A Continuous Coordinate System for the Plane by Triangular Symmetry

Abstract

The concept of the grid is broadly used in digital geometry and other fields of computer science. It consists of discrete points with integer coordinates. Coordinate systems are essential for making grids easy to use. Up to now, for the triangular grid, only discrete coordinate systems have been investigated. These have limited capabilities for some image-processing applications, including transformations like rotations or interpolation. In this paper, we introduce the continuous triangular coordinate system as an extension of the discrete triangular and hexagonal coordinate systems.The new system addresses each point of the plane with a coordinate triplet. Conversion between the Cartesian coordinate system and the new system is described. The sum of three coordinate values lies in the closed interval [−1, 1], which gives many other vital properties of this coordinate system. Symmetry 2019, 11, 191 2 of 17 of the grid. In Figure 1a, the first coordinate value is ascending right-upwardly, the second values are ascending into the right-downward direction, and the third one is ascending into the left-upward direction [7,8]. We should mention that this system could be seen as the extension of the oblique coordinate system using two independent integer values [9] by concerning the third value to obtain zero-sum for every triplet. The digital distance based on the neighborhood relation is computed in Reference [9]. Since the vectors describing the grid are not orthogonal, some geometric descriptions based on Cartesian coordinates are not very clear. However, to simplify the expressions of the constrained three-dimensional coordinate system is recommended. We should also mention that 0-sum triplets allowing real numbers were used in Reference [8] to describe rotations (that may not map the hexagonal grid to itself). In this way, a useful digitization operator is found. Her's system was mentioned and used in References [10,11] for various imaging-related disciplines.Symmetry 2019, 11, 191 3 of 17

Vector Arithmetic in the Triangular Grid

“…In this section, we show, first, how vector addition is related to translations on the traditional grids, i.e., on the square grid. Then, we recall the continuous coordinate system for the triangular grid with some of its crucial properties from [ 15 ].…”

“…Now we show the extension of the previous system to the whole plane [ 15 ]. The continuous coordinate system Ω uniquely addresses every point of the plane.…”

“…Then, Equation (1) is used to compute the corresponding coordinate values x and y for the given triplet ( i , j , k ). The conversion is recalled from [ 15 ].…”

“…Of course, the three values are not independent for the hexagonal and triangular grids, since they are also 2D grids. To work with various algorithms in computer graphics and image processing that may not map the grid into itself, one needs the continuous extension of this coordinate system as well, which was recently developed [ 15 ]. This continuous coordinate system for triangular grid (Ω) is used to describe every point of the plane with a unique coordinate triplet.…”

“…However, since the vectors of the grid fulfilling some constraints (e.g., at least one of the coordinates is an integer and the sum of the three coordinates is in the closed interval [−1,1]), the vector addition (that is closely connected to translations of images [ 16 ]) and other operations with these vectors are not straightforward. This is the topic of this paper: as a continuation of our earlier paper [ 15 ], we provide a procedure to add two (or more) vectors, to subtract vectors, and to compute the scalar product of a vector (with integer coefficient) on the continuous coordinate system for the triangular grid. Although we work with continuous coordinate system which uses real numbers, our work is essential in discrete mathematics, especially, in digital geometry to work, e.g., with digital images on the triangular grid.…”