# Discrete Cartesian Coordinate Transformations: Using Algebraic Extension Methods
## 1 Introduction  
Nowadays, objects corresponding to discrete Cartesian coordinates are used more and more frequently. A typical example in this respect is the tasks related to digital image processing [1, 2, 3]. In this case, the image is dissected into a set of pixels, each of which can be matched by two integers, which actually represent discrete Cartesian coordinates. For digital image processing, convolutional neural networks are now being increasingly used [4, 5, 6, 7]. One of the advantages of using such networks is the possibility of staggered training. This approach, in particular, allows analyzing an image as consisting of individual fragments of simple geometric figures, i.e., straight-line segments, fragments of circles, ellipses, parabolas, etc. [8, 9, 10]. When Cartesian coordinates with continuously varying coordinate values are used (hereinafter referred to as \"continuous coordinates\"), differentfragments of simple geometric figures can be reduced to each other using affine coordinate transformations. If discrete coordinates are used, especially in tasks involving the use of convolutional neural networks, the task becomes somewhat more complicated: ideally, a convolutional neural network oriented at dissecting an image into simple geometric elements should recognize, for example, any straight-line segments as the same geometric object, differing from each other only by coordinate transformations on the initial grid. The above studies can be referred to the field of discrete geometry [11; 12], various aspects of which have also been very intensively developed recently [13; 14; 15]. Another nontrivial area of application of discrete geometry becomes relevant due to the increasing use of unmanned aerial vehicles, including for military purposes [16; 17; 18]. As the current experience of using such vehicles shows, their vulnerability to electronic warfare methods becomes more and more significant. This fact forces us to pay attention to the use of UAV control channels protected physically. For this purpose, various approaches are currently proposed, including those using data series that are formed as a result of certain physical processes [19; 20] (e.g., geophysical [21]). Another method of forming a secure data channel is the use of optical fiber. This approach has been previously discussed in [22; 23; 24], and according to the open press, FVP drones controlled via fiber optics are finding increasing use in the military conflict in Ukraine. Another of the alternative options for protecting information (at least the part of it related to executable commands) involves identifying the location of the radio signal source, which is interpreted as \"one's own\" [25]. Let us take into account that in order to identify the location of a signal that is interpreted as \"one's own\", it is not necessary to determine its coordinates with great accuracy. It is enough to localize the area of its location. This allows one to move from the classical formulation of the problem to the formulation of the problem from the field of discrete geometry. Simplifying, a map on which the position of the control signal source is marked can be reduced to a discrete form with a sufficiently coarse partitioning into squares. These issues are discussed in more detail in Section 2; for now, let us return to the purely algebraic aspects of the problem at hand. Description of coordinate transformations can be realized by various methods, especially when two-dimensional problems are considered. In this case, in particular, the coordinates of a point in the plane can be given through a complex variable. On this basis, in particular, the theory of comfort transformations, which are widely used in problems of hydrodynamics, electrostatics, etc., is constructed [26; 27; 28]. In this case, some operators describing the coordinate transformation can be reduced to the operation of multiplication by a complex number [29]; however, the set of such operators is finite.