We propose a pattern theoretic approach for studying variability in shapes of facial surfaces. Our idea is to impose a specific, yet natural, coordinate system, called a curvilinear coordinate system, on facial surfaces. In this system, one coordinate xi1 measures the distance of a point from the tip of the nose and its level curves are called the facial curves. The other coordinate xi2 measures distances along these curves; level curves of this coordinate are orthogonal to the facial curves. To compare two facial surfaces we use elastic deformations that use stretching, shrinking, and bending to optimally register points across two surfaces. We will demonstrate this idea on Florida State University (FSU) 3D face database.