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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 back-face culling, partial Z-ordering 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.

     A---------------------------------a----------------------------------B

  |  |                                                                    |  |

  |  |                                            y                       |  |

  d1 |                                                                    |  b1

  |  |                               f'                                   |  |

  |  |                                                                    |  |

     |          C--------------------f-----------------------D            | 

  |  |          |                                            |            |  |

  |  |          |                    f"                      |            |  |

  |  d          |                                            |            b  |

  |  |          |                                            |            |  |

  |  |       e" e  e'                                     g' g  g"        |  |

  d2 |          |                                            |            |  b2

  |  |          |                                            |            |  |

  |  |          |                                            |            |  |

  |  |          E                                            F            |  |

  |  |                              x                                     |  |

  |  |                                                                    |  |

     G---------------------------------c----------------------------------H

                                                              

      ----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 A-H, the _faces_ are marked a-g.

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 left-hand-most 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 line-of-sight 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 one-dimensional Z-buffer 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 DOOM-speak).

Then put everything to the left of f in the left subtree, and vice-versa:

                            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 pseudo-code 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 pre-calculating 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 left-most pixel of the screen

vector  right_sightline                ; the right-most 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(vertex2-vertex1,player-vertex1)

  ; player is to the left of the nodeline

  if not left(vertex2-vertex1,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(vertex2-vertex1,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 hard-code it into some run-time 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         ==========================================
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