Figure 3.

Percentage of shape changes within the $xy$-plane (black) and along the z direction (gray). The proportion of shape effects β that lie along the $m^{th}$ dimension (m= 1, 2, or 3) is the sum of the squared effects over the k landmarks on the $m^{th}$ dimension normalized by the norm of the effect, ${\displaystyle {\sum }_{i=1}^k{\mathrm{\beta }}_{i,m}^2/\Vert \mathbf{\beta }\Vert }$. The figure may be understood as the proportion of changes that are embedded in the flat plane ($xy$) or that get out of this plane (z). Such parametrization is sensical here, despite the fact that shapes are invariant to rotation by definition, only because they were specifically oriented according to this specific coordinate system. (A) Principal components (PC), covariate, and QTL (quantitative trait loci) effects for 3D landmarks. (B) Principal components, covariate, and QTL effects for semilandmarks. Covariates are noted CS for the log of the centroid size, X for the direction of the cross, and G for gender.