# Geometric morphometrics

Morphometrics is a quantitative way of addressing the shape comparisons that have always interested biologists (Zelditch et al. 2004). In this sense, Geometric Morphometrics could be defined as the study of form in two or three dimensional spaces (Bookstein 1982) allowing in-depth investigation of morphological change.

Instead of focusing on multivariate analysis of a set of linear measurements between points, Geometric Morphometrics (GM) intends to study changes in size and shape, taking as a point of departure displacements in the plane (2D) or space (3D) of a set of morphometric landmarks. The two- or three- dimensional spatial relationship of these landmarks is always preserved throughout the whole analysis, which allows to accurately build shape and size of the specimen studied.

The set of analytical and graphical methods, collectively called Geometric Morphometrics (GM) was developed during the '80s, although it dates to biometricians from the nineteenth century.

The way to capture the geometry (and therefore the size and shape) of the object of study is by means of morphometric points or landmarks. The latter are characters based on coordinates from which it can be deduced the shape of the bodies and then study and compare using the appropriate methods for describing and analyzing the differences.

A landmark is a point in a two- or three- dimensional space which corresponds to the position of a particular trait on an object (Zelditch et al. 2004). The choice of landmarks depends on the biological question to be answered: depending on the purpose of the study, the homology of the landmarks can be purely operational (from a geometrical correspondence between the landmarks on the objects to biologically homologous structures). For example in osteological remains a landmark can be defined as the point which marks the trace of a muscle attachment, or the foramen which marks the passage of a neurovascular bundle. Landmarks with the same name, counterparts in the semantic sense of the term, are assumed to correspond in the same way all over the objects in the sample studied. Thus, the landmarks can be specific points on the object that show any phylogenetic, structural, functional or developmental significance (Lele and Richtsmeier 2001).

There are three types of landmarks called Type I, II and III by Bookstein (1991); traditional, confused or "fuzzy" and built, by Lele and Richstmeier (2001) and anatomical, mathematical and pseudolandmarks by Dryden and Mardia (1998). The coordinates of landmarks are Cartesian coordinates which locate the position of each landmark, on the plane (x, y) or space (x, y, z) with respect to an axis system.

By definition, Morphometry is the quantitative study of form.In these studies, the form of an object is decompose into size and shape, and is defined as the total geometric information that remains after removing the effects of translation, rotation and scale from the object (Bookstein 1991, Dryden and Mardia 1998). In traditional morphology, there is no single and precise definition of size: any definition is closely linked to the type of analysis to be carried out.

The size of the object of study can be defined as an area, length, weight, volume or centroid. In most tests, the size is represented by a single measure (eg total length, total weight) by a linear combination of measures (eg arithmetic mean) or the relationship between measures (eg area, volume, geometric mean). None of those ways of measuring the size is inherently better than another, but the choice of type of measure is very important to get relevant results according to the analysis used (Richtsmeier et al. 2002). The most commonly size estimator used is the centroid size. The centroid size equals the square root of the sum of the squared distances of landmarks from a set with a previously defined centroid or, in the same way, the square root of the sum of the variances of the landmarks around the centroid in x and y directions (Bookstein 1990, 1991).The centroid size is used because, in the absence of allometry, it is supposed to be uncorrelated with any variable in shape (Bookstein 1986) when the landmarks are distributed around its mean independently and have small and equal variances at each point and in each direction. The centroid size is the size measure used to scale a configuration of landmarks in such a way that it can be projected as a point in the space form of Kendall.

Shape is all the geometrical information that remains after removing the effects of translation, rotation and scale in the object(Bookstein 1991, Dryden and Mardia 1998, Kendall, 1981).