## 1 Introduction
In SAR mode (Synthetic Aperture Radar) the instrument is mounted on a slide and allows to obtain the mapping of the area investigated at different times and to get the deflection of each point. In both cases the precision of the displacement measurements is about 0.1 mm. This technique is increasingly used, but is still limited by the high cost of the instrumentation [24; 25]. Nowadays TLS is a widespread and reliable technique for monitoring bridges in static conditions [26]. Comparing the scans acquired in different epochs, we can get, e.g., the maps of deviations between the surface points of a bridge subject to various conditions (loads, temperature, etc.). As for dynamic monitoring, it is possible to take advantage of the high sampling rate of TLS; specifically, we can measure the deflections of a bridge superstructure dynamically. In this regard a point-surface based method has been proposed in [27]. We can see that, even if the coordinates of the single points obtained by a TLS are of low precision (\(\pm\)2 mm to \(\pm\)20 mm), a 2D/3D model of the entire point cloud can be effective for detecting the shape of a structure and its changes. Therefore, if the goal of a TLS survey is to model a line or a surface starting from a number of points, we can observe that the best interpolating line/surface has generally a rather better precision than any single point; for this reason, the reconstructed lines/surfaces show a better precision than the one declared by the manufacturers. In other words, a modeled surface will represent an object more precisely than the unmodeled observations. Starting from this remark, [PERSON] and [PERSON] [26] have developed a methodology devoted to the measurement of the deformation of structures. At the basis of the method are theoretical elements of mechanics of beam. For its implementation, least-squares are used. This method is based on obtaining analytical models that represent the physical bending of a structure and, in particular, of a beam. The models, derived from the governing differential equation for the elastic curve, are represented, in the case of simple static schemes, by low order polynomials. The coefficients, treated as unknown parameters, are estimated by solving a least squares procedure. Such a method is effectively applicable when two conditions are satisfied: (a) the theoretical model and the real structure are really similar and (b) the structure had not experienced events such as yielding of foundation or phenomena such as relaxation and creep. In this paper, in light of all the above considerations, a methodology has been developed that allows to obtain the displacements and, consequently, the elastic curve of a structure using a TLS. The instrument must be configured as a line scanner and able to provide line scans, along with the timestamp for the detected points [28]. The displacements at time \(t\) are given by the difference between the best fitting line of the points acquired at the same time for a bridge with a loaded deck, and the fitting line obtained for the unloaded bridge. These displacements provide the instant elastic curve of the structure subject to the mobile load present at the time \(t\). The procedure can also be used for vertical structures subjected to loads with a horizontal component (tall buildings, pylons, etc.). The main characteristics of the method are: (1) Ease of installation; (2) The accuracy required to monitor the deflection of the bridges; and (3) High acquisition rate (up to 120 lines per second) useful for monitoring phenomena characterized by rapid changes. The methodology for dynamic monitoring of displacements and the elastic curves of the structures presented below, is characterized by high precision, is easily implementable and can be effectively used within the SHM. It is worth mentioning that the steps of Structural Health Monitoring are: (1) Determination of damage existence; (2) Determination of damage's geometric location; (3) Quantification of damage severity and (4) Prediction of remaining life of the structure.