The relation between geometric morphometrics and functional morphology, as explored by Procrustes interpretation of individual shape measures pertinent to function.

A frequent concern in today's functional morphology is the relation of a landmark configuration to some a priori index or suite of indices of function. When an index is itself a generic mathematical or biomechanical shape function of landmark locations, meaning a dimensionless expression that has a nonzero gradient everywhere in the feasible region of morphospace, the question becomes sharper: how can we exploit it as a reference direction for representations within the realm of the customary geometric morphometric (GM) analyses? This article argues that the only valid approach to this problem is geometric, not statistical: to represent any such a priori index by way of its differential (its gradient) calculated as an explicit vector in the Procrustes dual space of the complete list of landmarks whether or not involved in the formulation of the index. Interpretation of the index follows by comparing its direction after this embedding with other interesting directions in the same shape space, such as principal warps, relative warps, group mean shape contrasts, specific form factors extracted independently, or directions corresponding to other functional indices. Here, I work an artificial but realistic example of this technique in complete detail: the construction of a Procrustes shape formula exactly aligned with a specific angle among three landmarks within an arbitrary configuration of six. A closing discussion traces the spirit of this intervention to comments by W. W. Howells and C. E. Oxnard, originally intended for anthropometric contexts other than GM, on the different purposes of systematics and functional morphology.