## 3 Results
### Principal Strain-Rates
From these representations, we can understand the magnitude and orientation of each principal component of the strain-rate tensor. More importantly, the trajectory patterns overlapped with the coordinates of the districts (in Figure 3) provide information about local regions of extension and contraction rates inside the urban description, where extension-rate patterns indicate differentiation and contraction-rate patterns indicate clustering--or homogenization. This can be observed from the color intensity: locations of high strain-rate are colored with great intensity, while the locations of low deformations are not visible since the white color represents null strain-rates. In the case of the first strain-rate component, which is shown in Figure 5 (top), we observe that the larger magnitude of extension-rate is localized in the middle of Nou Barris Sant Andreu, Sant Marti, Horta and Sants, and that it decreases near Eixample, Les Corts, and Ciutat Vella. Therefore, the main transformation is located at the group of nearby districts: Nou Barris, Sant Andreu, Sant Marti, and Horta. The extension-rate patterns are oriented from this group apart to Ciutat Vella, suggesting that there is a divergence of Ciutat Vella from the grouped districts. Indeed,  
Figure 5: The trajectory patterns of the principal strain-rates at the year 2003: first principal strain-rate (**top**); second principal strain-rate (**middle**); and third principal strain-rate (**bottom**). The color scale in these graphs determines the strain type: the red color is used to represent extension, while the blue color depicts contraction. the main extension pattern is oriented along the PC3 dimension and covers the grouped districts. The pattern which comprises the districts of Nou Barris, Sant Marti, and Sants and ends at Gracia and Example is of lesser importance. Contraction-rate, on the other hand, is expressed by the third principal strain-rate component, which by definition is orthogonal to the first and second principal strain-rates. The third principal strain-rate component is shown in Figure 5 (bottom), where we can appreciate this orthogonality by noticing that the trajectories of the third principal strain-rate are perpendicular to the extension-rate pattern. We observe that the contraction-rate trajectories are mostly homogeneous, with a minor importance among Sant Marti, Sant Andreu, Nou Barris and Horta districts, and completely declining at Sants and Gracia. This direct relation between extension and contraction is found in solids deformations, where it is ruled by the conservation of mass--or Poisson ratio [23]. Apart from the extension and contraction patterns of the mesh, locations of smaller strain-rates are represented by the second principal component. Considering Figure 5 (middle), we recognize that the orientation of this strain-rate component is concentrated in the middle of Les Corts, Sarria, Horta and Nou Barris, and that it is directed towards Example, fading at Ciutat Vella. This principal strain-rate component is certainly orthogonal to the first and second components, but it implies a strain-rate pattern that is two orders of magnitude smaller. In the previous lines, we have demonstrated the application of the trajectories diagrams of the principal strain-rate components as a powerful visualization technique of the three-dimensional strain-rate state of a data cloud. The strain-rate patterns can be used to analyze the system's development, for example, with the identification of regions with a special behavior: although there are some grouped districts in the case study, they are all separated at a high rate in the first and third principal components (spatial dimensions). Hence, those are differentiating themselves in the PC1 and PC3 description. On the contrary, low strain-rates can be an indication of stagnation, and, thus, an expression of inactivity where an abrupt change is unlikely to occur.