8.7: Geodesic Curvature of Curves on Surfaces

In lab 3 we resolved the accleration vector of a space curve X(t) into a linear combination of the tangent vector T(t) and the principal normal vector P(t); this was in the T(t), P(t), B(t) Frenet-Serret orthonormal frame. When dealing with curves on surfaces, it is more natural to express the accleration vector in terms of a different frame. If we let U(t) denote the normal vector to the normal plane, then the vectors T(t), U(t), and N(t) form an orthonormal frame where U(t) = N(t) × T(t). We saw in the first section of lab 8 that the projection of κP(t) onto N(t) gives us the normal curvature κ_N(t). Here we define the projection of κP(t) onto U(t) to be the geodesic curvature, κ_g. We then have

κP(t) = κ_gU(t) + κ_NN(t)

X''(t) = (X'(t))' = (s'(t)T(t))' = s''(t)T(t) + s'(t)T'(t)

             = s''(t)T(t) + s'(t)(κ_g(t)s'(t)U(t) + κ_N(t)s'(t)N(t))
         = s''(t)T(t) + κ_g(t)s'(t)²U(t) + κ_N(t)s'(t)²N(t)

From this equation for the acceleration vector X''(t), we can derive an explicit formula for calculating the geodesic curvature of a curve on a surface. First, take the cross-product of both sides with the velocity vector X'(t) to get:

X'(t) × X''(t) = κ_g(t)s'(t)³T(t) × U(t) + κ_N(t)s'(t)³T(t) × N(t)

κ_g(t) = (X'(t) × X''(t)) · N(t)/s'(t)³

Note that this terminology justifies our usage of the term geodesic curvature for a curve in the plane in labs 1 and 2.

κ(t)s'(t)²P(t) = κ_g(t)s'(t)²U(t) + κ_N(t)s'(t)²N(t),   and
κ(t)P(t) = κ_gU(t) + κ_N(t)N(t)

Exercises

1. For which t is κ_g(t) = 0? For which t is κ_N(t) = 0?

2. For a curve on the unit sphere, what can be said about κ_N(t)?