How flat can a horse be? Exploring 2D approximations of 3D crania in equids.

Quantitative analyses of morphological variation using geometric morphometrics are often performed on 2D photos of 3D structures. It is generally assumed that the error due to the flattening of the third dimension is negligible. However, despite hundreds of 2D studies, few have actually tested this assumption and none has done it on large animals, such as those typically classified as megafauna. We explore this issue in living equids, focusing on ventral cranial variation at both micro- and macro-evolutionary levels. By comparing 2D and 3D data, we found that size is well approximated, whereas shape is more strongly impacted by 2D inaccuracies, as it is especially evident in intra-specific analyses. The 2D approximation improves when shape differences are larger, as in macroevolution, but even at this level precise inter-individual similarity relationships are altered. Despite this, main patterns of sex, species and allometric variation in 2D were the same as in 3D, thus suggesting that 2D may be a source of 'noise' that does not mask the main signal in the data. However, the picture that emerges from this and other recent studies on 2D approximation of 3D structures is complex and any generalization premature. Morphometricians should therefore test the appropriateness of 2D using preliminary investigations in relation to the specific study questions in their own samples. We discuss whether this might be feasible using a reduced landmark configuration and smaller samples, which would save time and money. In an exploratory analysis, we found that in equids results seem robust to sampling, but become less precise and, with fewer landmarks, may slightly overestimate 2D inaccuracies.