The paper describes the development and application of a new approach for formulating an elliptic generation system on parametrically defined surfaces. The present derivation of the surface equations proceeds in two steps: First, conformal mapping of smooth surfaces onto rectangular regions is utilized to derive a first-order system of partial differential equations analogous to Beltrami's system for quasi-conformal mapping of planar regions. Second, a general elliptic generation system for three-dimensional surfaces, including forcing functions, is formulated based on Beltrami's system and quasi-conformal mapping. The resulting elliptic system is solved using an iterative method on arbitrary surfaces represented analytically by rational B-splines. The overall effect of this approach is a reliable and versatile elliptic method for generating and improving surface grids. Examples will be presented to demonstrate the application of the method in constructing practical grids.
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