Abuhmaidan Khaled, Aldwairi Monther, Nagy Benedek
Department of Computing and IT, Global College of Engineering and Technology, CPO Ruwi 112, Muscat Sultanate P.O. Box 2546, Oman.
College of Technological Innovation, Zayed University, 144534 Abu Dhabi, United Arab Emirates.
Entropy (Basel). 2021 Mar 20;23(3):373. doi: 10.3390/e23030373.
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 restrictions 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.
向量运算是（坐标）几何、物理及其他各种学科的基础。常用方法基于笛卡尔坐标系，它既适用于连续的平面/空间，也适用于数字矩形网格。三角形网格也是规则的，但它不是点格：它在向量加法下不封闭，这带来了挑战。三角形网格的点由零和与一和坐标三元组表示，保持了网格的对称性并反映了三角形的方向。该系统通过诸如至少一个坐标为整数且三个坐标之和在区间[-1,1]等限制扩展到平面。然而，向量运算仍然不直接；通过纯粹相加两个这样的向量，结果可能不满足上述条件。另一方面，对于数字网格的各种应用，例如在图像处理、制图和物理模拟中，人们需要进行向量运算。在本文中，我们提供了给出连续坐标系中向量的和、差及标量积的公式。我们的工作对于诸如计算三角形网格上图像的离散旋转或插值等应用至关重要。
