Isometric transformations of the dual of the hexagonal lattice

Nagy, Benedek

doi:10.1109/ispa.2009.5297709

Cited by 29 publications

(19 citation statements)

References 8 publications

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“…Without loss of generality we discuss and find the number of shortest paths between trixels P1(0,0,0) and P2(x,y,z) with the condition x,y > 0 and z < 0 (or x > 0 and y,z < 0). Based on translations and mirroring the grid to itself (see [26] for details), any respective position of any two trixels can be mapped to P1 and P2 with this condition. The trixels p(p(1),p(2),p(3)) and q(q(1),q(2),q (3)) are in m-neighbor relation (m ∈ {1,2,3}) if:…”

Section: Preliminariesmentioning

confidence: 99%

“…Without loss of generality we discuss and find the number of shortest paths between trixels P1(0,0,0) and P2(x,y,z) with the condition x,y > 0 and z < 0 (or x > 0 and y,z < 0). Based on translations and mirroring the grid to itself (see [26] for details), any respective position of any two trixels can be mapped to P1 and P2 with this condition. We note here that, when working with a given neighborhood and also at neighborhood sequences, in the second condition the sign ≤ is used and the neighborhood relation is having the extensive property, that is all m-neighbors are also (m − 1)-neighbors (for m > 1).…”

Section: Preliminariesmentioning

confidence: 99%

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On the Number of Shortest Weighted Paths in a Triangular Grid

Nagy

Khassawneh

2020

Mathematics

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Counting the number of shortest paths in various graphs is an important and interesting combinatorial problem, especially in weighted graphs with various applications. We consider a specific infinite graph here, namely the honeycomb grid. Changing to its dual, the triangular grid, paths between triangle pixels (we abbreviate this term to trixels) are counted. The number of shortest weighted paths between any two trixels of the triangular grid is discussed. For each trixel, there are three different types of neighbor trixels, 1-, 2- and 3-neighbours, depending the Euclidean distance of their midpoints. When considering weighted distances, the positive values α, β and γ are assigned to the ‘steps’ to various neighbors. We gave formulae for the number of shortest weighted paths between any two trixels in various cases by the respective weight values. The results are nicely connected to various numbers well-known in combinatorics, e.g., to binomial coefficients and Fibonacci numbers.

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Section: Preliminariesmentioning

confidence: 99%

“…Without loss of generality we discuss and find the number of shortest paths between trixels P1(0,0,0) and P2(x,y,z) with the condition x,y > 0 and z < 0 (or x > 0 and y,z < 0). Based on translations and mirroring the grid to itself (see [26] for details), any respective position of any two trixels can be mapped to P1 and P2 with this condition. We note here that, when working with a given neighborhood and also at neighborhood sequences, in the second condition the sign ≤ is used and the neighborhood relation is having the extensive property, that is all m-neighbors are also (m − 1)-neighbors (for m > 1).…”

Section: Preliminariesmentioning

confidence: 99%

On the Number of Shortest Weighted Paths in a Triangular Grid

Nagy

Khassawneh

2020

Mathematics

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“…The triangular grid is built by triangles in two different orientations. Moreover, it is not a point lattice since some of the grid vectors, i.e., the vectors connecting the midpoints of triangles with different orientations, do not translate the grid into itself [12]. However, the triangular grid gives a valid alternative for applications in image processing.…”

Section: Introductionmentioning

confidence: 99%

“…The coordination is vital because it is a leading tool in making a simple, easily usable, and effectively programmable system with integer numbers (coordinates). The isometric transformations are described in Reference [12]. However, up to now, there was no such extension of this coordinate system that is able to address the entire plane.…”

Section: Introductionmentioning

confidence: 99%

A Continuous Coordinate System for the Plane by Triangular Symmetry

Nagy¹,

Abuhmaidan²

2019

Symmetry

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The concept of the grid is broadly used in digital geometry and other fields of computer science. It consists of discrete points with integer coordinates. Coordinate systems are essential for making grids easy to use. Up to now, for the triangular grid, only discrete coordinate systems have been investigated. These have limited capabilities for some image-processing applications, including transformations like rotations or interpolation. In this paper, we introduce the continuous triangular coordinate system as an extension of the discrete triangular and hexagonal coordinate systems.The new system addresses each point of the plane with a coordinate triplet. Conversion between the Cartesian coordinate system and the new system is described. The sum of three coordinate values lies in the closed interval [−1, 1], which gives many other vital properties of this coordinate system. Symmetry 2019, 11, 191 2 of 17 of the grid. In Figure 1a, the first coordinate value is ascending right-upwardly, the second values are ascending into the right-downward direction, and the third one is ascending into the left-upward direction [7,8]. We should mention that this system could be seen as the extension of the oblique coordinate system using two independent integer values [9] by concerning the third value to obtain zero-sum for every triplet. The digital distance based on the neighborhood relation is computed in Reference [9]. Since the vectors describing the grid are not orthogonal, some geometric descriptions based on Cartesian coordinates are not very clear. However, to simplify the expressions of the constrained three-dimensional coordinate system is recommended. We should also mention that 0-sum triplets allowing real numbers were used in Reference [8] to describe rotations (that may not map the hexagonal grid to itself). In this way, a useful digitization operator is found. Her's system was mentioned and used in References [10,11] for various imaging-related disciplines.Symmetry 2019, 11, 191 3 of 17

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“…The generation of output primitives such as line, circle and ellipse on hexagonal grid is presented by Prabukumar etal [13][14][15][16]. Nagy presented the neighbourhood sequences on hexagonal grid [9][10][11].…”

Section: Introductionmentioning

confidence: 99%

An alternate line drawing algorithm on hexagonal grid

Manoharan

Ray

2013

Proceedings of the 6th ACM India Computing Convention

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In this paper an algorithm to scan convert a line using method of deviation on hexagonal grid is proposed. The method of deviation computes pixel nearest to analog line using only integer arithmetic. The proposed algorithm is compared favorably with the latest line drawing algorithm on a hexagonal grid. The main advantage of the proposed approach is that the same may be applied to an arbitrary curve. Owing to this approach we may visualize design ideas through animation and photorealistic renderings, and simulate how a design will perform in the real world in CAD tool with less aliasing artifact.

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Isometric transformations of the dual of the hexagonal lattice

Cited by 29 publications

References 8 publications

On the Number of Shortest Weighted Paths in a Triangular Grid

On the Number of Shortest Weighted Paths in a Triangular Grid

A Continuous Coordinate System for the Plane by Triangular Symmetry

An alternate line drawing algorithm on hexagonal grid

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