The paper describes the development and application of a new approach for formulating an elliptic generation system on parametdinates is well known; see Thompson et al. [18]. Since rically defined surfaces. The present derivation of the surface equaelliptic partial differential equations determine a function tions proceeds in two steps: First, conformal mapping of smooth in terms of its values on the entire closed boundary of a surfaces onto rectangular regions is utilized to derive a first-order region, such a system can be used to generate the interior system of partial differential equations analogous to Beltrami's system for quasi-conformal mapping of planar regions. Second, a gen-values of a surface grid from the values on the sides. An eral elliptic generation system for three-dimensional surfaces, inimportant property is the inherent smoothness in the solucluding forcing functions, is formulated based on Beltrami's system tions of elliptic systems. As a consequence of smoothing, and quasi-conformal mapping. The resulting elliptic system is slope discontinuities on the boundaries are not propagated solved using an iterative method on arbitrary surfaces represented into the field. Variational methods can also be used in the analytically by rational B-splines. The overall effect of this approach is a reliable and versatile elliptic method for generating and improvconstruction of curvilinear coordinate systems; see Knupp ing surface grids. Examples will be presented to demonstrate the and Steinberg [7]. These methods have been extended to application of the method in constructing practical grids. แฎ 1996 surface grid generation by Knupp [8] and Saltzman .