Gaussian Curvature

(Com S 477/577 Notes)

Yan-Bin Jia

Nov 4, 2014

We have learned that the two principal curvatures (and vectors) determine the local shapeof a point on a surface. One characterizes the rate of maximum bending of the surface and thetangent direction in which it occurs, while the other characterizes the rate and tangent directionof minimum bending. The rate of surface bending along any tangent direction at the same pointis determined by the two principal curvatures according to Eulers formula.

In this lecture, we will first look at how the local shape at a surface point can be approximatedusing its principal curvatures and direction. Then we will look at how to characterizes the rateof change of a vector defined on a surface with respect to a tangent vector. Our main focus willnevertheless be on two new measures of the curving a surface its Gaussian and mean curvatures that turn out to have greater geometrical significance than the principal curvatures.

1 Geometric Interpretation of Principal Curvatures

The values of the principal curvatures and vectors at a point p on a surface patch tell us aboutthe shape near p. To see this, we apply a rigid motion followed by a reparametrization.1 Morespecifically, we move the origin to p and let the tangent plane to at p be the xy-plane with thex-axis and y-axis along the directions of the two principal vectors, which correspond to principalcurvatures 1 and 2, respectively. Furthermore, we let the values of both parameters at the originbe zero, that is,

(0, 0) = 0. (1)

Without any ambiguity, we still denote the new parametrization by .Let us determine the function z = z(x, y) that describes the local shape. The unit principal

vectors can be expressed in terms of the partial derivatives:

(1, 0, 0) = 1u + 1v,

(0, 1, 0) = 2u + 2v.

So can any point (x, y, 0) in the tangent plane:

(x, y, 0) = x(1, 0, 0) + y(0, 1, 0)

The material is adapted from the book Elementary Differential Geometry by Andrew Pressley, Springer-Verlag,

2001.1The shape does not change under any rigid motion or reparametrization.

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= x(1u + 1v) + y(2u + 2v)

= su + tv, (2)

wheres = x1 + y2 and t = x1 + y2. (3)

Let us evaluate (s, t) at the parameter values s and t, applying Taylors theorem with higher orderterms in s and t neglected:

(s, t) = (0, 0) + su + tv +1

2(s2uu + 2stuv + t

2vv)

= (x, y, 0) +1

2(s2uu + 2stuv + t

2vv), (by (1) and (2))

All derivatives are evaluated at the origin p. Neglecting the second order terms added to x and y,the coordinates of (s, t) is (x, y, z), where

z = (s, t) n=

1

2(Ls2 + 2Mst+Nt2)

=1

2(s t)

(L MM N

)(s

t

).

x

z

y

u1

u2

p

Writing

T1 =

(11

)and T2 =

(22

),

we have from (3):

(s

t

)= xT1 + yT2.

Thus,

z =1

2(xT1 + yT2)

tF2(xT1 + yT2)

=1

2

(x2T1F2T1 + xy(T t1F2T2 + T t2F2T1) + y2T t2F2T2

)

=1

2(1x

2 + 2y2),

since T tiF2Tj = i if i = j or 0 otherwise. Hence the shape of a surface near the point p has aquadratic approximation determined by its principal curvature 1 and 2. It is an elliptic paraboloiddescribed by the equation z = 1

2(1x

2 + 2y2).

2 Covariant Derivative

We slightly abuse the notation n to represent a function that assigns to every point p on the surfaceS the normal n(p) at the point. Since n is continuous, it is a vector field on S, and referred to as

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the normal vector field. Similary, t1 and t2 are also vector fields on S that continuously assign toevery point two orthogonal principal vectors.

At the point p, a vector field Z typically changes differently in different tangential directions.The rate of change along a tangent w is charaterzied by its covariant derivative along w. Morespecifically, we let (t) be a curve on S that has initial velocity (0) = w. Consider restriction ofZ to . Then, the covariant derivative of Z with respect to w is defined to be

wZ = dZ((t))dt

t=0

.

In particular, consider the u-curve (u) = (u, v0) passing through p = (u0, v0) at velocityw = u(u0, v0). We have

wZ = dZ((u))du

u=u0

=dZ((u, v0))

du

u=u0

= Zu(u0, v0).

Reparametrize (u) as a unit-speed curve (s). Clearly,

ds

du(0) = (u0) = u(u0, v0).

At p, let x = (0) = u(u0, v0)/u(u0, v0). The covariant derivative with respect to the unitvector x is

xZ = dZ((s))ds

s=0

=dZ((u(s)))/du

ds/du

u=u0

=Zu(u0, v0)

u(u0, v0) .

In the Darboux frame T -V -U at p of a surface curve, where T is the curve tangent, U thesurface normal, and V = U T , it holds that U = nT gV . Denote T as u. Then U is thecovariant derivative along u. The normal curvature at p in the direction u is

n(u) = U T = un u.

This is the definition of the normal curvature in [1, p. 196]. Consequently, the principal curvaturesare

1 = n(t1) = t1n t1,2 = n(t2) = t2n t2.

It can be shown that tin tj = 0 if i 6= j.

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3 Gaussian and Mean Curvatures

Let 1 and 2 be the principal curvatures of a surface patch (u, v). The Gaussian curvature of is

K = 12,

and its mean curvature is

H =1

2(1 + 2).

To compute K and H, we use the first and second fundamental forms of the surface:

Edu2 + 2Fdudv +Gdv2 and Ldu2 + 2Mdudv +Ndv2.

Again, we adopt the matrix notation:

F1 =(

E FF G

)and F2 =

(L MM N

).

By definition, the principal curvatures are the eigenvalues of F11F2. Hence the determinant of this

matrix is the product 12, i.e., the Gaussian curvature K. So

K = det(F11F2) = det(F1)1 det(F2) = LN M2

EG F 2 .. (4)

The trace of the matrix is the sum of its eigenvalues, thus, twice the mean curvature H. Aftersome calculation, we obtain

H =1

2trace(F11F2) = 1

2

LG 2MF +NEEG F 2 . (5)

An equivalent way to obtain K and H uses the fact that the principal curvatures are also theroots of

det(F2 F1) = 0,which expands into a quadratic equation

(EG F 2)2 (LG 2MF +NE)+ LN M2 = 0.

The product K and the sum 2H of the two roots, can be determined directly from the coefficients.The results are the same as in (4) and (5).

Conversely, given the Gaussian and mean curvatures K and H, we can easily find the principalcurvatures 1 and 2, which are the roots of

2 2H+K = 0,

i.e., H H2 K.Example 1. We have considered the surface of revolution (see Example 1 in the notes titled SurfaceCurvatures)

(u, v) = (f(u) cos v, f(u) sin v, g(u)),

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where we assume, without loss of generality, that f > 0 and f2 + g2 = 1 everywhere. Here a dot denotesd/du. The coefficients of the first and second fundamental forms were determined:

E = 1, F = 0, G = f2, L = f g f g, M = 0, N = f g.So the Gaussian curvatures is

K =LN M2EG F 2 =

(f g f g)f gf2

=(f g f g)g

f.

Meanwhile, differentiate f2 + g2 = 1:f f + gg = 0.

Thus,

(f g f g)g = f2f f g2= f(f2 + g2)= f .

So the Gaussian curvature gets simplified to

K = ff.

Example 2. Here we compute the Gaussian and mean curvatures of a Monge patch z = f(x, y). Namely,the patch is described by (x, y) = (x, y, f(x, y)). First, we obtain the first and second derivatives:

x = (1, 0, fx), y = (0, 1, fy), xx = (0, 0, fxx), xy = (0, 0, fxy), yy = (0, 0, fyy).

Immediately, the coefficients of the first fundamental form are determined

E = 1 + f2x , F = fxfy, G = 1 + f2

y .

So is the unit normal to the patch:

n =x yx y =

(fx,fy, 1)1 + f2x + f

2y

.

With the normal n, we obtain the coefficients of the second fundamental form:

L = xx n = fxx1 + f2x + f

2y

,

M = xy n = fxy1 + f2x + f

2y

,

N = yy n = fyy1 + f2x + f

2y

.

Plug the expressions for E,F,G,L,M,N into (4) and (5). A few more steps of symbolic manipulation yield:

K =LN M2EG F 2 =

fxxfyy f2xy(1 + f2x + f

2y )

2,

H =1

2

LG 2MF +NEEG F 2 =

fxx(1 + f2

y ) 2fxyfxfy + fyy(1 + f2x)2(1 + f2x + f

2y )

3/2.

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4 Classification of Surface Points

The Gaussian curvature is independent of the choice of the unit normal n. To see why, supposen is changed to n. Then the signs of the coefficients of L,M,N change, so do the signs of bothprincipal curvatures 1 and 2, which are the roots of det(F2 F1). Their product K = 12is unaffected. The mean curvature H = (1 + 2)/2, nevertheless, has its sign depending on thechoice of n.

The sign of K at a point p on a surface S has an important geometric meaning, which is detailedbelow.

1. K > 0 The principal curvatures 1 and 2 have the same sign. The normal curvature inany tangent direction t is equal to 1 cos

2 + 2 sin2 , where is the angle between t and

the principal vector corresponding to 1. So has the same sign as that of 1 and 2. Thesurface is bending away from its tangent plane in all tangent directions at p. The quadraticapproximation of the surface near p is the paraboloid

z =1

2(1x

2 + 2y2).

We call p an elliptic point of the surface.

2. K < 0 The principal curvatures 1 and 2 have opposite signs at p. The quadraticapproximation of the surface near p is a hyperboloid. The point is said to be a hyperbolicpoint of the surface.

3. K = 0 There are two cases:

(a) Only one principal curvature, say, 1, is zero. In this case, the quadratic approximationis the cylinder z = 1

22y

2. The point p is called a parabolic point of the surface.

(b) Both principal curvatures are zero. The quadratic approximation is the plane z = 0.The point p is a planar point of the surface. One cannot determine the shape of thesurface near p without examining the third or higher order derivatives. For example, apoint in the plane and the origin of a monkey saddle z = x3 3xy2 (shown below) areboth planar points, but they have quite different shapes.

A torus is the surface swept by a circle originally in the yz-plane and centered on the y-axis ata distance greater than its radius from the origin, when the circle revolves about the z-axis. It is

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a good example which has all three types of points. At points on the outer half of the torus, thetorus bends away from from its tangent plane; hence K > 0. At each point on the inner half, thetorus bends toward its tangent plane in the horizontal direction, but away from it in the orthogonaldirection; hence K < 0. On the two circles, swept respectively by the top and bottom points of theoriginal circle, every point has K = 0.

z

y

x

pp

A surface S is flat if its Gaussian curvature is zero everywhere. A plane is flat. Let it be thexy-plane with the parametrization (x, y, 0). We can easily show that the plane has zero Gaussiancurvature. A circular cylinder, treated in Example 3 of the notes Surface Curvatures, has oneprincipal curvature equal to zero and the other equal to the inverse of the radius of its cross section.So a circular cylinder is also flat, even though it is so obviously curved.

A surface is minimal provided its mean curvature is zero everywhere. Minimal surfaces haveGaussian curvature K 0. This is because H = (1 + 2)/2 = 0 implies 1 = 2.

5 The Gauss Map

The standard unit normal n to a surface patch measures the direction of its tangent plane. Thechange rate of n in a tangent direction, i.e., the normal curvature, indicates the degree of variationof surface geometry in that direction at the point. To make the notion of change of geometryindependent of any tangent direction, we can measure by the rate of change of n per unit area.

Note that n is a point of the unit sphere S2 centered at the origin. The Gauss map from asurface patch (u, v) : U R3 to the unit sphere S2 sends a point p = (u, v) to the point n(u, v)of S2. The Gauss map may be a many-to-one mapping since multiple points on the patch can havethe same unit normal.

unit sphere

Gauss map

q

p

(R)

n(q)

n(p) n(p)

n(q) N(R)

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Let R U be a region. The amount by which n varies over the corresponding region (R)on the surface is measured by the area of the image region N(R) on the unit sphere. The rate ofchange of n per unit area is the limit of the ratio of the area AN (R) of N(R) to the area A(R) ofthe surface region (R), as R shrinks to a point. To be more precise, we consider R to be a closeddisk of radius centered at (u, v) U . This ratio is

lim0

AN(R)A(R) .

It can be shown [2, pp. 166168] that the above ratio is the absolute value of the Gaussian curvatureat p, i.e.,

lim0

AN (R)A(R) = |K|.

The integral of the Gaussian curvature K over a surface S,

S

KdS,

is called the total Gaussian curvature of S. It is the algebraic area of the image of the region on theunit sphere under the Gauss map. Note the use of the word algebraic since Gaussian curvaturecan be either positive or negative,

Suppose the patch S = (u, v) is defined over the domain [a, b] [c, d]. Then the total Gaussiancurvature is computed as d

c

ba

K(u, v)EG F 2 dudv.

Example 3. If the Gaussian curvature K of a surface S is constant, then the total Gaussian curvatureis KA(S), where A(S) is the area of the surface. Thus a sphere of radius r has total Gaussian curvature1

r2 4pir2 = 4pi, which is independent of the radius r.

Example 4. Without any computation, we can determine that an ellipsoid also has total curvature 4pi. The

Gauss map is bijective (one-to-one and onto) since every point on the ellipsoid has a distinct normal. The

image region covers the unit sphere. Because the Gaussian curvature is everywhere positive on the ellipsoid,

the area of the unit sphere, 4pi, is the total Gaussian curvature of the ellipsoid.

References

[1] B. ONeill. Elementary Differential Geometry. Academic Press, Inc., 1966.

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

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