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Curvature of Curves and Surfaces

The Curvature of Curves

The Frenet-Serret equations are a convenient framework for analyzing curvature. In these equations T(s) represents the unit tangent to the curve as a function of path length s. N(s) is the unit normal to the curve and the vector cross product of T(s) and N(s) is the unit binormal B(s). The remarkable thing is that the derivatives of these unit vectors can be expressed neatly in terms of the unit vectors themselves. In particular,

dT/ds = k(s)N(s).

where k(s) is the curvature. For three dimensional curves

dN/ds = -k(s)T(s) + τ(s)B(s)
dB/ds = -τN(s).

where τ is called the torsion coefficient.

In matrix form these equations are:

| dT/ds |  |  0  k(s)  0   || T(s) |
| dN/ds |  =| -k(s)   0    τ(s) || N(s) |
| dB/ds |  |   0 -τ(s)  0   || B(s) |


For a curve in a plane B(s) is constant so dB/ds=0 and thus the torsion coefficient τ must be zero. Therefore for a curve in a plane the equations reduce to:

dT/ds = k(s)N(s)
dN/ds = -k(s)T(s)

(To be continued.)

The Curvature of Surfaces

There are several differenct concepts of the curvature of a surface. The most important is the one called Gaussian curvature. Gaussian curvature may be defined in several different but equivalent ways. One elegant way is in terms of a Shape Operator that expresses the rate of change of the unit normal vector to a surface with respect to vectors in the tangent plane to the surface. The Gaussian curvature is the determinant of this shape operator.

Another very elegant method of defining the Gaussian curvature of a surface is in terms of differential forms. The defining equation for Gaussian curvature in these terms is the very abstruse equation, that will have to be explained in detail later,

1,2 = -Kζ12

The easy part of the explanation is that K is the Gaussian curvature. A frame field of unit vectors (E1(p), E2(p), E3(p)) is presumed in which E3(p) is the unit normal to the surface at point p of the surface. E1(p) and E2(p) are therefore necessarily unit vectors in the plane tangent to the surface at point p. ζ1(p) and ζ2(p) are the dual 1-forms of E1(p) and E1(p). A 1-form is a linear functional defined on the set of tangent vectors to E3, three dimensional space. In particular,

ζi(p)(v) = v·Ei(p) for i=1,2,3

The symbol ^ stands for the wedge product of 1-forms. Finally the matrix {ωi,j }is the connection form for the frame field of the Ei(p)'s. The components of the connection form are 1-forms and the differential of one of them is a 2-form.

Illustrations of the Equation

The Case of a Sphere

Let Ui be the unit vector in the direction of increasing Cartesian coordinate xi. The Cartesian coordinates expressed in terms of spherical coordinates are:

x1 = ρcos(φ)cos(θ)
x2 = ρcos(φ)sin(θ)
x3 = ρsin(φ)

The frame field for a sphere, the unit vectors in the directions of increasing φ, θ and ρ, is then given by

E1 = -sin(φ)cos(θ)U1 -sin(φ)sin(θ)U2 + cos(θ)U3
E2 = -sin(θ)U1 + cos(θ)U2
E3 = cos(φ)cos(θ)U1 + cos(phi;)sin(θ)U2 + sin(φ)U3

The unit vector in the direction of increaseing ρ is the unit normal vector for the sphere.

The dual 1-forms are then

ζ1(v) = ρdφ(v)
ζ2(v) = ρcos(φ)dθ(v)

The wedge product of these dual 1-forms is

ζ1(v) = ρ2cos(φ)dφ(v)dθ(v)

where dφ(v) is equal to the component of the vector v in the direction of increasing φ and likewise for dθ(v).

The relevant component of the connection form for the sphere is

ω1,2(v) = -sin(φ)dθ(v)
and thus
1,2(v) = cos(φ)dφdθ

Comparing dω1,2(v) and ζ1(v) we see that the Gaussian curvature of a sphere has to be 1/ρ2.

The General Formulation

For a surface given in terms of two parameters u and v the dual 1-forms are

ζ1 = fdu
ζ2 = gdv
and thus
ζ12 = fgdudv

where f and g are functions of the point on the surface.

The formula for Gaussian curvature then implies

uvK(u,v)dudv = -∫v1,2(u,v) - ω1,2(u0,v0)]dv

For the sphere ω1,2 is equal to sin(φ)dθ so the integral for φ from π/2 to φ0 and θ from 0 to 2π is then -2π[1-sin(φ0)], which is known from other techniques.

For more on this topic see the Gauss-Bonnet Theorem.

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