# Bisector-Based Tracking of In Plane Subpixel Translations and Rotations
## 2 Conceptual Approach  
Let us consider a simple object in Figure 1 consisting of two nonparallel straight segments \(s_{1}\) and \(s_{2}\). The straight lines that best fit to these segments are \(l_{i}\), with \(i=1\), 2, and they fulfil Equation (1):  
\[y=\tan(\theta_{i})x+b_{i}=m_{i}x+b_{i}, \tag{1}\]where \(b_{i}\) are the \(y\)-intercepts of each line, and \(\theta_{i}\) are the angles with respect the abscises axis. The straight lines intersect in a point \((x_{c},y_{c})\) that can be used to locate the position of the object [14]. As we said above, by only tracking the lines' intersection, we are not able to distinguish orientation changes, and thus the object's movement will only be partially described. Therefore, we need an extra parameter that provides information on the orientation of the assembly formed by the two segments. To this end, the trivial solution is selecting the angle between them, e.g., \((\theta_{1}-\theta_{2})\) in Figure 1. Obtaining all of the describing parameters, i.e., the crossing point and the angle between segments, implies computing the four unknown parameters in (1). Although valid, this procedure has two main drawbacks. The first is that it considers the two segments as independent from one another, so it does not inform about the shape invariance of the target as a whole. The second drawback is that a combination between two independent adjustments may increase the errors, thus making an unreliable method for tracking small movements. Here, we propose an alternative method that overcomes those drawbacks. If the object is rigid and its shape does not change, the Equation in (1) can be transformed and linked to reduce the number of unknowns, and then the accuracy of the tracking algorithm would be increased. Following with Figure 1, the straight lines define two pairs of opposite angles in the plane. These angles are divided into two equal parts by the lines called angle bisectors. The bisectors can be used as axes of a new Cartesian system due to the fact that they perpendicularly intersect. Hence, they form a coordinates system centred at the intersection point \((x_{c},y_{c})\) that moves with the object. Therefore, our proposal is not solving the Equation in (1) but to find the new coordinates system formed by the bisectors and track its movement. In that way, we reduce the unknown to three: the coordinates of the intersection point \((x_{c},y_{c})\), or equivalently, how much the original coordinates system must be translated to match the intersection with the origin, and the angle (\(\beta\)), which means how much the original coordinates system must be rotated to match the axes with the bisectors. Therefore, the aim now is to compute vector \((x_{c},y_{c},\,\beta)\) that locates \((X^{\prime},\,Y^{\prime})\) in position and orientation. A particular feature of the Cartesian system formed by the bisectors is that the axes act as reflecting axes of symmetry, i.e., the specular reflection of one of the lines relative to one of the axes coincides with the other line. This can be observed in Figure 2, where we have represented the rotated and re-centred version of the scheme depicted in Figure 1. There, \(\xi_{1}\), the reflection of \(s_{1}\) with respect to \(X^{\prime}\)  
Figure 1: Two nonparallel segments (blue lines), best fitting straight lines to each segment (dashed blue lines), intersection \((x_{c},y_{c})\), and bisectors (red lines). matches with the line \(l_{2}\). Therefore, both straight lines \(l_{1}\) and \(l_{2}\) referred to (X\({}^{\prime}\), Y\({}^{\prime}\)) are equivalent if we change the sign of one of the coordinates to one of them (2). Thus, this feature can be used as a condition to link Equation (1) and to obtain (\(x_{c}\), \(y_{c}\), \(\beta\)). \[