Forty Years After

Abstract

The coordinate construction in Part I was based on basic theorems in algebraic-geometry. Alternative formulations have been developed by Floater and Warren. These nonrational alternatives include kernels based on contour integrals for elements with curved sides. Much effort has been directed toward areal coordinates which are degree-one interpolation functions adequate for graphics application. Higher degree interpolants apply to patchwork polynomial approximation such as that fundamental to finite element computation. The simplicity of the algebraic-geometry approach with the GADJ adjoint construction supports use of this for convex polygons and well-set elements with curved sides. The mean-value approach seems best for areal coordinate construction for nonconvex polygons. The significance of all these approaches is that one is no longer restricted to triangle and rectangle elements. Element geometry for which polynomial interpolants may be constructed is now quite broad. Speed of arithmetic and input–output, large memory of modern computers, and extensive software enhance application of sophisticated algorithms.