# Types of directions

A*direction*in CQL denotes a set of 1 or more basic directions on the chessboard. The name of a direction can also be used as a filter that translates a set of squares along certain directions by a certain length.

There are eight *basic directions*:

basic direction |
---|

`up` |

`down` |

`right` |

`left` |

`northeast` |

`northwest` |

`southwest` |

`southeast` |

`up`

is in the direction of increasing rank.

There are also five *compound directions*, which represent two or more
basic directions:

compound direction | basic directions represented |
---|---|

`diagonal` | `northeast` , `northwest` , `southeast` , `southwest` |

`orthogonal` | `up` , `down` , `left` , `right` |

`vertical` | `up` , `down` |

`horizontal` | `left` , `right` |

`anydirection` | `orthogonal` , `diagonal` |

A *direction keyword* is the name of a direction, like
`northeast`

.

## direction filters

A *direction filter* consists of a direction keyword, one of:

`up`

`down`

`left`

`right`

`northeast`

`northwest`

`southeast`

`southwest`

`diagonal`

`orthogonal`

`vertical`

`horizontal`

- anydirection

followed by an optional range followed by a set filter:

directionrangesetdirectionset

For example,

orthogonal ♔ diagonal 2 5 ♙♟ up 1 ♟

If *range* is
absent it is taken to be `1 7`

.

A direction filter represents the set of squares by shifting the set of squares represented by its argument, in the direction indicated by its direction keyword, a number of times indicated by its range. For example:

up 1 d4 ≡ d5 up 2 d4 ≡ d6 up 3 d4 ≡ d7 up 1 3 d4 ≡ d5-7 up d4 ≡ d5-8 down 1 d4 ≡ d3 down d4 ≡ d1-3 left 1 [d4,e5] ≡ [c4,d5] left [d4,e5] ≡ [a-c4,a-d5] right a1-8 ≡ b-h1-8 right 1 a-h2 ≡ b-h2 right 6 7 a-h2 ≡ [g2,h2]

The *range* in the direction filter can include negative numbers. These denote a direction opposite
to the direction of the direction keyword in the direction filter.

up -2 d4 ≡ d2 right -1 1 d4 ≡ c-e4

Each compound direction keyword is likewise associated with a direction filter. These represent the union of the sets corresponding to the basic directions in the compound direction filter. For example:

vertical 3 d4 ≡ {up 3 d4} or {down 3 d4} ≡ [d7,d1]diagonal 1 d4 ≡ [e5,c5,c3,e3]

orthogonal d4 ≡ [a-c4,d5-8,e-h4,d1-3]

Although a compound direction filter can have a range with negative bounds, it is unnecessary to include this, since each compound direction filter includes the opposite direction of each of its basic direction elements:

horizontal 0 1 d4 ≡ c-e4 horizontal -1 0 d4 ≡ c-e4 horizontal -1 1 d4 ≡ c-e4Note that these examples can be used for any sets, not just squares. For example, suppose the current position is the start position:

Then the following relations would hold:

up 1 ♙ ≡ a-h3up ♘ ≡ [b2-8,g2-8]

right 1 ♖♛ ≡ [b1,b8]

right 1 △ attackedby ♕ ≡ right 1 [c-e2,e1,c1] ≡ [d-f2,f1,d1]

These can be combined with other primitives, including direction filters. For example, to find
squares a knight's hop away from a `♕`

, use:

`flip up 2 right 1 ♕`

If the current position were
the starting position, for example, this expression would be represent `[f2,e3,c3,b2]`

Similarly,

diagonal diagonal h1is the set of light squares:

diagonal diagonal h1 ≡ diagonal diagonal 1 7 h1 ≡ diagonal [g2,f3,e4,d5,c6,b7,a8] ≡ diagonal 1 7 [g2,f3,e4,d5,c6,b7,a8]This last expression is the set of light squares, because every light square can be reached by moving a distance from 1 to 7 along a diagonal direction from one of these squares. For instance:

f1 ≡ southwest 1 g2 h8 ≡ northeast 3 d5

## directions in `ray`

Directions are also used as a parameter to the ray filter.
# Examples

Directions are used throughout the examples: in Qq-rotations.cql (`right`

filter); in chameleon.cql (`right`

; staircase-sort.cql (`up`

and `right`

); in quadrupled-pawns2.cql (`vertical`

); in bristol-universal.cql (`anydirection`

).
Let us consider the use of directions in the staircase-sort.cql file. This file finds a number of studies using queen staircases, one of which is

(found from CQL file: staircase-sort.cql)

The arrows in the diagram trace the path of the white queen.

If we consider a filter from the CQL file

♕――up 1 ♕Suppose there is only one white queen on the position, say on

`e4`

. Then
up 1 ♕ ≡ up 1 e4 ≡ e5

Thus,

♕――up 1 ♕ ≡ ♕e4――e5

In other words, the queen moves one square up the board.

Similarly,

♕――right 1 ♕ ≡ ♕e4――f4

that is, the queen moves one square to the right.

Lets call these movements `Q_UP`

and `Q_RIGHT`

respectively.

Basically,

(♕――up 1 ♕ ♕――right 1 ♕) ≡ (Q_UP Q_RIGHT)This means a

`Q_UP`

move followed by a `Q_RIGHT`

move.
Let's call a `Q_UP`

followed by a `Q_RIGHT`

a *stair*.

The `+`

at the end means `one or more repetitions`

(♕――up 1 ♕ ♕――right 1 ♕)+

means one or more successive stairs.

The

♕――up 1 ♕ ?at the end is just

Q_UP ?where the

`?`

means "optional". Thus, the whole sequence means:
*one or more stairs followed optionally by* `Q_UP`

.

This would define an *up* staircase, like this (assuming the queen is `a1`

)

To get a staircase in any direction, put flip in front of the above staircase's filter. (If you reflect the above staircase starting at `a1`

about the horizontal bisector, you get the original staircase starting at `a8`

).