Spiral Path FOV

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Okay; Basically it uses a table of squares, and a queue of pointers to the table elements. In the table, it keeps track of the minimum and maximum lit angle reaching the corresponding square, plus static precomputed information such as the angles from the origin to the square corners.

There are two things that can happen to a square; you can either pass light to it, or you can pass light from it. When you're passing light to it, you record (or expand) the minimum and maximum lit angles. When you're passing light from it, you mark the square as lit/seen, take the minimum and maximum angle of the light that's reaching it, and if it's transparent, pass light to the squares that are adjacent to it and further from the center.

The tricky bit with spiral-path is using the queue of pointers correctly, so no pointer to a square that's not lit/seen is ever put there, while at the same time putting the squares into the queue an the order that guarantees that all light will be passed to each before any light is passed from it. Each square can get light passed to it once, or twice; the first time light is passed to it, a pointer to it gets added to the processing queue. The second time light gets added to it, if there is a second time, its minimum and maximum lit angles are adjusted.

It has to be a queue of pointers (or the moral equivalent) because you have to be able to get at the squares two ways: First you have to be able to find it in constant time by its coordinates when you're adding light to it. Second, you've got to be able to get at it in constant time when you're taking it off the front of the queue. So basically, you have your choice whether to implement the queue as a queue of coordinates into the table or as a queue of literal pointers.

Spiral-path consists in loading the queue with the initial four squares, (one space east, north, west, and south of the center, in that order), marking each with the minimum and maximum angle of light reaching it (nominally equal to its minimum and maximum angles), then

   repeat:
   take one square off the front of the queue.*1
   mark it visible/lit.
   If it's within the sight radius and not opaque then
      pass light from it
          (this may add things to the end of the queue,
      or add light to things that are already in the queue)
   : until the queue is empty.

Efficiency note:

*1 There can never be more elements in the queue than twice the sight diameter, so you can use a static array of fixed size, plus head and tail indexes that get incremented modulo its length, for the queue. You halt processing when head is equal to tail.

The reason it's called spiral-path is that when you're passing light from a square, it MUST be passed to new squares in the correct order, or else you will wind up passing light from something before all its light has been passed to it. And the correct order, where light/vision is unobstructed, is a spiral.

The path in which squares actually get added to the queue (where unobstructed) is: (digits, then letters, beginning from *)

      I
     J8H
    K927G
   LA3*16F
    MB45E
     NCD
      0

etc... spiraling out from the center. Squares 1,2,3,4 get added in initialization. Squares 5,6,7 get added when passing light from square 1. Passing light from square 2 adds light to square 7 which is already on the queue, then adds squares 8 and 9 to the queue. Passing light from square 3 adds light to square 9 which is already on the queue, then adds squares A and B to the queue. Passing light from square 4 adds light to B, then adds square C to the queue, then adds light to square 5 which is already on the queue. Passing light from square 5 adds D and E to the queue. Passing light from square 6 adds light to E which is already on the queue, then adds F and G to the queue. And so on.

The tricky bit was making sure things got added in the right order so that all the light that 5 was going to get, got to it before light was passed from it. With the added condition that you don't know in advance which squares light will actually come through to a given square, and that cone effects or directional light can start on any angle, not just with square 1, that turns out to depend on the order of additions being a spiral.

So... when passing light from square S, you figure out from the minimum and maximum lit angles of S which of its neighbors it will pass light to, and then add whichever of those neighbor squares you need to add in this order:

If S is on the east axis:

   3
   S2
   1

If S is in the northeast quadrant:

    2
    S1

If S is on the North axis:

    2
   3S1

If S is in the northwest quadrant:

    1
   2S

If S is on the West axis:

    1
   2S
    3

If S is in the southwest quadrant:

   1S
    2

If S is on the South Axis:

   1S3
    2

If S is in the southeast quadrant:

    S2
    1

That way, no matter which squares get skipped (because no light is getting to them) the squares that get added will get added in such an order that none of them ever passes light to a square that light has already been passed from, and no square that light has been passed from will ever be re-added to the queue.

There is one more fiddly bit: Along one of your axes, you'll have the "zero line" of your angle measurement. So the code to pass light to and from squares on that axis has to take into account the wraparound in angles.

There's no need for floating-point in it; you're keeping track of angles, but all you have to do is look them up in tables and compare them to each other;It's simple enough to multiply the angles by a million and round them off to integers.

Finally, there's the corner patchup: If you want spiral- path to fill in the corners of your rooms so you get

########              #######
.......#              .......#
.......#  instead of  .......#,

add that an opaque square whose maximum-angle corner is lit, passes an infinitely narrow beam of light through that corner to the *last* square it would otherwise pass light to.

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