*Article* **A Continuous Coordinate System for the Plane by Triangular Symmetry**

#### **Benedek Nagy \* and Khaled Abuhmaidan**

Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, via Mersin 10, Turkey, Famagusta 99450, North Cyprus; kabuhumaidan@yahoo.com

**\*** Correspondence: nbenedek.inf@gmail.com

Received: 20 December 2018; Accepted: 31 January 2019; Published: 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
