Vector Arithmetic in the Triangular Grid

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.