Knight's distance in digital geometry

In this paper we have extended the knight's moves in chess to introduce super-knight's moves in digital geometry.

ABSTBACT An analysis of paths and distances in n dimensions is carried out using variable neigbborhood sequences. A symbolic expression for the distance function between any two points in this quantized space is derived. An algorithm for finding the shortest path is presented. The necessary and sufficient condition for such distance functions to satisfy the properties of a metric has been derived. Certain practical and efficient methods to check for metric properties are also presented.

arXiv (Cornell University), 2023

Two algorithms for construction of all closed knight's paths of lengths up to 16 are presented. An approach for classification (up to equivalence) of all such paths is considered. By applying the construction algorithms and classification approach, we enumerate both unrestricted and non-intersecting knight's paths, and show the obtained results.

A generalized distance measure called m-neighbor distance in n-D quantized space is presented.

wellbehaved, where p is the length of the sequence B. We have also reviewed the other weaker and stronger conditions of triangularity and have introduced a new weaker condition here to reformulate the strategy for deciding the metricity of a d(B).

A generalized distance measure called t-Cost-m-Neighbour (tCmN) distance in n-D grid point space is presented.

Digital distance geometry (DDG) is the study of distances in the geometry of digitized spaces. This was introduced approximately 25 years ago, when the study of digital geometry itself began, for providing a theoretical background to digital picture processing algorithms. In this survey we focus our attention on the DDG of arbitrary dimensions and other related issues and compile an up-to-date list of references on the topic.

A new class of distance functions has been defined in n-D, where the distance between neighboring points may be more than unity. A necessary and sufficient condition for such distance functions to satisfy the properties of a metric has been derived. These metrics, called t-cost distance, give the length of the shortest t-path between two points in n-D digital space. Some properties of their hyperspheres are also studied. Suitability of these distances as viable alternative to Euclidean distance in n-D has been explored using absolute and relative error criteria. It is shown that lower dimension (2-D and 3-D) distance measures presently used in digital geometry can be easily derived as special cases. Finally most of these results have been extended for the natural generalization of integral costs to real costs.

Journal of Discrete Mathematics, 2014

The classic puzzle of finding a closed knight's tour on a chessboard consists of moving a knight from square to square in such a way that it lands on every square once and returns to its starting point. The 8 × 8 chessboard can easily be extended to rectangular boards, and in 1991, A. Schwenk characterized all rectangular boards that have a closed knight's tour. More recently, Demaio and Hippchen investigated the impossible boards and determined the fewest number of squares that must be removed from a rectangular board so that the remaining board has a closed knight's tour. In this paper we define an extended closed knight's tour for a rectangular chessboard as a closed knight's tour that includes all squares of the board and possibly additional squares beyond the boundaries of the board and answer the following question: how many squares must be added to a rectangular chessboard so that the new board has a closed knight's tour? Schwenk's Theorem. An m × n chessboard with m ≤ n has a closed knight's tour in all but the following cases:

Discrete Applied Mathematics, 1997