Surface Curvatures∗ (Com S 477/577 Notes)

Home , Darboux frame, Principal curvature, Surface

Yan-Bin Jia

Oct 23, 2017

1 Curve on a Surface: 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 nˆ at the same point. The three vectors γ˙ , nˆ γ˙ , and nˆ 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 nˆ and · nˆ γ˙ : × γ¨ = κnnˆ + κgnˆ γ˙ (1) × Here κn is called the normal curvature and κg is the geodesic curvature of γ.

nˆ

γ¨ γ˙ nˆ γ˙ × φ κn κg γ

Since nˆ and nˆ γ˙ are orthogonal to each other, (1) implies that × κn = γ¨ nˆ and κg = γ¨ (nˆ γ˙ ). · · × Since γ is unit-speed, its curvature is κ = γ¨ k k = κnnˆ + κgnˆ γ˙ k × k 2 2 = κn + κg. (2) q ∗The material is adapted from the book Elementary Diﬀerential Geometry by Andrew Pressley, Springer-Verlag, 2001.

1 Let ψ be the angle between the principal normal nˆ γ of the curve, in the direction of γ¨, and the surface normal nˆ. We have κn = κnˆ γ nˆ = κ cos ψ. (3) · Therefore, equation (2) implies κg = κ sin ψ. ± 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 κg = 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˙ , nˆ γ˙ , and nˆ 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¨ . k k × k k Let us rename the three unit vectors γ˙ , nˆ γ˙ , nˆ as T , V , U, respectively. Their derivatives × must be respectively orthogonal to themselves. Differentiation of U T = 0 yields · ′ ′ U T = U T · − · = U (κnU + κgV ) (by (1)) − · = κn. − ′ ′ The vector U has a second component — along V . The geodesic torsion, defined to be τg = U V , − · describes the negative rate of change of U in the direction of V . With τg we characterize this change rate completely as below: ′ U = κgT τgV. − − 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 0 κg κn T  V  =  κg 0 τg   V  (4) − U κn τg 0 U    − −    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 κn = 0. ′ In some sense, the surface bends less along γ than it does along a general curve. In (4), T = κgV . ′ ′ Thus, κ = κg and V is aligned with the principal normal T / T (assuming κg = 0). Consequently, k k 6 the Darboux frame T -V -U coincides with the Frenet frame everywhere.

2 3 The Second Fundamental Form

Let σ be a surface patch in R3 with standard unit normal

σu σv σu σv nˆ = × = × 2 . (5) σu σv √EG F k × k − 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) − 1 2 2 3 = σu∆u + σv∆v + σuu(∆u) + 2σuv∆u∆v + σvv(∆v) )+ O (∆u +∆v) . 2 The ﬁrst order terms are tangent to the surface, hence perpendicular to nˆ. 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 nˆ, namely, it is

1 2 2 L(∆u) + 2M∆u∆v + N(∆v) , (6) 2 where

L = σuu nˆ · σuu (σu σv) = · × by (5) σu σv k × k det(σ σ σ ) = uu u v , √EG F 2 − det(σuv σu σv) M = σuv nˆ = , (7) · √EG F 2 − det(σvv σu σv) N = σvv nˆ = . · √EG F 2 − Recall that a unit-speed space curve α(s) can be approximated around s = 0 up to the second 1 2 order as α(0) + sˆt(0) + 2 κ(0)s nˆ(0), where ˆt(0) and nˆ(0) are the tangent and principal normal, respectively, and κ(0) the curvature. The expression (6) for the surface is analogous to the curvature 1 2 term 2 κ(0)s nˆ(0) for a curve. In particular, the expression

2 2 Ldu + 2Mdudv + Ndv 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 Example 1. Consider a surface of revolution σ(u, v) = (f(u)cos v,f(u) sin v,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 +g ˙ 2 = 1, where a dot denotes diﬀerentiation with respect to u.→ We perform the following calculations:

σu = (f˙ cos v, f˙ sin v, g˙),

σv = ( f sin v,f cos v, 0), − 2 2 E = σu σu = f˙ +g ˙ = 1, · F = σu σv = 0, · 2 G = σv σv = f , · σu σv = ( fg˙ cos v, fg˙ sin v,ff˙), × − − σu σv = f, k × k σu σv nˆ = × = ( g˙ cos v, g˙ sin v, f˙), σu σv − − k × k σuu = (f¨cos v, f¨sin v, g¨),

σuv = ( f˙ sin v, f˙ cos v, 0), − σvv = ( f cos v, f sin v, 0), − − L = σuu nˆ = f˙g¨ f¨g,˙ · − M = σuv nˆ = 0, · N = σvv nˆ = fg.˙ · Thus, the second fundamental form is 2 2 (f˙g¨ f¨g˙)du + fgdv˙ . −

4 4 Principal Curvatures

Let γ(t)= σ(u(t), v(t)) be a unit-speed curve on a surface patch σ with the standard unit normal nˆ. Below we obtain the normal curvature of the curve:

κn = γ¨ nˆ · d = γ˙ nˆ dt · d = nˆ (σuu˙ + σvv˙) · dt

= nˆ σuu¨ + σvv¨ + (σuuu˙ + σuvv˙)u ˙ + (σuvu˙ + σvvv˙)v ˙ · 2 2 = Lu˙ + 2Mu˙v˙ + Nv˙ , (8) where L, M, N are coefficients of the second fundamental form defined (7). In the last step above, we used the fact that nˆ is normal to both σu and σv, i.e., nˆ σu = nˆ σv = 0. · · Equation (8) states that the normal curvature of γ(t) depends on u, v,u ˙, andv ˙. The first two quantities specify the location of the point on the surface σ, and are thus curve independent. Let uˆ be the unit tangent vector so that γ˙ = uˆ. Take the dot products of the equation uˆ = σuu˙ + σvv˙ with σu and σv separately, yielding

Eu˙ + F v˙ = uˆ σu, · F u˙ + Gv˙ = uˆ σv, · where E,F,G are the coefficients of the first fundamental form of the surface patch. Since σu and σv are linearly independent, the coefficient matrix in the above linear system inu ˙ andv ˙ is non-singular. We solve the system to obtain −1 u˙ E F uˆ σu = · v˙ F G uˆ σv · This implies thatu ˙ andv ˙ 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 uˆ must have the same normal curvature at this point. Subsequently, we refer to κn as the normal curvature of the surface σ at the point p = σ(u, v) in the tangent direction of uˆ. It measures the curving of the surface in that direction. Generally, nˆ the surface bends at different rates in different tangent directions. The tangent uˆ and the normal nˆ at p defines a plane that cuts a uˆ curve α out of the patch. This curve is called the normal section of σ in the uˆ direction. Since the principal normal nˆ γ of the α normal section is related to the surface normal nˆ by nˆ γ = nˆ. ± Equation (3) implies that the curvature of the normal section is σ the normal curvature κn at the point or its opposite, depending on the choice of the surface normal nˆ at the point.

5 To analyze the normal curvature κn further, we make use of the ﬁrst and second fundamental forms: 2 2 2 2 Edu + 2F dudv + Gdv and Ldu + 2Mdudv + Ndv . For convenience, we introduce two symmetric matrices

E F L M 1 = and 2 = F F G F MN

The tangent vector of the unit-speed curve γ(t) = σ(u(t), v(t)) is γ˙ =u ˙σu +v ˙σv. Introducing T = (u, ˙ v˙)t, where t denotes the transpose operator, equation (8) can be rewritten as

t κn = T 2T. (9) F

Since σu and σv span the tangent plane, consider two tangent vectors:

ˆt1 = ξ1σu + η1σv and ˆt2 = ξ2σu + η2σv.

We obtain their inner product:

ˆt1 ˆt2 = (ξ1σu + η1σv) (ξ2σu + η2σv) · · = Eξ1ξ2 + F (ξ1η2 + ξ2η1)+ Gη1η2

t E F = T1 T2, (10) F G where ξ1 ξ2 T1 = and T2 = . η1 η2 The principal curvatures of the surface patch σ are the roots of the equation

L κE M κF det( 2 κ 1)= − − = 0. (11) F − F M κF N κG − −

From the linear independence of σu and σv, it is easy to show that matrix 1 is always invertible. F −1 Equation (11) essentially states that the principal curvatures are the eigenvalues of 1 2. −1 T F F Let κ be a principal curvature of 1 2, and T = (ξ, η) the corresponding eigenvector. That −1 F F ( 1 2)T = κT implies F F ( 2 κ 1)T = 0. (12) F − F The unit tangent vector ˆt in the direction of ξσu + ησv is called the principal vector corresponding to the principal curvature κ.

Theorem 1 Let κ1 and κ2 be the principal curvatures at a point p of a surface patch σ. Then

(i) κ1, κ2 R; ∈ (ii) if κ1 = κ2 = κ, then 2 = κ 1 and every tangent vector at p is a principal vector. F F (iii) if κ1 = κ2, then the two corresponding principal vectors are perpendicular to each other 6

6 For a proof of the theorem, we refer the reader to [2, 133–135]. Intuitively, the principal vectors give the directions of maximum and minimum bending of the surface at the point, and the principal curvatures measure the bending rates. In case (ii), the point is umbilic. In this case, the surface bends the same amount in all directions at p (thus all directions are principal).

Example 2. A sphere bends the same amount in every direction. Take the unit sphere in Example 9 in the notes “Surfaces”, for instance, with the parametrization

σ(θ, φ) = (cos θ cos φ, cos θ sin φ, sin θ),

We found previously that 2 E =1, F =0, G = cos θ. Since a sphere is a surface of revolution, we can plug in the result from Example 1, with f(θ) = cos θ and g(u) = sin θ:

L = f˙g¨ f¨g˙ − = sin θ( sin θ) ( cos θ)cos θ − − − − = 1, M = 0, N = fg˙ 2 = cos θ.

Hence the principal curvatures are the roots of

1 κ 0 det( 2 k 1)= − 2 2 =0. F − F 0 cos θ κ cos θ −

Hence κ = 1. And every tangent direction is a principal vector.

Example 3. Consider a cylinder with the z-axis as its axis and circular cross sections of unit radius. The parametrization is given as σ(u, v) = (cos v, sin v,u).

ˆ t1 ˆ nˆ t2 p

The coeﬃcients of the ﬁrst and second fundamental forms can be computed as

E =1, F =0, G =1,L =0,M =0, N =1.

The principal curvatures are roots of 0 κ 0 − 0 1 κ −

7 −1 So we obtain κ1 = 0 and κ2 = 1. The eigenvectors Ti = (ξi, ηi), i = 1, 2 of 1 2 are found from solving the equation F F ( 2 κi 1)Ti =0. F − F t t The results are T1 = λ1(1, 0) and T2 = λ2(0, 1) for any non-zero λ1, λ2 R. Hence the principal vector ∈ ˆt1 is along the direction of 1σu +0σv , i.e., ˆt1 = (0, 0, 1). The principal vector ˆt2 is along the direction of 0σu +1σv = ( sin v, cos v, 0), i.e., ˆt2 = ( sin v, cos v, 0). − − A curve γ on the surface σ is a principal curve if its velocity γ′ always points in a principal direction, that is, the direction of a principal vector. At every point on a principal curve, the normal curvature is a maximum or minimum. The next ﬁgure shows some principal curves on the ellipsoid 2 2 x y 2 + + z = 1. 12 5

5 Euler’s Formula

Suppose the two principal curvatures κ1 = κ2 at p on the surface σ. Then by Theorem 1(iii), the 6 two corresponding principal vectors ˆt1 = ξ1σu + η1σv and ˆt2 = ξ2σu + η2σv must be orthogonal t t to each other. Denote by T1 = (ξ1, η1) and T2 = (ξ2, η2) . Replace the T in (12) with Tj, j = 1, 2, t multiply both sides of the equation by Ti to the left, and move the second resulting term to the right hand side of the equation. This yields

t t T 2Tj = κjT 1Tj, i, j = 1, 2. i F i F Meanwhile, the orthogonality of the two principal vectors implies that

t ˆt1 ˆt2 = T1 1T2 = 0, from (10). · F

8 To summarize, we have t t κi, if i = j, T 2Tj = κiT 1Tj = (13) i F i F 0, otherwise With the principal curvatures and vectors at p, we can evaluate the normal curvature in any direction.

Theorem 2 Let κ1, κ2 be the principal curvatures, and ˆt1,ˆt2 the two corresponding principal vectors of a patch σ at p. The normal curvature of σ in the direction uˆ = cos θˆt1 + sin θˆt2 is 2 2 κn = κ1 cos θ + κ2 sin θ.

t Proof Let uˆ = ξσu + ησv and T = (ξ, η) . We ﬁrst look at the special case κ1 = κ2 = κ. By Theorem 1(ii), uˆ = ξσu + ησv is a principal vector. The normal curvature in the direction uˆ is

t κn = T 2T (by (9)) Ft = κT 1T (by (12)) F = κuˆ uˆ = κ. (14) · Meanwhile, we have 2 2 2 2 κ1 cos θ + κ2 sin θ = κ(cos θ + sin θ)= κ. So the theorem holds when the point is umbilic. Assume κ1 = κ2. Therefore by Theorem 1(iii), ˆt1 and ˆt2 are perpendicular to each other. Let 6 t ˆti = ξiσu + ηiσv, and Ti = (ξi, ηi) .

Thus,

uˆ = cos θ(ξ1σu + η1σv)+sin θ(ξ2σu + η2σv)

= (ξ1 cos θ + ξ2 sin θ)σu + (η1 cos θ + η2 sin θ)σv.

So we have uˆ = ξσu + ησv, where

ξ = ξ1 cos θ + ξ2 sin θ, η = η1 cos θ + η2 sin θ,

The above is written succinctly as T = cos θT1 + sin θT2. By equation (9) the normal curvature in the uˆ direction is

t t κn = (cos θT1 + sin θT2) 2(cos θT1 + sin θT2) 2 t F t t 2 t = cos θ T1 2T1 + cos θ sin θ (T1 2T2 + T2 2T1)+sin θ T2F2T2 2 F 2 F F = κ1 cos θ + κ2 sin θ.

The last step above followed from the equation (13).

Theorem 2 implies that κ1 and κ2 are the maximum and minimum of any normal curvatures at the point. Equivalently, among all tangent directions at the point, the geometry varies the most in one principal direction while the least in the other.

9 References

[1] B. O’Neill. Elementary Diﬀerential Geometry. Academic Press, Inc., 1966.

[2] A. Pressley. Elementary Diﬀerential Geometry. Springer-Verlag London, 2001.