The measurement of form and variation in form: An application of three‐dimensional quantitative morphology by finite‐element methods

Abstract

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.

Biomechanics of phalangeal curvature

“…Note that FEA is not the same method as finite element scaling analysis, which adapts FEA to the measurement of form difference between two structures in a D'Arcy Thompsonian fashion (Cheverud, 1983).…”

Craniofacial Growth in Apert Syndrome as Measured by Finite-Element Scaling Analysis

“…. 1984, are currently being applied with success to the study of biological form change [Skalak et al" 1982: Cheverud et al" 1983: Grayson et al. 1985: Lavelle.…”

Transverse Dentoskeletal Features of Anterior Open Bite in the Mixed Dentition

“…[12][13][14][15][16] A recent morphometric approach to the comparison of configurations of landmarks in 2 or more specimens is known as thin-plate spline (TPS) analysis, as developed by Bookstein. 17 TPS analysis enables the construction of transformation grids that capture the differences in shape and are available for visual interpretation. For a more detailed review of theoretical base, calculation procedures, and assumptions and limitations of TPS morphometrics, see Bookstein,17,18 Rohlf and Marcus, 19 and Dryden and Mardia.…”