We study the problem of analyzing variability in shapes of facial
surfaces. The main difficulty in this problem comes from the lack of
a canonical coordinate system to compare faces. Our idea is to
impose a specific, yet natural, coordinate system, called a Darcyan
curvilinear coordinate system, on facial surfaces. This system is
intrinsic to the surfaces and it deforms with them. Here, one
coordinate $\xi_1$ measures the distance from the tip of the nose
and the other coordinate $\xi_2$ measures distances along level
curves of $\xi_1$. Using the Darcyan system, we develop tools for
matching, comparing and deforming surfaces under an elastic metric.
The central idea is to find optimal matches between level curves of
$\xi_1$ across faces, and to use an elastic (Riemannian) metric on
the space of closed curves to define and compute geodesic paths
between matched curves. Together these geodesics provide optimal
elastic deformations between faces and an elastic metric for
comparing facial shapes. We demonstrate this idea using examples
from FSU face database.