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001-es BibID:	BIBFORM012651
Első szerző:	Nagy Benedek (informatikus, matematikus)
Cím:	Distences based on neighborhood sequences in the triangular grid / Nagy Benedek
Dátum:	2009
Megjegyzések:	In computers the discrete plane/space is used since this world is digital. The discrete space is usually defined by a grid. There are three regular grids in two dimensions: the square grid, the hexagonal grid and the triangular grid. The square grid is the most used due to the simplicity of the Cartesian coordinate frame. There are two types of neighbors defined naturally [52, 53]: the city-block and the chessboard neighborhoods. The hexagonal grid is the simplest one, since there is only one natural neighborhood among the hexagons of the grid. The triangular grid is a little bit sophisticated (there are three types of natural neighborhoods), however it has some nice and interesting properties. In the digital space the usage of the usual (Euclidean) distance may lead to some strange phenomena [25]. Instead, path-based, so-called digital distances can be used based on the neighborhood structure of the grid. Digital distances are frequently used in computerized applications of geometry, e.g., in image processing, in computer graphics. There are two main approaches to define digital distances: distances based on neighborhood sequences in which the used types of neighbors is varied along a path; and weighted distances, where various types of steps have various weights (lengths). In the present chapter distances based on neighborhood sequences on the triangular grid are detailed. First, an effective coordinate system is presented to the grid. By this system the grid can handle as easily, as the square grid by the Cartesian frame. After the definitions of neighborhood sequences, paths and distances, some results are detailed: a greedy algorithm that provides a shortest path, formula to compute the distance from a point to another point defined by a given neighborhood sequence. Interesting properties of these distances, such as non-metrical distances are shown (triangular inequality and symmetry may be violated). A necessary and sufficient condition to define metrical distances is proved. Some details on digital circles based on these distances are also presented. Finally some further directions of research and open problems close the chapter.
Tárgyszavak:	Természettudományok Matematika- és számítástudományok könyvfejezet
Megjelenés:	Computational Mathematics: Theory, Methods and Applications / ed. Peter G. Chareton. - p. 1-39. -
Internet cím:	Intézményi repozitóriumban (DEA) tárolt változat
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