Current options for visualization of local deformation in modern shape analysis applied to paleobiological case studies

In modern shape analysis, deformation is quantified in different ways depending on the algorithms used and on the scale at which it is evaluated. While global affine and non-affine deformation components can be decoupled and computed using a variety of methods, the very local deformation can be considered, infinitesimally, as an affine deformation. The deformation gradient tensor F can be computed locally using a direct calculation by exploiting triangulation or tetrahedralization structures or by locally evaluating the first derivative of an appropriate interpolation function mapping the global deformation from the undeformed to the deformed state. A suitable function is represented by the thin plate spline (TPS) that separates affine from non-affine deformation components. F, also known as Jacobian matrix, encodes both the locally affine deformation and local rotation. This implies that it should be used for visualizing primary strain directions (PSDs) and deformation ellipses and ellipsoids on the target configuration. Using C = FTF allows, instead, one to compute PSD and to visualize them on the source configuration. Moreover, C allows the computation of the strain energy that can be evaluated and mapped locally at any point of a body using an interpolation function. In addition, it is possible, by exploiting the second-order Jacobian, to calculate the amount of the non-affine deformation in the neighborhood of the evaluation point by computing the body bending energy density encoded in the deformation. In this contribution, we present (i) the main computational methods for evaluating local deformation metrics, (ii) a number of different strategies to visualize them on both undeformed and deformed configurations, and (iii) the potential pitfalls in ignoring the actual three-dimensional nature of F when it is evaluated along a surface identified by a triangulation in three dimensions.