Janice L. Pappas, Daniel J. Miller

The surface geometry of an organism represents the boundary of its three-dimensional (3D) form and can be used as a proxy for the phenotype. A mathematical approach is presented that describes surface morphology using parametric 3D equations with variables expressed as x, y, z in terms of parameters u, v. Partial differentiation of variables with respect to parameters yields elements of the Jacobian representing tangent lines and planes of every point on the surface. Jacobian elements provide a compact size-free summary of the entire surface, and can be used as variables in principal components analysis to produce a morphospace. Mollusk and echinoid models are generated to demonstrate that whole organisms can be represented in a common morphospace, regardless of differences in size, geometry, and taxonomic affinity. Models can be used to simulate theoretical forms, novel morphologies, and patterns of phenotypic variation, and can also be empirically-based by designing them with reference to actual forms using reverse engineering principles. Although this study uses the Jacobian to summarize models, they can also be analyzed with 3D methods such as eigensurface, spherical harmonics, wavelet analysis, and geometric morphometrics. This general approach should prove useful for exploring broad questions regarding morphological evolution and variation.