Perimeter-based Fourier analysis: a new morphometric method applied to the trilobite cranidium

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Abstract
A new Fourier method is presented to quantify shapes too complex to be described by conventional polar Fourier analysis. The length along a closed curve serves as the independent variable. The centroid of the curve is determined and for each point on the curve two different dependent variables are defined, based on: 1) the angle defined by the starting point, the centroid, and the point on the curve; and 2) the radial distance from the centroid to the point on the curve. The method is used to describe the trilobite cranidium, and 12 harmonic coefficients are found to summarize 99 percent of the shape information contained in the cranidial outline. In an application to trilobite evolution during the Cambrian and Ordovician, it is found that higher taxa of trilobites become progressively more distinct morphologically. This result is in agreement with previous qualitative observations, and is attributable to an increase in morphologic dispersion among higher taxa, but not to a decrease in morphologic dispersion within higher taxa.