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Computing LOS For Large Areas - Gordon Lipford.
This article was originally written for Darren Hebden's RLNews

Summary and genesis:
    In most roguelike games,  there is a need to calculate which dungeon
    grids are visible in a direct line of sight to the player.  Closely
    tied to this is a need to calculate whether or not a dungeon grid is
    'lit' by a player light source of potentially variable radius.

    Using ray-tracing to calculate LOS is a well known and excellent
    method to address both these needs,  if the radius involved is
    reasonably small.  The problem with ray-traced LOS algorithms
    lies only with their memory consumption,  which increases as
    O(radius^3).   As per most LOS algorithms,  every cell within
    the radius of computation must be visited at least once;  ray-traced
    LOS visits each cell only once and so takes O(radius^2) to
    complete.

    Using shadow casting,  one may reduce the memory consumption to
    O(radius),  while keeping computation time O(radius^2) - though
    probably still slower than ray-tracing.

    Two other features of shadow casting that may prove useful:
    * radius may be changed dynamically;  very little recomputation
      necessary.
    * shadows may be 'relaxed' such that a blocker takes up less
      than the entire cell - normal calculations assume that a blocker
      takes up the entire (square) cell.   This involves very little
      modification of the algoritm itself,  and in fact may be
      parameterized such that it can be changed dynamically.

General method:
    This algoritm is given for a single octant (45 degrees of a
    circle,  or one eighth of a sqaure).  It is left as an exercise
    for the reader to translate the algorithm to other octants;
    however,  four arrays and some pseudo-code are given in appendix
    B.

    The player is treated as a geometric point in the center of
    cell (0,0).
    Each grid cell (defined below) may contain something which
    blocks the player's line of sight.

    Given a LOS radius of N,  we traverse the octant from (0,0)
    to (N,N) one column at a time,  from cell (0,y) to (y,y) within
    the column.  We examine each cell to see if it is blocked.
    If so, we perform two calculations:
    1.  Find the inverse slope of a line from the center of
        (0,0) to the upper left corner of the cell, and
    2.  Find the inverse slope of a line from the center of
        (0,0) to the lower right corner of the same cell.


                              ____________
                              |          |
                              |          |
                              |  (2,2)   |
                              |   U.     |
                   ___________|_UU_______|
                   |         UUU#########|
                   |       UU |##########|
                   |    UUU   |# (2,1) ##|
                   |  UU      |##########|
        ___________|UU________|##########|LLLL..
        |        UUU          |  LLLLLLLLL
        |      UU  |   LLLLLLLLLL        |
        |  (0UULLLLLLLL(1,0)  |  (2,0)   |
        |          |          |          |
        |__________|__________|__________|


    In this example,  (2,1) is blocked.  The line corresponding to
    the upper bound of the shadow cast by the blocker at (2,1)
    has a slope of 1.5 / 1.5 = 1.0.  The inverse of this slope
    is also 1.0.
    Similarly for the lower bound,  slope is 0.5 / 2.5 = 0.2.
    The inverse slope of 0.2 is 5.0.

    Note that any grid blocker in cell 0 will generate a
    _negative_ lower slope.  When this happens,  assign some
    arbitrarily large value.

    The result of #1 is assigned to the 'upper shadow max', and
    the result of #2 is assigned to the 'lower shadow max' (see
    below).

    The algorithm then uses these values to determine whether or
    not other cells farther out than the blocker are in it's
    shadow or not.   In this example,  the shadow will 'grow'
    upwards at a rate of 1 grid every 1.0 steps (ie every
    step),  and the lower bound of the shadow will 'decay',
    exposing the cell(s) to light,  at the rate of 1 grid
    every 5.0 steps.

    Note that the use of floating point arithmetic is NOT
    necessary for this algorithm.  All formulas are presented
    using floating point in order not to confuse the algorithm
    with its implementation.

    Also note that the 'relaxing' of the algorithm can be done
    at precisely this point by assuming the blocking cell to
    occupy less than the full grid square.

Algorithm terms:
    Within the algorithm,  a Cell is a working representation of
    a grid square.
    Each cell has several properties:
    - upper shadow count    (numeric)
    - upper shadow maximum  (numeric)
    - lower shadow count    (numeric)
    - lower shadow maximum  (numeric)
    - visible               (boolean)
    - lit                   (boolean)
    - lit_delay             (boolean)

    The first four of these will be referred to in the specific
    algorithm as Cell[n].upper_max, Cell[n].up_count, Cell[n].low_max,
    and Cell[n].low_count.

    To 'initialize' a Cell,  set all integer values to 0, and all
    boolean values to 'true' except for lit_delay.

    A Cell has 'reached' it's upper maximum iff the upper maximum is
    non-zero,  count+0.5 is >= the maximum,  and count-0.5 <= the maximum.

    Similarly for a Cell 'reaching' it's lower maximum.

Specific Algorithm:

allocate an array of Cells.  The array is one dimensional and
indexed from 0 to N (so it has a size of N+1).  No cell initialization
is necessary at this time;  it occurs within the inner loop in very
specific places.  There is no need to re-initialize anything between
octant calculations.

begin function los_octant:

boolean variable VISIBLE_CORNER
// this is necessary for esthetic reasons.  Without this variable
// and associated hack, a dead-end will appear as follows:
//       ######
//       .@....#
//       ######
// Although this is geometrically correct (at least if the blockers
// occupy the entire grid cell),  it is more pleasing to see this:
//       #######
//       .@....#
//       #######

boolean variable BLOCKER
// convenience: does the current cell represent a grid square that
// blocks LOS?

numeric UP_INC
numeric LOW_INC
numeric SOUTH
// always CELL-1.  Convenience.

// Cell (0,0) is assumed to be lit and visible in all cases.
initialize Cell[0]
VISIBLE_CORNER = false

// now for the main double loop:
for each COLUMN in (1.. N)
for each CELL in (0.. COLUMN)

assign TRUE to BLOCKER iff the object at grid (COLUMN, CELL) will
block the players LOS

UP_INC = 1
LOW_INC = 1
SOUTH = CELL - 1

// STEP 1 - inherit values from immediately preceeding column
//          light up from lit_delay if appropriate
//          'steal' lower bound shadow from 'south' cell if
//          if it lit
if CELL < COLUMN
    if Cell[CELL].lit_delay
        if not BLOCKER
            if Cell[SOUTH].lit
                if Cell[SOUTH].low_max <> 0
                    Cell[CELL].lit = false
                    Cell[CELL].low_max = Cell[SOUTH].low_max
                    Cell[CELL].low_count = Cell[SOUTH].low_count
                    Cell[SOUTH].low_max = 0
                    Cell[SOUTH].low_count = 0
                    LOW_INC = 0
                else
                    Cell[CELL].lit = true
                endif
            endif
        endif
        Cell[CELL].lit_delay = false
    endif
else
    initialize Cell[CELL]
endif

// STEP 2 - check for blocker
//          a dark blocker in a shadows edge will be visible
//          (but still dark)
if BLOCKER
    if Cell[CELL].lit OR (CELL > 0 AND Cell[SOUTH].lit) OR VIS_CORNER
        VIS_CORNER = Cell[CELL].lit
        Cell[CELL].lit = false			// blockers are always dark
        Cell[CELL].visible = true		// but always visible if we get here..

        calculate temporary UPPER and LOWER values for this grid position

        if UPPER < Cell[CELL].up_max OR Cell[CELL].up_max == 0
            // new upper shadow
            Cell[CELL].up_max = UPPER
            Cell[CELL].up_count = 0;
            UP_INC = 0
        endif
        if LOWER > Cell[CELL].low_max OR Cell[CELL].lower == 0
            // new lower shadow
            Cell[CELL].low_max = LOWER
            Cell[CELL].low_count = -1
            LOW_INC = 0
            if LOWER <= 3		// somewhat arbitrary, but looks right
                Cell[CELL].lit_delay = true
            endif
        endif
    else
        Cell[CELL].visible = false
    endif
else
    Cell[CELL].visible = false
endif

// STEP 3 - add increments to upper and lower counts
add UP_INC to Cell[CELL].up_count
add LOW_INC to Cell[CELL].low_count

// STEP 4 - look south to see if we've been overtaken by shadow
if CELL > 0
    if Cell[SOUTH] has 'reached' upper maximum
        if Cell[CELL] has NOT 'reached' upper maximum
            Cell[CELL].up_max = Cell[SOUTH].up_max
            Cell[CELL].up_count = Cell[SOUTH].up_count
            subtract Cell[SOUTH].up_max from Cell[CELL].up_count
        endif
        Cell[CELL].lit = false
        Cell[CELL].visible = false
    endif

    // STEP 5 - erase current lower shadow if one is active in the
    //          cell to our south

    if Cell[SOUTH] has 'reached' lower maximum
        Cell[CELL].low_max = Cell[SOUTH].low_max
        Cell[CELL].low_count = Cell[SOUTH].low_count
        subtract Cell[SOUTH].low_max from Cell[CELL].low_count
        set both Cell[SOUTH].low_max and Cell[SOUTH].low_count to 0
    endif
    if Cell[SOUTH].low_max <> 0 OR (Cell[SOUTH].low_max == 0 AND NOT Cell[SOUTH].lit)
        Cell[CELL].low_count = Cell[CELL].low_max + 10
    endif
endif

// STEP 6 - light up if we've reached lower max (ie come out of shadow)
if Cell[CELL] has 'reached' lower maximum
    Cell[CELL].lit = true
endif

// STEP 7 - apply 'lit' value
This step in the algorithm is entirely dependent on how the map grid
is implemented.   The basic criterion for a lit square is as follows:
if Cell[CELL].lit OR (BLOCKER AND Cell[CELL].vislble)
    // do something appropriate
else
    // the cell is not visible by the player.
endif

end for : CELL
end for : COLUMN
end function : los_octant

---------------------------------------------------------------------------
Appendix A: possible refinements

If an entire column is dark (ie Cell[CELL].lit is FALSE for each cell
in the column),  there is no need to continue calculations - every
subsequent column will be dark and NOT visible.

It may be possible to reconstruct certain parts of this algorithm so
that needless computation is avoided.  This is left as an exercise in
order to make the basic algorithm as readable as possible.

At no point does the algorithm assume sequential evaluation of
expression terms.    It may be possible to rewrite certain tests to
take advantage of this compiler feature.

The above algorithm processes a square area,  but can be trivially
modified to process any other area posessing eightfold symmetry.
---------------------------------------------------------------------------
Appendix B: octant translation

Any cell with co'ordinates (X,Y) may be translated to another
octant with the following transformation (given in C-like code):

int xxcomp[8] = { 1, 0, 0, -1, -1, 0, 0, 1 }
int xycomp[8] = { 0, 1, -1, 0, 0, -1, 1, 0 }
int yxcomp[8] = { 0, 1, 1, 0, 0, -1, -1, 0 }
int yycomp[8] = { 1, 0, 0, 1, -1, 0, 0, -1 }

tx = X * xxcomp[o] + Y * xycomp[o]
ty = X * yxcomp[o] + Y * yycomp[o]

Where o is the octant number, ranging from 0 to 7.
---------------------------------------------------------------------------
Appendix C: the author
I (Gordon Lipford) am a happily employed father of 2 boisterous boys
who enjoys designing and coding games as a hobby.  I am drawn to
Roguelikes and their abstract representation of a game world  - allowing
my imagination free reign to visualize the environment.

Please send all correspondence to lipford@sympatico.ca.   If that fails,
you may try my work address: lipford@ca.ibm.com.
Copyright 2001 Steve Register.