By F. L. Bookstein
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Extra resources for The Measurement of Biological Shape and Shape Change
Example text
53 Fig. --Two-arc conic spline to digitization of the outline of a pelecypod, Pisidium com£ressum, from Burch (1975:14). The open area at the top is a project~on of the hinge obscuring the line of closure. and ratios of rather large reach. Figure IV-10 shows a specimen of this form quite closely fit by either two or three conic arcs splined together--the optional third arc makes it possible to represent the little bump above the inion, at the far left. There is then no need for landmarks at all in this part of the skull, but the whole shape itself, as smoothly reconstructed by the conic spline, may be used as a function-valued measure in analysis of growth patterns, racial differences, evolution of the hominid form, and the like.
Lestrel proceeds further and submits these Fourier coefficients to a multivariate analysis, emerging with two principal components of the measurement vector. On a plot of one principal axis against the other he locates specimens and species. The technique allows two landmarks, one the starting point of the series, one the point chosen as center of coordinates. All the objections listed in the preceding paragraph apply to this method, and 30 also one further crucial difficulty. Each analysis depends fundamentally on the landmark used for the center of polar coordinates.
P to R normal to the curve. These will be equal for the two conics impinging at P if and only if the gradients of the two scalar fields are equal at P. But to specify that these two gradients be equal is just a p~ir of linear constraints, for the gradients are linear forms in the coefficients of the conics. To wit: if the two conics be Qi(x,y) = Aix2 + Bixy + Ciy2 + Dix + EiY + F = 0, i = 1, 2, and if P have the coordinates (r,s), then the two consistency criteria are 0, We must also insist that both conics actually pass through the point (r,s) at which we are specifying their tangents, and this is two more constraints: We can estimate both conics, all twelve coefficients, simultaneously by minimizing their total squared error-of-fit subject to the normalization A12+B12/2+C12 = 2 and the four constraints preceding.