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                 | A Simple Explanation of BSP Trees |

            Written for the PC-GPE by Mark Feldman
            e-mail address : u914097@student.canberra.edu.au

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| BSP Trees |

Binary Space Partition trees are handy for drawing 3D scenes where the
positions of objects are fixed and the user's viewing coordinate changes
(flight simulators being a classic example).

BSP's are an extention of the "Painter's Algorithm". The painter's algorithm
works by drawing all the polygons (or texture maps) in a scene in back-to-
front order, so that polygon's in the background are drawn first, and
polygons in the foreground are drawn over them. The "classic" painter's
algorithm does have a few problems however:
1) polygon's will not be drawn correctly if they pass through any other
2) it's difficult and computationally expensive calculating the order that
the polygons should be drawn in for each frame
3) the algorithm cannot handle cases of cyclic overlap such as the
following :
                           ___       ___
                          |   |     |   |
                        __|   |_____|___|___
                       |  |   |             |
                       |__|   |_____________|
                          |   |     |   |
                        __|___|_____|   |___
                       |            |   |   |
                       |____________|   |___|
                          |   |     |   |
                          |___|     |___|

In this case it doesn't matter which order you draw the polygon's it still
won't look right!

BSP's help solve all these problems.

Ok, so let's get down to business. BSP's work by building an ordered tree of
all the objects in a scene. Let's imagine we live in a 2D world and we have
a scene like this:

             |                                    |
             |                                    |
             |              --------------------  |
             |                    line 1          |
             |           \                        |
             |             \                      |
             |               \ line 2             |
             |                 \                  |
             |                   \                |
             |   --------          \              |
             |   line 3              \            |
             |                                    |

                              viewpoint (assume the viewpoint is the
                                         origin for this example)

Now if we were to draw this scene using the painters algorithm we would
draw line 1 first, then line 2, finally line 3. Using BSP's we can figure
out the order beforehand and create a tree. First we note that any
arbitrary point &ltx,y&gt can be on either of it's 2 sides or on the line (which
can be regarded as being on either of the sides). When we build our tree, we
take a line and put all the lines on one side of it to the left and all the
nodes on the other side on the right. So for the above example could wind up
with the following tree:


Alternatively, we could also wind up with this tree:

       / \
      3   1

Notice that line 2 is the head node, line 3 is on the same side of line 2
as the origin is and line 1 is on the opposite side.

Now, I hear you say "but hang on a tic, what if line 3 is the head node? What
side of it is line 2 on?". Yes boys and girls, there had to be a catch
somewhere and this is it. What you have to do here is split line 2 into
*TWO* lines so that each portion falls nice and neatly onto either side of
line 3, so you get a tree like this:

      / \
    2a   2b

The lines 2a and 2b are portions of the original line 2. If you draw *BOTH*
of them on the screen it will look as though you've drawn the entire original

You don't have to worry about balancing a BSP tree, since you have to
traverse every node in it every time you draw the scene anyway. The trick
is to figure out how to organise the tree so that you get the *least* number
of polygon splits. I tackled this by looking at each polygon yet to be
inserted into the tree, calculating how many splits it will cause if it
is inserted next and selecting the one which will cause the fewest. This
is a very slow way of going about things, O(N^2) I think, but for most games
you only need to sort the tree once when you are developping the game and not
during the game itself.

Extending these concepts to 3D is pretty straight-forward. Let's say that
polygon 1 is at the top of the BSP tree and we want to insert polygon 2. If
all the points in polygon 2 fall on one side or the other of polygon 1 then
you insert it into polygon 2's left or right node. If some points fall on
one side and the rest fall on the other, then you have to figure out the line
of intersection formed by the planes that each polygon lies in and split
polygon 2 along this line. Each of these 2 new polygons will then fall on
either side of polygon 1.

To draw the objects in a BSP tree you start at the top node and figure out
which side of the object your view coordinate is on. You then traverse the
node for the *other* side, draw the current object, then traverse the node
for the side the view coordinate is on.

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