Near-optimal adaptive polygonization

Consider a triangulation of the xy plane, and a general surface z=f(x, y). The points of the triangle, when lifted to the surface, form a linear spline approximation to the surface. We are interested in the error between the surface and the linear approximant. In fact, we are interested in building triangulations in the plane such that the induced linear approximant is near-optimal with respect to a given error. We describe a new method, which iteratively adds points to a "Delaunay-like" triangulation of the plane. We locally approximate f by a quadratic surface and utilize this surface to establish an edge-flipping criterion for a convex quadrilateral that enables us to minimize the error between the surface and the triangulation.