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,

dω_1,2 = -Kζ₁^ζ₂
 

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,

ζ_i(p)(v) = v·E_i(p) for i=1,2,3
 

The symbol ^ stands for the wedge product of 1-forms. Finally the matrix {ω_i,j }is the connection form for the frame field of the E_i(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 U_i be the unit vector in the direction of increasing Cartesian coordinate x_i. 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^ζ(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

ω_1,2(v) = -sin(φ)dθ(v)
and thus
dω_1,2(v) = cos(φ)dφdθ
 

Comparing dω_1,2(v) and ζ₁^ζ(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
ζ₁^ζ₂ = fgdudv
 

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

The formula for Gaussian curvature then implies

∫_u∫_vK(u,v)dudv = -∫_v[ω_1,2(u,v) - ω_1,2(u₀,v₀)]dv
 

For the sphere ω_1,2 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.