Some approaches toward the mathematical modeling of the craniofacial complex

It is increasingly recognized by quantitative morphologists that the use of the conventional metrical approach (CMA), consisting of linear distances, angles, and ratios, to describe the size and shape of complex morphological forms as well as the shape changes between forms, is a procedure that is inherently deficient. Conventional metrics were designed to measure regular geometric objects and were never intended to describe the shape of complex irregular forms. A number of new methods have been developed to describe complex irregular shapes that attempt to circumvent the difficulties with CMA. These procedures fall into two categories, those dependent on homologous-point representations and those based on boundary representations. Three of these new methods, 1) biorthogonal grids, 2) finite element scaling, and 3) elliptical Fourier analysis, are reviewed here. While all three methods represent major improvements over CMA as shape and shape change descriptors, they must be considered as promising preliminary numerical models rather than final definitive pronouncements. Each method has constraints that preclude it from being the method of choice. It is apparent that a generalized model for the characterization of complex morphological forms remains to be developed.