Southeastern Section - 64th Annual Meeting (19–20 March 2015)

Paper No. 10
Presentation Time: 1:00 PM-5:00 PM


MIKE, Joshua1, SCHWARTZ, Fernando1 and SUMRALL, Colin D.2, (1)Department of Mathematics, University of Tennessee, Knoxville, TN 37996, (2)Department of Earth and Planetary Sciences, University of Tennessee, 306 EPS Building, 1412 Circle Drive, Knoxville, TN 37996-1410,

One of the greatest challenges in paleobiology is discriminating species. This is usually accomplished by comparing shape. Various shape comparison techniques have been employed over the years, ranging from simple linear regression over a set of measurements, to landmark analysis of 3D scans. Unfortunately, all of these methodologies ignore valuable fine-scale geometric information such as curvature, thus limiting their power. For example, it is particularly difficult to detect the presence of concavities or convexities using the aforementioned techniques; these geometric features are, more often than not, crucial for distinguishing species.

New techniques from the emerging field of computational geometry offer solutions to these limitations. By considering state-of-the-art tools inspired by the mathematical field of differential geometry, we are able to compare a prescribed mix of global and fine-scale structures of shapes. More precisely, we implement a landmark-free notion of “distance” between 3D surface scans, which gives a quantitative measure of their shape-dissimilarity. Using this information we are able to cluster samples, and ultimately discriminate between species –all this based on an optimal range of pre-selected geometric features associated to their shape.

We apply computational geometry techniques to 3D scans of mixed populations of blastoid Pentremites, including both pyriform and godoniform taxa. Basing the analysis on a particular combination of geometric features such as curvature and area density, we are able to separate out these form taxa. Our novel method allows us to recover the “ground truth,” which corresponds to two clusters equivalent to the empirically assigned groups. Our result provides strong experiential evidence supporting the use of these techniques for similar studies in different taxa.

Previous studies using 3D landmark analyses have produced similar clustering results. Although our procedure is still time consuming because of the scanning phase, it is considerably foreshortened because it does not require an expert choosing landmarks on the 3D scan. Our techniques also reduce the potential error resulting from user bias, and open the door to incorporating fine-structure, curvature-based features into the analysis.