BSP Trees
Binary Space Partitioning Trees
This article explains how BSP (binary space partitioning) trees can be used in a game such as DOOM as part of the rendering pipeline to perform backface culling, partial Zordering and hidden surface removal.
To explain the use of BSP trees, it is best to start with an example. Consider
a very simple DOOM level.
AaB
   
  y  
d1   b1
  f'  
   
 CfD 
     
   f"   
 d   b 
     
  e" e e' g' g g"  
d2     b2
     
     
  E F  
  x  
   
GcH
c1 c2 c3

The level consists of a room within a room. The player cannot go outside of the
area within the square ABHG.
First some definitions (sorry :)
The _vertices_ are marked AH, the _faces_ are marked ag.
We define a _line_ by using an ordered pair of vertices, so that
a = (A,B) e = (E,C) f = (C,D) g = (F,D)
We say a point is to the _left_ of a line if it is to the left of the vector
between its two vertices, taken in order.
So, in the above example, nothing is to the left of line a; everything is to
the right of it. Note that this depends upon our defintion of line a, and if
we had defined a = (B,A) then everything would be to the left of line a.
A _face_ is a side of a line which is visible to the player. Wall e above, for
example, has two faces (marked e' and e"). Not all walls have two faces  if
the player can never see one side of a wall it only has one.
A face is fully defined by an ordered pair of vertices and an ordered pair of
faces  a left face and a right face.
The BSP tree for the example above might look like this:
f
/ \
/ \
/ \
a,d1,b1 e
/ \
/ \
/ \
d2,c1 g
/ \
/ \
/ \
c2 c3,b2

Each node contains a line. Everything to the left of that line is in the left
subtree, and everything to the right of that line is in the right subtree.
Note that face d is neither completely to the right of nor to the left of face
f. To accomodate this, we split it up into two halves, and put one half into
the left subtree and one half into the right subtree. Thus, we have to generate
new faces in order to build the BSP tree.
I will explain how the BSP tree is created later. Firstly, I will give the
algorithm used to render a picture using the tree.
Suppose the player is standing at position 'x', and looking North.
We start at the top of the tree at line f. We are standing to the right of line
f, so we go down the LEFT of the tree. This is because we want the furthest
polygons first.
We come to the lefthandmost terminating node. We write down the faces here
in our notepad. "a,d1,b1".
Since we've come to a terminator, we back up a level. Back to the top, but we
have to go down the right subtree yet. Firstly, though, we look at face f  the
deciding face for this node. We've got everything behind it in our list, we've
yet to look at anything in front of it, but we must put it into our list.
Note that face f has two sides  f' and f". Since we already know we're on the
right of line f, we know that we can only see its right side  so we write
f" in our notepad. It now says a,d1,b1,f".
Note, though, that if we were looking south (i.e. our lineofsight vector
points away from face f) then we could not see either face f or anything on
the other side of face f  in this case, we just don't bother going any further
down the tree.
Now we go down the subtree and come to node e. We are on the right of e, so we
go down the left subtree and get a terminal node  we just write d2,c1 in our
notepad.
Back up, decide on which side of e to put in. We decide e'. The notepad now
says a,d1,b1,f",d2,c1,e'.
Down the right subtree to node g. We're on the left, so down the right subtree
to c3,b2, up, check g (we're on the left = g'), back down to the final node,
get c2, up, up, up, and we're done.
The notepad ends up saying:
a d1 b1 f" d2 c1 e' c3 b2 g' c2
If we draw these walls, in this order, then we will get the correct scene. I
would recommend using a onedimensional Zbuffer to get finer granularity than
the painter's algorithm provides, before plotting the walls. Note also that
some walls are behind you  however, since you need to calculate their z
coordinates for the perspective transform, you can merely discard faces with
negative z values.
Creating the BSP tree
The BSP tree almost creates itself. The only difficulty is knowing when to stop
recursing. Notice that the terminal nodes are just put into the list  so a
sufficient condition for a group of faces to form a terminal node is that they
can be drawn in a set order without any mistakes occuring in the drawing. That
is, if wherever the player can stand, the group of walls will never obscure
each other.
So let us begin: Choose face f (the choice is fairly arbitrary  it is best
to choose faces which don't split many other faces up. However, in this case
it is unavoidable). Split up faces d and b, because they straddle the line f.
(The line you are splitting along is known as the _nodeline_ in DOOMspeak).
Then put everything to the left of f in the left subtree, and viceversa:
f
/ \
/ \
/ \
a,d1,b1 b2,c,d2,e,g

We can terminate the left node  because walls a,d1 and b1 form a convex
shape, they can never overlap each other from any point of view. However, on
the other side, face e can obscure face d2 from certain viewpoints (our example
viewpoint above, for one) so we divide along side e. This causes side c to be
split, but side a is not split because it's not in our current list of sides.
The next level is:
f
/ \
/ \
/ \
a,d1,b1 e
/ \
/ \
/ \
d2,c1 b2,c2,g

Now, c1 and d2 never overlap, so we have another terminal node. We next divide
along line g, splitting c2 into c2 and c3, and the last nodes are terminals
(a node with one face in is always terminal :).
This is the basic idea behind a BSP tree  to give an example how effective it
is, consider standing at point y and looking North. Because you're looking
away from face f, you don't bother recursing down the entire left subtree. This
then very quickly gives you the ordered list of faces: a,d1,b1.
Refinements
If at each node we define a bounding box for each subtree, such that every line
in a subtree is contained by its corresponding bounding box, then we can cut
some invisible polygons (ones which lie to the left or right of the screen) out
by comparing each bounding box with the cone of vision  if they don't
intersect, then you don't go down the whole subtree. DOOM does this, allowing
it to store an *entire* level in one huge BSP tree.
Here's some pseudocode to traverse the tree. The function left() returns TRUE
if the second input vector is to the left of the first input vector. This is
a simple dot product, and by precalculating the slope of the nodeline can be
done with one multiply and one subtract.
vector player ; player's map position
vector left_sightline ; vector representing a ray cast through
; the leftmost pixel of the screen
vector right_sightline ; the rightmost pixel of the screen
structure node
{
vector vertex1
vector vertex2
node left_subtree
node right_subtree
face left_face
face right_face
box bounding_box
bool terminal_node
face terminal_node_faces[lots]
}
recurse(node input)
if (cone defined by left and right sightlines does not intersect the node's
bounding box)
return
fi
if node.terminal_node
; terminal node  add faces to list
add(node.terminal_node_faces)
return
fi
if left(vertex2vertex1,playervertex1)
; player is to the left of the nodeline
if not left(vertex2vertex1,right_sightline)
; sight points right  we are looking at the face
recurse(node.right_subtree)
add(node.left_face)
fi
; now go down the left subtree
recurse(node.left_subtree)
else
; player is to the right of the nodeline
if left(vertex2vertex1,left_sightline)
; sight points left  we are looking at the face
recurse(node.left_subtree)
add(node.right_face)
fi
; now go down the right subtree
recurse(node.right_subtree)
fi
return
end recurse

This isn't anywhere near a decent implementation  the data structures, for
example, leave a *lot* to be desired :)
It should be possible to encode all the functions inline; in fact, it would be
feasible to take a BSP tree and hardcode it into some runtime generated code
which you just call to recurse the tree ... but I'm just a hacker at heart ;)
Anyway, I hope this helps answer some peoples' questions on this subject. If
you have any more questions, please don't hesitate to email me.
Catch you later,
Contact the Author: Eddie xxx ee@datcon.co.uk
===========================================================================
Official Archimedes convertor of : Hear and remember, see and understand,
Wolfenstein 3D and proud of it!! : do and forget.
=================================: Something like that, anyway.
ee@datcon.co.uk ==========================================
