## 2 Methods
### Principal Strain-Rates  
Up to this point, we have demonstrated how to calculate the elemental strain-rate. Now, our purpose is to identify the data cloud transformation throughout the visualization of the strain-rate patterns. That is, we need to identify the extrema strain-rates and their orientations. In a formal sense, this is the well-known _eigenvalues and eigenvectors_ problem, which is stated as: if \(T\) is a linear transformation from a vector space \(V\) over a field \(F\) into itself, and \(v\) is a vector in \(V\) that is not the zero vector, then \(v\) is an eigenvector of \(T\) if \(T(v)\) is a scalar multiple of \(v\). Knowing that by definition the second-order strain-rate tensor is a linear operator from a vector field into another first-order tensor field, the previous definition applied to the strain-rate tensor leads to:  
\[\left[\mathbf{E}\left(K,t^{n}\right)-\mathbf{I}\lambda\left(K,t^{n}\right)\right]\mathbf{n} \left(K,t^{n}\right)= 0, \tag{12}\]  
where \(\mathbf{I}\) is the \(3\times 3\) identity tensor; \(\mathbf{n}\left(K,t^{n}\right)\in\mathbb{R}^{3}\) is a normalized (non zero), i.e., unit, vector called eigenvector; and \(\lambda\left(K,t^{n}\right)\in\mathbb{R}\) is the eigenvalue associated with the eigenvector. In other words, an eigenvector is a vector that changes by only a scalar factor when the strain-rate tensor is applied to it, resulting in a vector parallel to itself. By solving Equation (12),one obtains three different eigenvalues \(\lambda_{1}\left(K,t^{n}\right),\lambda_{2}\left(K,t^{n}\right),\lambda_{3} \left(K,t^{n}\right),\) and three eigenvectors \(\mathbf{n}_{1}\left(K,t^{n}\right),\mathbf{n}_{2}\left(K,t^{n}\right),\mathbf{n}_{3}\left(K,t^{n}\right),\) associated with each eigenvalue. The eigenvalues and eigenvectors describe the principal magnitudes and orientations of the strain-rate tensor: since the diagonal components of the strain-rate tensor \(E_{11}\left(K,t^{n}\right),E_{22}\left(K,t^{n}\right),\) and \(E_{33}\left(K,t^{n}\right)\) have different values in different reference systems, one finds with the set of eigenvalues the extreme--maximum and minimum--possible values that any of these components may take. Indeed, the maximum and minimum stress-rates--and their orientations--are related to the maximum and minimum eigenvalues. In this work, we follow the notation in which positive values for the eigenvalues represent the extension-rate and negative values represent contraction-rate. Hence, \(\lambda_{1}\left(K,t^{n}\right)\) is the maximum and positive eigenvalue meaning extension-rate, \(\lambda_{3}\left(K,t^{n}\right)\) is the minimum and negative eigenvalue meaning contraction-rate, and \(\lambda_{2}\left(K,t^{n}\right)\) is either extension or contraction rate, but in smaller magnitude. Hence, with the extrema strain-rates at the elemental level, we can reveal the deformation trend of the data cloud, and, above all, locating which regions suffer the most abrupt change in the time-span. We also propose to draw the family of curves--trajectories--that are instantaneously tangent to \(\lambda_{1}\boldsymbol{n}_{1}\left(K,t^{n}\right),\lambda_{2}\boldsymbol{n}_ {2}\left(K,t^{n}\right),\) and \(\lambda_{3}\boldsymbol{n}_{3}\left(K,t^{n}\right)\) in the complete mesh \(\Omega\). This results in a graph plot for each of the \(\lambda\boldsymbol{n}\left(K,t^{n}\right)\) fields that can be easily represented in a two-dimensional framework. These tangent lines can be colored according to the eigenvalue magnitude such that they completely illustrate the main patterns of change inside the data cloud. Note, nevertheless, that \(\lambda\boldsymbol{n}\left(K,t^{n}\right)\) is a composition of a vector using tensor components. Those differ in formal definition, but we use this concept merely for visualization purposes.