Knight's distance in digital geometry

In the game of chess, the move of a knight has been particularly interesting because of its tricky nature. This had earlier lead to the well-known problem of a knight 's tour over a chessboard. Recently Yamashita and lbaraki [4] have pointed out that knight ' s moves should define a highly interesting and peculiar distance function in two dimensional digital geometry, but the functional form and properties of this distance function have not been studied so far. Let us call it the knight's distance. It has often been presumed that the knight 's move is erratic in nature and thus the thought of its use as a distance summari ly dismissed. But our observat ion is that this mot ion does not only give an interesting distance function, but also has an underlying uniform nature despite local aberrations. The other properties of this distance regarding digital disks and circles are also interesting. It is not our intention to propose that the knight 's distance is the best measure of distance, but to disprove the normally held idea that it defies all concepts of uniformity. That is, it may not be the best distance function, but it is a usable one. Now let us see what is meant by the knight 's distance. Let Z be the set of integers, and thus Z 2 ~ Z × Z = ~xlx = (x(l) ,x(2)) A x(1) ,x(2)eZ}, where A stands for logical ANt), defines the digital plane (grid). Now if a knight is placed in the cell x e Z 2 then in the next step it can move to any of the eight possible cells y, where 0 ' x ) ~ [ ( + 1, + 2), ( + 2, + 1)} (see Figure 1). A succession of such movements from u to v in Z 2 defines a knight's path where the number of moves taken by the knight gives the length of the path. Obviously there are infinitely many paths from u to v, each having a length of its own. However, the one with the least value of length is called the minimal (shortest) path and the length of this minimal path is the knight 's distance between u and v, denoted by dknight(U, I~). Yamashi ta and lbaraki [4] observed that this is a special case of the class of distances defined by ne ighbourhood sequences. In this paper we formulate a closed functional form for the knight 's distance. An algori thm is presented which can trace one of the minimal knight ' s paths between any pair of points on the grid Z 2. The properties of the circles and disks of knight 's distances are explored. The knight 's t ransform is defined and an algori thm is given for this transform.