for practical reasons (the application of standard statistical methods), the distribution of objects in either of these spaces is approximated by their distribution in a tangent space of the space under consideration, at the point corresponding to the mean shape of the sample (see Dryden and Mardia (1998) for more details).
1.6. The treatment of symmetry in morphometrics
Beyond their classical applications in systematics, paleontology, ecology and evolution, geometric morphometric methods have been extended over the last two decades to the analysis of the symmetry of shapes (Klingenberg and McIntyre (1998) and Mardia et al. (2000), and see Klingenberg (2015) for a review). As mentioned above, it is then possible to decompose the measured morphological variation into symmetric and asymmetric components, capturing the effects of distinct biological processes. When considering the symmetric organization of biological structures, two classes of symmetry are recognized by morphometricians: matching symmetry and object symmetry (Mardia et al. 2000; Klingenberg et al. 2002).
Matching symmetry describes the case where the repeated units that give the biological structure its symmetric architecture are physically disconnected from each other. This is, for example, the case for diptera wings that are attached individually to the second thoracic segment (bilateral symmetry with respect to the median plane of bilaterians), or for the arrangement of petals in many flowers (rotational symmetry). A configuration of landmarks is defined for the repeated unit (e.g. the wing) and digitized for all units (right and left wings). The configurations are then oriented in a comparable way (reflection of the left wings, so that they can be superimposed on the right wings) and analyzed using the Procrustes method (Figure 1.3(a)). In the shape space, a sample of n structures made up of m repeated units appears as m clouds of n points, whose relative overlap indicates the degree of symmetry of the structure. The more the repetition of the units is faithful to the symmetry group, the closer the different units of the same structure are and the more the clouds overlap (Figure 1.3(b)).
Figure 1.3. Morphometric analysis of matching symmetry. For a color version of this figure, see www.iste.co.uk/grandcolas/systematics.zip
COMMENT ON FIGURE 1.3. – a) In the case of bilateral symmetry (the shape is likened to a triangle to facilitate visualization), the unit on one side (blue) is reflected (I) to match its landmark configuration with that of the same unit on the other side (green) (II). The original and reflected units of all the individuals in the sample are then superimposed by the Procrustes method. b) Each individual is then defined by two points in the tangent space corresponding to each of its repeated units. The relative position of the points and clouds of points representing the two sides (blue and green here) describes the degree of deviation from perfect bilateral symmetry at the individual and sample levels. A principal component analysis allows the visualization of the tangent space with a reduced number of dimensions.
In the case of object symmetry, the symmetry operator is an integral part of the biological structure. The m repeated units are thus physically connected and their relative arrangement intervenes in the emergence of the symmetric pattern of the structure. This is, for example, the case for the right and left halves of the human face (bilateral symmetry with respect to the median plane passing through the skull) or for the arms of the starfish (rotational symmetry with respect to the antero-posterior axis passing through the center of the body). For object symmetry, a single configuration of landmarks is defined for the whole structure (and not one for each unit as in the case of matching symmetry). If q is the number of isometries of the symmetry group of the structure, then q copies of the configuration are created and transformed accordingly. All of the configurations and their copies are then superimposed (Figure 1.4(a)). In the shape space, a sample of n structures appears as q clouds of n points, where the relative overlap of the copies reflects the precision of the symmetric pattern of the structure (Figure 1.4(b)).
Figure 1.4. Morphometric analysis of object symmetry. For a color version of this figure, see www.iste.co.uk/grandcolas/systematics.zip
COMMENT ON FIGURE 1.4. – a) In the case of bilateral symmetry (the shape is likened to a triangle for easier visualization), the original structure (orange) has a plane of symmetry. A copy of the structure is generated (the bilateral symmetry group includes two isometries: identity and reflection), reflected (I) and relabeled to ensure the correspondence between homologous landmarks (II). The original and reflected structures of all the individuals in the sample are then superimposed by the Procrustes method. b) Each individual is then defined by two points in the tangent space corresponding to each of its copies (including the original). The relative position of the points and clouds of points representing the two sides (orange and green here) describes the degree of deviation of perfect bilateral symmetry at the individual and sample levels. A principal component analysis not only allows the visualization of the tangent space with a reduced number of dimensions, but also allows the separation of the symmetric and asymmetric components.
In both cases, the statistical exploration of the shape space allows the separation of the symmetric and asymmetric components of the shape variation. However, object symmetry has a specificity. It has been shown that the different symmetric and asymmetric components occupy complementary and orthogonal subspaces of the shape space (Kolamunnage and Kent 2003, 2005). Mathematically, the shape space is the direct sum of the symmetric and asymmetric subspaces. These components are easily separable and their morphological meaning can be interpreted by principal component analysis (Figure 1.5).
Figure 1.5. Morphometric analysis of object symmetry and partition of the symmetric and asymmetric components. For a color version of this figure, see www.iste.co.uk/grandcolas/systematics.zip
COMMENT ON FIGURE 1.5. – The arrangement of the copies of the symmetric structure, generated according to its symmetry group, is such that a principal component analysis automatically separates the symmetric and asymmetric components. Here, we consider a structure similar to a triangle with a plane (axis) of symmetry. The tangent space thus has two dimensions (see Figure 1.2). We observe two clouds of points corresponding to the shapes of the original structures (orange) and to the shapes of the reflected and relabeled structures (green). The principal component analysis defines two main components, which are perfectly separated: the symmetric component, distributed along PC2 (red histogram) and describing the inter-individual variation, obtained by averaging the pairs of points corresponding to the same individual; the asymmetric component, distributed along PC1 (green histogram) and describing the intra-individual variation, obtained by calculating the difference between its original or reflected shape (orange or green points) and its symmetric shape (red points), for each individual.