The measurement of form and variation in form: an application of three-dimensional quantitative morphology by finite-element methods.

D'Arcy Thompson developed a method of coordinates which allowed for a geometrical presentation of form and form change. While his grid transformations have received much attention, little work in the geometry of form and form change has occurred since. We present a three-dimensional nonhomogeneous finite-element scaling method which allows for the mathematical and geometrical measurement of form change in addition to the graphical representation of these deformations as D'Arcy Thompson grids. This allows a reconciliation between geometrical and statistical methods for analyzing form. The method involves quantification of the transformation of one form into another in three dimensions without special registration and contains algorithms for obtaining a mean form. The method is applied to an analysis of variation in cranial form among adult male rhesus macaques from the Cayo Santiago skeletal collection. Variation was greatest in the superior-inferior direction, followed by the anterior-posterior and medial-lateral directions. The upper facial region is particularly variable. An analysis of allometry relative to local size variation shows that the larger any particular region is, the relatively greater its height, narrower its width, and shorter its length. An analysis of allometry relative to overall size showed that the upper face is positively allometric, the occipital region is strongly negatively allometric, and the other regions are isometric. After within-group variation is characterized, as described here, between-group studies, such as growth series and phylogenetic series, can be performed.