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Vector Arithmetic in the Triangular Grid.

Khaled Abuhmaidan1, Monther Aldwairi2, Benedek Nagy3

  • 1Department of Computing and IT, Global College of Engineering and Technology, CPO Ruwi 112, Muscat Sultanate P.O. Box 2546, Oman.

Entropy (Basel, Switzerland)
|April 3, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces novel vector arithmetic formulas for triangular grids, overcoming challenges with standard coordinate systems. These methods enable precise calculations for applications in digital image processing and simulations.

Keywords:
coordinate systemsdigital geometrydiscretized translationsnonlinearitynontraditional gridtriangular gridtriangular symmetryvector additionvector arithmetic

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Area of Science:

  • Computational Geometry
  • Digital Image Processing
  • Scientific Simulation

Background:

  • Vector arithmetic is fundamental to geometry and physics, typically using Cartesian coordinates.
  • Triangular grids, while regular, pose challenges for vector addition due to not being a point lattice.
  • Existing methods for triangular grids struggle with straightforward vector arithmetic operations.

Purpose of the Study:

  • To develop and present formulae for vector arithmetic (sum, difference, scalar product) on a continuous coordinate system for triangular grids.
  • To address the limitations of standard coordinate systems in handling triangular grid vector operations.
  • To facilitate applications requiring precise vector calculations on triangular grids.

Main Methods:

  • Representation of triangular grid points using zero-sum and one-sum coordinate-triplets.
  • Expansion of the coordinate system to the plane with specific restrictions.
  • Derivation of novel formulae for vector addition, subtraction, and scalar multiplication.

Main Results:

  • Successfully derived formulae for fundamental vector arithmetic operations on the triangular grid coordinate system.
  • The provided formulae ensure results adhere to the defined coordinate system constraints.
  • Demonstrated the applicability of the formulae for essential computations like discrete rotations and interpolations.

Conclusions:

  • The developed vector arithmetic formulae provide a robust solution for triangular grid computations.
  • This work is crucial for advancing applications in digital image processing, cartography, and physical simulations on triangular grids.
  • Enables more efficient and accurate manipulation of data represented on triangular grids.