# Discrete Cartesian Coordinate Transformations: Using Algebraic Extension Methods
## 2 Background: Technical Aspect
(The hyperbola branches corresponding to the second pair are shown with blue lines.) The signal source coordinates correspond to the intersection of the corresponding hyperbola branches. It is impossible not to see that the problem under consideration is reduced to a geometric one. The intersection point of the above hyperbolas corresponds to the solution of a system of two quadratic equations. Such systems of equations have been considered in many works; it is recognized that their solution is one of many nontrivial problems of classical algebra [60; 61]. To solve applied problems of this kind, numerical methods are currently usually used [62; 63]. There is, however, a significant nuance. The numerical solution of even a relatively simple system of quadratic equations requires fairly serious computing resources. At a minimum, onboard computing systems must provide for calculations using fractional values, which significantly complicates their algorithmic basis. It is much easier to implement calculations that correspond to integers, even in the case when a specific integer corresponds to a variable of multivalued logic. Moreover, in this case, there are quite a few opportunities to make the computational procedures more efficient (e.g., using methods using residue number systems (RNS) [64; 65; 66]). In relation to the problem considered above, the transition-to-integer calculations correspond to the transition to a discrete coordinate grid. Indeed, to ensure the identification of a signal source as \"one's own\", there is no need to determine its coordinates with high accuracy. It is sufficient to determine the corresponding square on a discrete coordinate grid. This, in turn, means it is permissible to move from the equations describing the intersection of hyperbola branches in terms of continuity to their discrete analog. In particular, such equations can be rewritten in terms of Galois fields. To do this, it is sufficient to move to the coordinate grid schematically shown in Figure 2.  
This grid corresponds to the representation of the elements of the field \(GF(p^{2})\) in the form (1). One of the coordinates in this figure corresponds to the variable \(x_{1}\) and the second to the variable \(x_{2}\), and both variables belong to a discrete set, specifically the field \(GF(p)\). (The corresponding formulas are considered in the next section.) For illustrative purposes, the case of three UAVs forming a group is considered. The figure emphasizes that in order to ensure information protection by the considered method, it is enough to define the square where the operator is located (highlighted in blue in Figure 2). Classical methods of studying quadratic equations of two variables are associated with coordinate transformations. It is this approach that allows, among other things, to classify quadratic equations of the type under consideration (ellipse, parabola, and hyperbola). Consequently, the first task that arises when solving the problems under consideration in terms of finite algebraic structures is the task of optimizing the representations of coordinate transformations. One of the options for solving this problem is proposed in this paper. Figure 1: Illustration of the geometric problem corresponding to the determination of the operator’s coordinates. 1: UAVs forming a group. 2: Radio signal source. The branches of hyperbolas corresponding to two pairs of UAVs located at their focuses are represented by blue and red colors. Note also that the basis of the proposed approach is the fact that the \"map\" corresponding to real practical applications is always finite. This applies to the use of UAVs and many other applications. Thus, for digital image processing tasks generated by modern computers, their range of variation is also finite. This means that, in this case, it is acceptable to use Formula (1) to display the coordinates of each pixel.