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.
Highlights
The concept of the grid is essential and heavily used in digital geometry and in digital image processing
Most of the applications use the square grid because its orthogonal coordinate system, known as the Cartesian coordinate system (CCS), which fits very well to it
The Cartesian coordinate system allows real numbers to be used in such cases
Summary
The concept of the grid is essential and heavily used in digital geometry and in digital image processing. The hexagonal grid, tiling the plane by the same size regular hexagons (hexagonal pixels), has been used for decades in image processing applications [2], in cartography [3,4], in biological simulations [5], and in other fields, since the digital geometry of the hexagonal grid provides better results than the square grid in various cases It is used in various table and computer games based on its compactness. We introduce a continuous coordinate system for the plain based on the symmetry of the triangular grid, where every point of the 2D plane has its unique coordinate triplet. We use three coordinate values to describe the triangular grid as in Reference [15] and to address the points of the plane “between” and “around” the nodes of the dual grid.
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