Direction

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wind rose

The Direction is a information of the geometrical relation of two squares on a chessboard without the information of distance. Mathematically, direction may be uniquely specified by a unit vector, or equivalently by the angle with respect to a specified set of axes.

Move Directions

Directions on the chessboard are usually related to from- to-coordinates of moves, that is four orthogonal and four diagonal ray-directions plus the eight directions a knight may move. Following Compass roses give the directions and single step increments of an 8x8 Board.

Ray Directions

  northwest    north   northeast
  noWe         nort         noEa
          +7    +8    +9
              \  |  /
  west    -1 <-  0 -> +1    east
              /  |  \
          -9    -8    -7
  soWe         sout         soEa
  southwest    south   southeast

Knight Directions

        noNoWe    noNoEa
            +15  +17
             |     |
noWeWe  +6 __|     |__+10  noEaEa
              \   /
               >0<
           __ /   \ __
soWeWe -10   |     |   -6  soEaEa
             |     |
            -17  -15
        soSoWe    soSoEa

Directions between any Squares

In Degrees

Based on arctangent of the quotient of file- and rank-distance, one may calculate following 15 times 15 sheet with all possible angels which might occur between any squares in Degree (rounded to integer). Inner squares whose angle occur on farther squares are left blank, so each angle given is distinct, 144 from 224 (15x15-1), move directions are given in bold:

315	319	324	330	337	344	352	0	8	16	23	30	36	41	45
311		320	326	333	342	351	|	9	18	27	34	40		49
306	310		321	329	338	349	|	11	22	31	39		50	54
300	304	309		323		346	|	14		37		51	56	60
293	297	301	307				|				53	59	63	67
286	288	292					|					68	72	74
278	279	281	284				|				76	79	81	82
270	-	-	-	-	-	-	+	-	-	-	-	-	-	90
262	261	259	256				|				104	101	99	98
254	252	248					|					112	108	106
247	243	239	233				|				127	121	117	113
240	236	231		217		194	|	166		143		129	124	120
234	230		219	211	202	191	|	169	158	149	141		130	126
229		220	214	207	198	189	|	171	162	153	146	140		131
225	221	216	210	203	196	188	180	172	164	157	150	144	139	135

In Begrees

Enumerating all distinct integer degrees of the above rose in order but without any gaps, leaves a circle with 144 board degrees (Begrees) - only the 16 angels with modulo 9 equals zero are actually covered by move directions (bold). A possible application of Begrees is to select moves with a direction to an "interesting" area of the board, specified by some center- or gravity square.

126	127	131	134	136	139	143	0	1	5	8	10	13	17	18
125		128	132	135	138	142	|	2	6	9	12	16		19
121	124		129	133	137	141	|	3	7	11	15		20	23
118	120	123		130		140	|	4		14		21	24	26
116	117	119	122				|				22	25	27	28
113	114	115					|					29	30	31
109	110	111	112				|				32	33	34	35
108	-	-	-	-	-	-	+	-	-	-	-	-	-	36
107	106	105	104				|				40	39	38	37
103	102	101					|					43	42	41
100	99	97	94				|				50	47	45	44
98	96	93		86		76	|	68		58		51	48	46
95	92		87	83	79	75	|	69	65	61	57		52	49
91		88	84	81	78	74	|	70	66	63	60	56		53
90	89	85	82	80	77	73	_72	_71	_67	_64	_62	_59	_55	_54

Opposed to the compass-rose above, all inner squares are filled with Begrees as well.

126	127	131	134	136	139	143	0	1	5	8	10	13	17	18
125	126	128	132	135	138	142	0	2	6	9	12	16	18	19
121	124	126	129	133	137	141	0	3	7	11	15	18	20	23
118	120	123	126	130	135	140	0	4	9	14	18	21	24	26
116	117	119	122	126	132	138	0	6	12	18	22	25	27	28
113	114	115	117	120	126	135	0	9	18	24	27	29	30	31
109	110	111	112	114	117	126	0	18	27	30	32	33	34	35
108	108	108	108	108	108	108	+	36	36	36	36	36	36	36
107	106	105	104	102	99	90	72	54	45	42	40	39	38	37
103	102	101	99	96	90	81	72	63	54	48	45	43	42	41
100	99	97	94	90	84	78	72	66	60	54	50	47	45	44
98	96	93	90	86	81	76	72	68	63	58	54	51	48	46
95	92	90	87	83	79	75	72	69	65	61	57	54	52	49
91	90	88	84	81	78	74	72	70	66	63	60	56	54	53
90	89	85	82	80	77	73	_72	_71	_67	_64	_62	_59	_55	_54

Lookup

Since the computation is quite expensive, a two dimensional table with pre-calculated values might be used for that purpose. The enumerated type enumDir might contain only ray- or move directions, Degrees, Begrees, or a structure combining them.

enumDir arrDirection[64][64]; // 4 KByte

enumDir direction(enumSquare sq1, enumSquare sq2) {
   return arrDirection[sq1][sq2];
}

Lookup by 0x88 Difference

The 0x88 square relation permits a denser lookup approach. The difference of valid 0x88 coordinates of two squares is unique with respect to distance and direction. That way, one can greatly reduce the size of the lookup array to only 240 instead of 4096 elements. Some additional computation required, if one has to convert usual square coordinates to 0x88. If one already relies on 0x88 coordinates, it becomes even cheaper:

enumDir arrDirectionBy0x88Diff[240];

unsigned int x88diff(enumSquare sq1, enumSquare sq2) {
   return sq2 - sq1 + (sq2|7) - (sq1|7) + 120;
}

enumDir direction(enumSquare sq1, enumSquare sq2) {
   return arrDirectionBy0x88Diff[x88diff(sq1, sq2)];
}

See also

• 0x88 Square Relations
• Attacks on the otherwise empty Board
• Butterfly Boards
• Distance
• Flipping Mirroring and Rotating
• Intersection Squares
• Knight-Distance
• Manhattan-Distance
• Pieces versus Directions
• Vector Attacks

Forum Posts

• Extract direction (ray) informations from two squares by Mathieu Pagé, CCC, June 18, 2013 » Rays, Square Attacked By

External Links

• Direction from Wikipedia
• Relative direction from Wikipedia
• Windward and leeward from Wikipedia
• Starboard from Wikipedia
• Port (nautical) from Wikipedia
• Origins of the Compass Rose by Bill Thoen
• Boxing the compass from Wikipedia
• Weather Report - Directions, 73rd NDR-Jazzworkshop, IFA Berlin, September 03, 1971, YouTube Video

Joe Zawinul, Wayne Shorter, Miroslav Vitouš, Alphonse Mouzon, Dom Um Romão, & guests John Surman, Eje Thelin, Alan Skidmore

References

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