## 2 Formulation of the Problem
DIC helps in understanding the mechanism of cracking [26; 39] and can be used to monitor the deflection of a bridge [40] or to detect damage [23]. The works [41; 42] compared the DIC method with laser vibrometry in the application for dynamic measurements. Due to the fact that the result of DIC measurements is information about displacements or deformations of the studied area, this method was readily used to obtain data for the validation of numerical models [43]. Other examples of DIC usage are described in [44; 45; 46; 47]. On the basis of the numerical examination of a two-story steel portal frame (see Figure 2a), the hypothesis was formulated that observation of rotations of elements during dynamic tests can be useful for the detection of damage in nodes. However, numerical studies are not the sole subject of consideration, but they underlie the research presented here. In this paper, 2D DIC was used--indirectly--as a rotation gauge. Figure 1: Scheme of the proposed procedure. The dynamic measurement of the angle of rotation is difficult or impossible to perform using the usual equipment. Even in static examination (e.g., in measurements of rotational stiffness of  
Figure 2: **(a)** The examined structure before testing, **(b)** the node under examination, **(c)** a scheme of the structure with its dimensions and cross sections, where the observed connection is marked with a red oval, **(d)** realized scenarios of the anomalies’ appearances, **(e)** a connection prepared for Digital Image Correlation (DIC) examinations (with a speckled pattern), **(f)** a picture registered by the camera, with the net of measured points (on every crossing of red lines). connections in steel structures), it is rather complicated [48; 49], but the changes in the angle of rotation can also be indirectly measured by observation of changes in the shape of the selected area. During the motion of the element recorded in photos, all types of displacements are combined (vertical and horizontal displacements as well as rotations). When using Cartesian coordinates, the description of a number of subsequent planar transformations (translations, rotations, scaling, etc.) requires that single transformation matrices (describing subsequent transformations) have to be combined through addition and/or multiplication. In computer graphics, this problem is avoided by the application of so-called _homogeneous_ coordinates [50], which allow for a description of all the transformations using a single \(3\times 3\) transformation matrix. Each point with Cartesian coordinates \((x_{C},y_{C})\) is represented by the \((x,y,W)\) homogeneous coordinates (for \(W=const\), the points are located on the same projective plane). Transformation from homogeneous to Cartesian coordinates is simple:  
\[ \begin{array}{l}x_{C}=x/W\\ y_{C}=y/W\end{array} , \tag{1}\]  
but there is no unique transformation in the other direction, since there are many possible projective planes (values of W). Inspired by affine transformation in homogeneous coordinates, the connection between the new location of points can be estimated according to Equation (2), if only the transformation is affine and the coefficient matrix is known. \[\left[ \begin{array}{c}x^{\prime}\\ y^{\prime}\\ 1\end{array} \right]=\left[ \begin{array}{ccc}a&b&c\\ d&e&f\\ 0&0&1\end{array} \right]\left[ \begin{array}{c}x\\ y\\ 1\end{array} \right] \tag{2}\]  
where  
\(\{x^{\prime},y^{\prime},1\}\) -- new homogeneous coordinates;  
\(\{x,y,1\}\) -- old homogeneous coordinates;  
\(a,b,c,d,e,f\) -- transformation matrix coefficients. The problem can be inverted: when the old and new homogeneous coordinates of three points are known, the coefficients of the transformation matrix (the so-called _affine parameters_) can be calculated and describe the transformation that has taken place (neglecting the type of motion). Making a decision about the loosening of connectors based on six transformation coefficients means working in a 6D space, which is quite a complicated task.