Anti-Diagonals

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Anti-Diagonals are the diagonals from south-east to north-west on a chess board, the main anti-diagonal is h1\a8. There are 15 anti-diagonals, with line-length from 1 to 8. An Anti-Diagonal is monochrome, all their squares are either white or black.

=Square Mapping Notes= A 90 degree rotation of the Chessboard, as well as flipping vertically (reversed ranks) or (exclusive) mirroring horizontally (reversed files), change the roles of diagonals and anti-diagonals. However, we define the main diagonal on the chess board from a1/h8 and the main anti-diagonal from h1\a8. Whether the square difference of neighbored squares on a diagonal or anti-diagonal is either 7 or 9, depends on the square mapping. We further rely on little-endian rank-file mapping.

=Square Difference= Within a 0..63 square index range and the mentioned square mapping (a1 = 0), the difference of two neighbored squares (if any) on an anti-diagonal is seven.

=Enumeration= If we follow an anti-diagonal from south-east (h1) to north-west (a8) step by step, we increment the rank, but decrement the file, which yields in same sum. Thus, adding rank and file indices enumerates all Anti-Diagonals.
 * Square a1 (file- and rank index 0) is therefor anti-diagonal with index 0 and length 1.
 * The main anti-diagonal h1\a8 with index 7 and length 8.
 * Square h8 is the 15th anti-diagonal with index 14 and length 1.
 * All even indices are the anti-diagonals with dark squares

rank + file

=Alternative Enumeration= Some alternative enumeration of anti-diagonals to make the main-diagonal index 0, by xoring the sum with 7 (which complements the lower three bits of the sum). This yields in a 0..15 range with 8 as gap or Nexus in the center of the range:

(rank + file) ^ 7

=Two Squares on a Anti-Diagonal= Two Squares are on the same Anti-Diagonal, if sum of file and rank distance is zero bool squaresOnSameAntiDiagonal(int sq1, int sq2) { return ((sq2 - sq1) & 7) + ((sq2>>3) - (sq1>>3)) == 0; } The alternative approach, to test whether the square difference is divisible by 7, would take extra conditions, since the outer squares of the main-diagonal (63-0) would wrongly classified otherwise


 * Two Squares on a Diagonal
 * Two Squares on a File
 * Two Squares on a Rank

=See also=
 * Diagonals
 * Ranks
 * Files
 * Squares
 * Intersection Squares

=References= Up one Level