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Symmetry 2019, 11(2), 191; https://doi.org/10.3390/sym11020191

A Continuous Coordinate System for the Plane by Triangular Symmetry

1
Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, via Mersin 10, Turkey, Famagusta 99450, North Cyprus
2
*Correspondence: [email protected]
*
Author to whom correspondence should be addressed.
Received: 20 December 2018 / Revised: 24 January 2019 / Accepted: 31 January 2019 / Published: 9 February 2019
(This article belongs to the Special Issue Symmetry in Applied Mathematics)
PDF [996 KB, uploaded 9 February 2019]

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.
Keywords: barycentric coordinate system; coordinate system; hexagonal grid; triangular grid; tri-hexagonal grid; transformations barycentric coordinate system; coordinate system; hexagonal grid; triangular grid; tri-hexagonal grid; transformations
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Nagy, B.; Abuhmaidan, K. A Continuous Coordinate System for the Plane by Triangular Symmetry. Symmetry 2019, 11, 191.

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