Geodesic torsion of curves on surfaces

We defined the geodesic curvature and the normal curvature of curves on surfaces in the preceding sections of this lab. There is one more function that is used in the study of curves on surfaces in three-space. This is called the geodesic torsion, t_g(t) , of a curve on a surface. The nature of this quantity is quite abstract, even more so than the torsion of space curves. However, we provide one common definition in the following paragraph.

If we have a curve X(t) on a surface X(u,v) , we know that at any given point X(t₀) on that curve there is a geodesic G₀ going through that point with direction X^'( t₀) . The geodesic torsion t_g(t₀) of X(t) on X(u,v) at t=t₀ is defined to be the torsion of G₀ at X(t₀) . In the next section we provide a little more information about the usefulness of the geodesic torsion. For the moment, a demo to illustrate t_g is in order.

Need to put an IMG here!This simple demo is identical to the previous demo which displayed the function graph of the geodesic curvature except that we display the geodesic torsion instead.