Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane

Abuhmaidan, Khaled; Nagy, Benedek

doi:10.3390/math8010029

Cited by 9 publications

(3 citation statements)

References 13 publications

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“…It is easy to see that only even vectors (i.e., triplets with 0 sum, translate the grid into itself. Detailed descriptions about isomeric transformations of the grid can be seen in, for example, Abuhmaidan & Nagy, 2020; Nagy, 2009). Theorem The weighted distances are translation invariant for vectors translating the grid into itself , i.e., for a vector

t = ()(,,, t ()(, 1), t ()(, 2), t ()(, 3))

with

t ()(, 1) + t ()(, 2) + t ()(, 3) = 0

d ()(, p, q; α, β, γ) = d ()(, p^{'}; q^{'}, α, β, γ)

where

p^{'} = p + t, q^{'} = q + t

with

p + t = ()(,,, p ()(, 1) + t ()(, 1), p ()(, 2) + t ()(, 2), p ()(, 3) + t ()(, 3))

and similarly for

q + t

.Proof Let

Π

be a smallest weighted path from

p

q

.…”

Section: Properties Of the Distancesmentioning

confidence: 99%

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Weighted distances and distance transforms on the triangular tiling

Nagy

2023

Transactions in GIS

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Digital geometry is a field in the intersection of discrete mathematics and geometry having various applications including geographical information systems (GIS). In digital spaces, in grids, distances can be defined based on steps in paths in somewhat similarly as in graph theory. However, the grids have more definite structures, thus one may obtain more concrete results, for example, close formulae, than on arbitrary graphs. In this article, the weighted (also called chamfer) distances, and based on them, the distance transform are investigated on the regular triangular grid. Three types of neighborhood relations are used on the grid, and therefore, three weights are used to define a distance function. Natural conditions are used on the weights such as they are positive and a larger step (in the usual and also in the Euclidean sense) cannot have a smaller weight than a smaller one. Some properties of the weighted distances are discussed; for example, they are proven to be metrics. We also give algorithms and formulae that compute the weighted distance of any point pair on a triangular grid. Algorithm for weighted distance transform is provided based on wave‐front propagation. Therefore, these new distance functions are ready for further applications in GIS, in image processing tasks, in computer vision, in graphics, in networking, and also in other applied fields.

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t = ()(,,, t ()(, 1), t ()(, 2), t ()(, 3))

with

t ()(, 1) + t ()(, 2) + t ()(, 3) = 0

d ()(, p, q; α, β, γ) = d ()(, p^{'}; q^{'}, α, β, γ)

where

p^{'} = p + t, q^{'} = q + t

with

p + t = ()(,,, p ()(, 1) + t ()(, 1), p ()(, 2) + t ()(, 2), p ()(, 3) + t ()(, 3))

and similarly for

q + t

.Proof Let

Π

be a smallest weighted path from

p

q

.…”

Section: Properties Of the Distancesmentioning

confidence: 99%

Section: Propertie S Of the D Is Tan Ce Smentioning

confidence: 99%

Weighted distances and distance transforms on the triangular tiling

Nagy

2023

Transactions in GIS

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“…Moreover, this system can be seen as an extension of the discrete coordinate systems of the hexagonal and triangular grids. However, since the vectors of the grid fulfilling some constraints (e.g., at least one of the coordinates is an integer and the sum of the three coordinates is in the closed interval [−1,1]), the vector addition (that is closely connected to translations of images [ 16 ]) and other operations with these vectors are not straightforward. This is the topic of this paper: as a continuation of our earlier paper [ 15 ], we provide a procedure to add two (or more) vectors, to subtract vectors, and to compute the scalar product of a vector (with integer coefficient) on the continuous coordinate system for the triangular grid.…”

Section: Introductionmentioning

confidence: 99%

Vector Arithmetic in the Triangular Grid

Abuhmaidan¹,

Aldwairi

Nagy

2021

Entropy

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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.

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Non-traditional 2D Grids in Combinatorial Imaging – Advances and Challenges

Nagy

2023

Lecture Notes in Computer Science

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Bijective, Non-Bijective and Semi-Bijective Translations on the Triangular Plane

Cited by 9 publications

References 13 publications

Weighted distances and distance transforms on the triangular tiling

Weighted distances and distance transforms on the triangular tiling

Vector Arithmetic in the Triangular Grid

Non-traditional 2D Grids in Combinatorial Imaging – Advances and Challenges

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