**5. Conclusions**

The presented continuous coordinate system is an extension of some previously known discrete coordinate systems, e.g., of the symmetric coordinate frame for the triangular grid. This extension is needed and helpful for various applications, where the grid points are not necessarily mapped to grid points, e.g., arbitrary angled rotations, zooming or interpolation of images. We should also mention translations of images [25] since the triangular grid is not a point lattice. Mathematical morphology operators are also based on local translations [21]. Thus, our coordinate system provides a new tool for the research direction as well. The proposed system addresses each point of the 2D (triangular) plane. Conversion to and from the Cartesian coordinate system is provided. These mappings are inverses of each other. Thus, the new coordinate system is ready to use in various applications including those operations that do not necessarily map the grid to itself.

**Author Contributions:** B.N. and K.A. both designed research and wrote the paper. Both authors have read and approved the final manuscript.

**Funding:** The research was completed under the research project BAPC-04-18-03 of the Eastern Mediterranean University.

**Acknowledgments:** Comments of the anonymous reviewers are gratefully acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.
