), = ( f sinv, f cosv,0), σ uv σ vv = ( f cosv, f sinv,0), L = σ uu ˆn = f g fġ, M = σ uv ˆn = 0, N = σ vv ˆn = fġ. Thus, the second fundamental form is ( f g fġ)du 2 +fġdv 2. 4

5 4 Independence of Normal Curvature to a Curve Let γ(t) = σ(u(t),v(t)) be a unit-speed curve on the surface patch σ. Below we obtain the normal curvature of the curve: κ n = γ ˆn = d γ ˆn dt = ˆn d dt (σ u u+σ v v) ( ) = ˆn σ u ü+σ v v +(σ uu u+σ uv v) u+(σ uv u+σ vv v) v = L u 2 +2M u v +N v 2, (8) where L,M,N are coefficients of the second fundamental form defined (7). In the last step above, we used the fact that ˆn is normal to both σ u and σ v, i.e., ˆn σ u = ˆn σ v = 0. Equation (8) states that the normal curvature of γ(t) depends on u, v, u, and v. The first two quantities specify the location of the point on the surface σ, and are thus curve independent. Let û be the unit tangent vector so that γ = û. Take the dot products of the equation û = σ u u+σ v v with σ u and σ v separately, yielding E u+f v = û σ u, F u+g v = û σ v, where E,F,G are the coefficients of the first fundamental form of the surface patch. Since σ u and σ v are linearly independent, the coefficient matrix in the above linear system in u and v is non-singular. We solve the system to obtain ) ) 1 ) (û σu ( u v = ( E F F G û σ v This implies that u and v are independent of the parametrization of γ. By (8), we conclude that any two unit-speed curves passing through the same point in the same direction û must have the same normal curvature at this point. Subsequently, we refer to κ n as the normal curvature of the surfaceσ atthepointp = σ(u,v)inthetangentdirectionofû. It measures the curving of the surface in that direction. Generally, the surface bends at different rates in different tangent directions. ˆn The tangent û and the normal ˆn at p defines a plane that cuts a û curve α out of the patch. This curve is called the normal section of σ in the û direction. Since the principal normal ˆn γ of the α normal section is related to the surface normal ˆn by ˆn γ = ±ˆn. The normal curvature equation σ κ n = κˆn γ ˆn implies that the curvature of the normal section is the normal curvature κ n at the point or its opposite, depending on the choice of the surface normal ˆn at the point. 5

6 References [1] B. O Neill. Elementary Differential Geometry. Academic Press, Inc., [2] A. Pressley. Elementary Differential Geometry. Springer-Verlag London,