Document Type

Article

Source of Publication

Entropy

Publication Date

3-1-2021

Abstract

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using re-strictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid.

Publisher

MDPI AG

Volume

23

Issue

3

Disciplines

Physical Sciences and Mathematics

Keywords

Coordinate systems, Digital ge-ometry, Discretized translations, Nonlinearity, Nontraditional grid, Triangular grid, Triangular symmetry, Vector addition, Vector arithmetic

Scopus ID

85103489027

Creative Commons License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

Indexed in Scopus

yes

Open Access

yes

Open Access Type

Gold: This publication is openly available in an open access journal/series

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