The mesh control parameters discussed in the preceding sections are used as input to an ordinary program for automatic triangulation in two-dimensional domains. The program must be able to (i) generate anisotropic meshes, and (ii) accept as input arbitrarily subdivided boundary. The first requirement is obvious, the second stems from the fact that singularities in the surface parameterizations need to be handled specially, and a subdivision of the boundary needs to be prescribed independently of the mesh control function.
Let us denote the Cartesian axes in the parametric space. The mesh control function is given at a given point in the axes , which are the principal axes of the stretch. These are local Cartesian coordinates at the point , i.e. these axes describe the desired stretch only in a certain small neighborhood around the point . It is assumed that the mesh size along the axis is always smaller than or equal to the mesh size along the axis , i.e. the mesh is elongated (flows along) the axis . The mesh control information is assumed to be composed of the mesh size in the direction of the first principal stretch, the angle between the and axes, and the ratio .