1. About this document
    2. Acknowledgements
    3. How can you contribute?
    4. About the pseudo C++ code
    5. What is a BSP Tree?
    6. How do you build a BSP Tree?
    7. How do you partition a polygon with a plane?
    8. How do you remove hidden surfaces with a BSP Tree?
    9. How do you compute analytic visibility with a BSP Tree?
   10. How do you accelerate ray tracing with a BSP Tree?
   11. How do you perform boolean operations on polytopes with a BSP
   12. How do you perform collision detection with a BSP Tree?
   13. How do you handle dynamic scenes with a BSP Tree?
   14. How do you compute shadows with a BSP Tree?
   15. How do you extract connectivity information from BSP Trees?
   16. How are BSP Trees useful for robot motion planning?
   17. How are BSP Trees used in DOOM?
   18. How can you make a BSP Tree more robust?
   19. How efficient is a BSP Tree?
   20. How can you make a BSP Tree more efficient?
   21. How can you avoid recursion?
   22. What is the history of BSP Trees?
   23. Where can you find sample code and related online resources?
   24. References

   About this document
       The purpose of this document is to provide answers to Frequently
       Asked Questions about Binary Space Partitioning (BSP) Trees. This
       document will be posted monthly to It is
       also available via WWW at the URL:
       The most recent newsgroup posting of this document is available
       via ftp at the URL:
       Requesting the FAQ by mail
       You can request that the FAQ be mailed to you in plain text and
       HTML formats by sending e-mail to
       with a subject line of "SEND BSP TREE [what]". The "[what]" should
       be replaced with any combination of "TEXT" and "HTML".
       Respectively, these will return to you a plain text version of the
       FAQ, and an HTML formatted version of the FAQ viewable with Mosaic
       or Netscape.
       Copyrights and distribution
       This document is maintained by Bretton Wade, a graduate student at
       the Cornell University Program of Computer Graphics.
       This document, and all its associated parts, are Copyright &copy
       1995, Bretton Wade. All rights reserved. Permisson to distribute
       this collection, in part or full, via electronic means (emailed,
       posted or archived) or printed copy are granted providing that no
       charges are involved, reasonable attempt is made to use the most
       current version, and all credits and copyright notices are
       retained. If you make a link to the WWW page, please inform the
       maintainer so he can construct reciprocal links.
       Requests for other distribution rights, including incorporation in
       commercial products, such as books, magazine articles, CD-ROMs,
       and binary applications should be made to
       Warranty and disclaimer
       This article is provided as is without any express or implied
       warranties. While every effort has been taken to ensure the
       accuracy of the information contained in this article, the
       author/maintainer/contributors assume(s) no responsibility for
       errors or omissions, or for damages resulting from the use of the
       information contained herein.
       The contents of this article do not necessarily represent the
       opinions of Cornell University or the Program of Computer
       Last Update: 07/05/95 03:46:05
       About the contributors
       This document would not have been possible without the selfless
       contributions and efforts of many individuals. I would like to
       take the opportunity to thank each one of them. Please be aware
       that these people may not be amenable to recieving e-mail on a
       random basis. If you have any special questions, please contact
       Bretton Wade ( or before trying to contact anyone
       else on this list.
          + Bruce Naylor (
          + Richard Lobb (
          + Dani Lischinski (
          + Chris Schoeneman (
          + Philip Hubbard (
          + Jim Arvo (
          + Kevin Ryan (
          + Joseph Fiore (
          + Lukas Rosenthaler (
          + Anson Tsao (
          + Robert Zawarski (
          + Ron Capelli (
          + Eric A. Haines (
          + Ian CR Mapleson (
          + Richard Dorman (
          + Steve Larsen (
          + Timothy Miller (
          + Ben Trumbore (
          + Richard Matthias (
          + Ken Shoemake (
          + Seth Teller (
          + Peter Shirley (
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          + Robert Schmidt (
       If I have neglected to mention your name, and you contributed,
       please let me know immediately!
       Last Update: 07/05/95 15:42:30
   How can you contribute?
       Please send all new questions, corrections, suggestions, and
       contributions to
       Last Update: 03/29/95 14:12:10
   About the pseudo C++ code
       The general efficiency of C++ makes it a well suited language for
       programming computer graphics. Furthermore, the abstract nature of
       the language allows it to be used effectively as a psuedo code for
       demonstrative purposes. I will use C++ notation for all the
       examples in this document.
       In order to provide effective examples, it is necessary to assume
       that certain classes already exist, and can be used without
       presenting excessive details of their operation. Basic classes
       such as lists and arrays fall into this category.
       Other classes which will be very useful for examples need to be
       presented here, but the definitions will be generic to allow for
       freedom of interpretation. I assume points and vectors to each be
       an array of 3 real numbers (X, Y, Z).
       Planes are represented as an array of 4 real numbers (A, B, C, D).
       The vector (A, B, C) is the normal vector to the plane. Polygons
       are structures composited from an array of points, which are the
       vertices, and a plane.
       The overloaded operator for a dot product (inner product, scalar
       product, etc.) of two vectors is the '|' symbol. This has two
       advantages, the first of which is that it can't be confused with
       the scalar multiplication operator. The second is that precedence
       of C++ operators will usually require that dot product operations
       be parenthesized, which is consistent with the linear algebra
       notation for an inner product.
       The code for BSP trees presented here is intended to be
       educational, and may or may not be very efficient. For the sake of
       clarity, the BSP tree itself will not be defined as a class.
       Last Update: 04/30/95 15:45:19
   What is a BSP Tree?
       Overview A Binary Space Partitioning (BSP) tree represents a
       recursive, hierarchical partitioning, or subdivision, of
       n-dimensional space into convex subspaces. BSP tree construction
       is a process which takes a subspace and partitions it by any
       hyperplane that intersects the interior of that subspace. The
       result is two new subspaces that can be further partitioned by
       recursive application of the method.
       A "hyperplane" in n-dimensional space is an n-1 dimensional object
       which can be used to divide the space into two half-spaces. For
       example, in three dimensional space, the "hyperplane" is a plane.
       In two dimensional space, a line is used.
       BSP trees are extremely versatile, because they are powerful
       sorting and classification structures. They have uses ranging from
       hidden surface removal and ray tracing hierarchies to solid
       modeling and robot motion planning.
       An easy way to think about BSP trees is to limit the discussion to
       two dimensions. To simplify the situation, let's say that we will
       use only lines parallel to the X or Y axis, and that we will
       divide the space equally at each node. For example, given a square
       somewhere in the XY plane, we select the first split, and thus the
       root of the BSP Tree, to cut the square in half in the X
       direction. At each slice, we will choose a line of the opposite
       orientation from the last one, so the second slice will divide
       each of the new pieces in the Y direction. This process will
       continue recursively until we reach a stopping point, and looks
       like this:
+-----------+      +-----+-----+      +-----+-----+
|           |      |     |     |      |     |     |
|           |      |     |     |      |  d  |     |
|           |      |     |     |      |     |     |
|     a     |  ->  |  b  X  c  |  ->  +--Y--+  f  |  -> ...
|           |      |     |     |      |     |     |
|           |      |     |     |      |  e  |     |
|           |      |     |     |      |     |     |
+-----------+      +-----+-----+      +-----+-----+
       The resulting BSP tree looks like this at each step:
      a                  X                  X           ...
                       -/ \+              -/ \+
                       /   \              /   \
                      b     c            Y     f
                                       -/ \+
                                       /   \
                                      e     d
       Other space partitioning structures
       BSP trees are closely related to Quadtrees and Octrees. Quadtrees
       and Octrees are space partitioning trees which recursively divide
       subspaces into four and eight new subspaces, respectively. A BSP
       Tree can be used to simulate both of these structures.
       Last Update: 05/16/95 01:18:59
   How do you build a BSP Tree?
       Given a set of polygons in three dimensional space, we want to
       build a BSP tree which contains all of the polygons. For now, we
       will ignore the question of how the resulting tree is going to be
       The algorithm to build a BSP tree is very simple:
         1. Select a partition plane.
         2. Partition the set of polygons with the plane.
         3. Recurse with each of the two new sets.
       Choosing the partition plane
       The choice of partition plane depends on how the tree will be
       used, and what sort of efficiency criteria you have for the
       construction. For some purposes, it is appropriate to choose the
       partition plane from the input set of polygons. Other applications
       may benefit more from axis aligned orthogonal partitions.
       In any case, you want to evaluate how your choice will affect the
       results. It is desirable to have a balanced tree, where each leaf
       contains roughly the same number of polygons. However, there is
       some cost in achieving this. If a polygon happens to span the
       partition plane, it will be split into two or more pieces. A poor
       choice of the partition plane can result in many such splits, and
       a marked increase in the number of polygons. Usually there will be
       some trade off between a well balanced tree and a large number of
       Partitioning polygons
       Partitioning a set of polygons with a plane is done by classifying
       each member of the set with respect to the plane. If a polygon
       lies entirely to one side or the other of the plane, then it is
       not modified, and is added to the partition set for the side that
       it is on. If a polygon spans the plane, it is split into two or
       more pieces and the resulting parts are added to the sets
       associated with either side as appropriate.
       When to stop
       The decision to terminate tree construction is, again, a matter of
       the specific application. Some methods terminate when the number
       of polygons in a leaf node is below a maximum value. Other methods
       continue until every polygon is placed in an internal node.
       Another criteria is a maximum tree depth.
       Pseudo C++ code example
       Here is an example of how you might code a BSP tree:
struct  BSP_tree
   plane     partition;
   list      polygons;
   BSP_tree  *front,
   This structure definition will be used for all subsequent example
       code. It stores pointers to its children, the partitioning plane
       for the node, and a list of polygons coincident with the partition
       plane. For this example, there will always be at least one polygon
       in the coincident list: the polygon used to determine the
       partition plane. A constructor method for this structure should
       initialize the child pointers to NULL.
void    Build_BSP_Tree (BSP_tree *tree, list polygons)
   polygon   *root = polygons.Get_From_List ();
   tree->partition = root->Get_Plane ();
   tree->polygons.Add_To_List (root);
   list      front_list,
   polygon   *poly;
   while ((poly = polygons.Get_From_List ()) != 0)
      int   result = tree->partition.Classify_Polygon (poly);
      switch (result)
         case COINCIDENT:
            tree->polygons.Add_To_List (poly);
         case IN_BACK_OF:
            backlist.Add_To_List (poly);
         case IN_FRONT_OF:
            frontlist.Add_To_List (poly);
         case SPANNING:
            polygon   *front_piece, *back_piece;
            Split_Polygon (poly, tree->partition, front_piece, back_piece);
            backlist.Add_To_List (back_piece);
            frontlist.Add_To_List (front_piece);
   if ( ! front_list.Is_Empty_List ())
      tree->front = new BSP_tree;
      Build_BSP_Tree (tree->front, front_list);
   if ( ! back_list.Is_Empty_List ())
      tree->back = new BSP_tree;
      Build_BSP_Tree (tree->back, back_list);
   This routine recursively constructs a BSP tree using the above
       definition. It takes the first polygon from the input list and
       uses it to partition the remainder of the set. The routine then
       calls itself recursively with each of the two partitions. This
       implementation assumes that all of the input polygons are convex.
       One obvious improvement to this example is to choose the
       partitioning plane more intelligently. This issue is addressed
       separately in the section, "How can you make a BSP Tree more
       Last Update: 05/08/95 13:10:25
   How do you partition a polygon with a plane?
       Partitioning a polygon with a plane is a matter of determining
       which side of the plane the polygon is on. This is referred to as
       a front/back test, and is performed by testing each point in the
       polygon against the plane. If all of the points lie to one side of
       the plane, then the entire polygon is on that side and does not
       need to be split. If some points lie on both sides of the plane,
       then the polygon is split into two or more pieces.
       The basic algorithm is to loop across all the edges of the polygon
       and find those for which one vertex is on each side of the
       partition plane. The intersection points of these edges and the
       plane are computed, and those points are used as new vertices for
       the resulting pieces.
       Implementation notes
       Classifying a point with respect to a plane is done by passing the
       (x, y, z) values of the point into the plane equation, Ax + By +
       Cz + D = 0. The result of this operation is the distance from the
       plane to the point along the plane's normal vector. It will be
       positive if the point is on the side of the plane pointed to by
       the normal vector, negative otherwise. If the result is 0, the
       point is on the plane.
       For those not familiar with the plane equation, The values A, B,
       and C are the coordinate values of the normal vector. D can be
       calculated by substituting a point known to be on the plane for x,
       y, and z.
       Convex polygons are generally easier to deal with in BSP tree
       construction than concave ones, because splitting them with a
       plane always results in exactly two convex pieces. Furthermore,
       the algorithm for splitting convex polygons is straightforward and
       robust. Splitting of concave polygons, especially self
       intersecting ones, is a significant problem in its own right.
       Pseudo C++ code example
       Here is a very basic function to split a convex polygon with a
void Split_Polygon (polygon *poly, plane *part, polygon *&front, polygon *&back
   int   count = poly->NumVertices (),
         out_c = 0, in_c = 0;
   point ptA, ptB,
   real sideA, sideB;
   ptA = poly->Vertex (count - 1);
   sideA = part->Classify_Point (ptA);
   for (short i = -1; ++i < count;)
      ptB = poly->Vertex (i);
      sideB = part->Classify_Point (ptB);
      if (sideB > 0)
         if (sideA < 0)
            // compute the intersection point of the line
            // from point A to point B with the partition
            // plane. This is a simple ray-plane intersection.
            vector v = ptB - ptA;
            real   sect = - part->Classify_Point (ptA) / (part->Normal () | v);
            outpts[out_c++] = inpts[in_c++] = ptA + (v * sect);
         outpts[out_c++] = ptB;
      else if (sideB < 0)
         if (sideA > 0)
            // compute the intersection point of the line
            // from point A to point B with the partition
            // plane. This is a simple ray-plane intersection.
            vector v = ptB - ptA;
            real   sect = - part->Classify_Point (ptA) / (part->Normal () | v);
            outpts[out_c++] = inpts[in_c++] = ptA + (v * sect);
         inpts[in_c++] = ptB;
         outpts[out_c++] = inpts[in_c++] = ptB;
      ptA = ptB;
      sideA = sideB;
   front = new polygon (outpts, out_c);
   back = new polygon (inpts, in_c);
   A simple extension to this code that is good for BSP trees is to
       combine its functionality with the routine to classify a polygon
       with respect to a plane.
       Note that this code is not robust, since numerical stability may
       cause errors in the classification of a point. The standard
       solution is to make the plane "thick" by use of an epsilon value.
       Last Update: 07/05/95 15:42:30
   How do you remove hidden surfaces with a BSP Tree?
       Probably the most common application of BSP trees is hidden
       surface removal in three dimensions. BSP trees provide an elegant,
       efficient method for sorting polygons via a depth first tree walk.
       This fact can be exploited in a back to front "painter's
       algorithm" approach to the visible surface problem, or a front to
       back scanline approach.
       BSP trees are well suited to interactive display of static (not
       moving) geometry because the tree can be constructed as a
       preprocess. Then the display from any arbitrary viewpoint can be
       done in linear time. Adding dynamic (moving) objects to the scene
       is discussed in another section of this document.
       Painter's algorithm
       The idea behind the painter's algorithm is to draw polygons far
       away from the eye first, followed by drawing those that are close
       to the eye. Hidden surfaces will be written over in the image as
       the surfaces that obscure them are drawn. One condition for a
       successful painter's algorithm is that there be a single plane
       which separates any two objects. This means that it might be
       necessary to split polygons in certain configurations. For
       example, this case can not be drawn correctly with a painter's
                          |      |
          +---------------|      |--+
          |               |      |  |
          |               |      |  |
          |               |      |  |
          |      +--------|      |--+
          |      |        |      |
       +--|      |--------+      |
       |  |      |               |
       |  |      |               |
       |  |      |               |
       +--|      |---------------+
          |      |
   One reason that BSP trees are so elegant for the painter's algorithm
       is that the splitting of difficult polygons is an automatic part
       of tree construction. Note that only one of these two polygons
       needs to be split in order to resolve the problem.
       To draw the contents of the tree, perform a back to front tree
       traversal. Begin at the root node and classify the eye point with
       respect to its partition plane. Draw the subtree at the far child
       from the eye, then draw the polygons in this node, then draw the
       near subtree. Repeat this procedure recursively for each subtree.
       Scanline hidden surface removal
       It is just as easy to traverse the BSP tree in front to back order
       as it is for back to front. We can use this to our advantage in a
       scanline method method by using a write mask which will prevent
       pixels from being written more than once. This will represent
       significant speedups if a complex lighting model is evaluated for
       each pixel, because the painter's algorithm will blindly evaluate
       the same pixel many times.
       The trick to making a scanline approach successful is to have an
       efficient method for masking pixels. One way to do this is to
       maintain a list of pixel spans which have not yet been written to
       for each scan line. For each polygon scan converted, only pixels
       in the available spans are written, and the spans are updated
       The scan line spans can be represented as binary trees, which are
       just one dimensional BSP trees. This technique can be expanded to
       a two dimensional screen coverage algorithm using a two
       dimensional BSP tree to represent the masked regions. Any convex
       partitioning scheme, such as a quadtree, can be used with similar
       Implementation notes
       When building a BSP tree specifically for hidden surface removal,
       the partition planes are usually chosen from the input polygon
       set. However, any arbitrary plane can be used if there are no
       intersecting or concave polygons, as in the example above.
       Pseudo C++ code example
       Using the BSP_tree structure defined in the section, "How do you
       build a BSP Tree?", here is a simple example of a back to front
       tree traversal:
void    Draw_BSP_Tree (BSP_tree *tree, point eye)
   real   result = tree->partition.Classify_Point (eye);
   if (result > 0)
      Draw_BSP_Tree (tree->back, eye);
      tree->polygons.Draw_Polygon_List ();
      Draw_BSP_Tree (tree->front, eye);
   else if (result < 0)
      Draw_BSP_Tree (tree->front, eye);
      tree->polygons.Draw_Polygon_List ();
      Draw_BSP_Tree (tree->back, eye);
   else // result is 0
      // the eye point is on the partition plane...
      Draw_BSP_Tree (tree->front, eye);
      Draw_BSP_Tree (tree->back, eye);
   If the eye point is classified as being on the partition plane, the
       drawing order is unclear. This is not a problem if the
       Draw_Polygon_List routine is smart enough to not draw polygons
       that are not within the viewing frustum. The coincident polygon
       list does not need to be drawn in this case, because those
       polygons will not be visible to the user.
       It is possible to substantially improve the quality of this
       example by including the viewing direction vector in the
       computation. You can determine that entire subtrees are behind the
       viewer by comparing the view vector to the partition plane normal
       vector. This test can also make a better decision about tree
       drawing when the eye point lies on the partition plane. It is
       worth noting that this improvement resembles the method for
       tracing a ray through a BSP tree, which is discussed in another
       section of this document.
       Front to back tree traversal is accomplished in exactly the same
       manner, except that the recursive calls to Draw_BSP_Tree occur in
       reverse order.
       Last Update: 05/08/95 13:10:25
   How do you compute analytic visibility with a BSP Tree?
       Last Update: 05/20/95 22:56:51
   How do you accelerate ray tracing with a BSP Tree?
       Ray tracing a BSP tree is very similar to hidden surface removal
       with a BSP tree. The algorithm is a simple forward tree walk, with
       a few additions that apply to ray casting.
       Last Update: 04/30/95 15:45:19
   How do you perform boolean operations on polytopes with a BSP Tree?
       There are two major classes of solid modeling methods with BSP
       trees. For both methods, it is useful to introduce the notion of
       an in/out test.
       An in/out test is a different way of talking about the front/back
       test we have been using to classify points with respect to planes.
       The necessity for this shift in thought is evident when
       considering polytopes instead of just polygons. A point can not be
       merely in front or back of a polytope, but inside or outside.
       Somewhat formally, a point is inside of a polytope if it is inside
       of, or in back of, each hyperplane which composes the polytope,
       otherwise it is outside.
       Incremental construction
       Incremental construction of a BSP Tree is the process of inserting
       convex polytopes into the tree one by one. Each polytope has to be
       processed according to the operation desired.
       It is useful to examine the construction process in two
       dimensions. Consider the following figure:

A               B
 |             |
 |             |
 |      E      |        F
 |       +-----+-------+
 |       |     |       |
 |       |     |       |
 |       |     |       |
 +-------+-----+       |
D        |      C      |
         |             |
         |             |
        H               G
   Two polygons, ABCD, and EFGH, are to be inserted into the tree. We
       wish to find the union of these two polygons. Start by inserting
       polygon ABCD into the tree, choosing the splitting hyperplanes to
       be coincident with the edges. The tree looks like this after
       insertion of ABCD:

              -/  \+
              /    \
             /      *
          -/  \+
          /    \
         /      *
      -/  \+
      /    \
     /      *
  -/  \+
  /    \
 *      *
   Now, polygon EFGH is inserted into the tree, one polygon at a time.
       The result looks like this:

A               B
 |             |
 |             |
 |      E      |J       F
 |       +-----+-------+
 |       |     |       |
 |       |     |       |
 |       |     |       |
 +-------+-----+       |
D        |L    :C      |
         |     :       |
         |     :       |
        H      K        G

                      -/  \+
                      /    \
                     /      *
                  -/  \+
                  /    \
                 /      \
                CD       \
              -/  \+      \
              /    \       \
             /      \       \
            DA       \       \
          -/  \+      \       \
          /    \       \       \
         /      *       \       \
        EJ              KH       \
      -/  \+          -/  \+      \
      /    \          /    \       \
     /      *        /      *       \
    LE              HL              JF
  -/  \+          -/  \+          -/  \+
  /    \          /    \          /    \
 *      *        *      *        FG     *
                               -/  \+
                               /    \
                              /      *
                           -/  \+
                           /    \
                          *      *
   Notice that when we insert EFGH, we split edges EF and HE along the
       edges of ABCD. this has the effect of dividing these segments into
       pieces which are inside ABCD, and outside ABCD. Segments EJ and LE
       will not be part of the boundary of the union. We could have saved
       our selves some work by not inserting them into the tree at all.
       For a union operation, you can always throw away segments that
       land in inside nodes. You must be careful about this though. What
       I mean is that any segments which land in inside nodes of side the
       pre-existing tree, not the tree as it is being constructed. EJ and
       LE landed in an inside node of the tree for polygon ABCD, and so
       can be discarded.
       Our tree now looks like this:

A               B
 |             |
 |             |
 |             |J       F
 |             +-------+
 |             |       |
 |             |       |
 |             |       |
 +-------+-----+       |
D        |L    :C      |
         |     :       |
         |     :       |
        H      K        G

                      -/  \+
                      /    \
                     /      *
                  -/  \+
                  /    \
                 /      \
                CD       \
              -/  \+      \
              /    \       \
             /      \       \
            DA       \       \
          -/  \+      \       \
          /    \       \       \
         *      *       \       \
                        KH       \
                      -/  \+      \
                      /    \       \
                     /      *       \
                    HL              JF
                  -/  \+          -/  \+
                  /    \          /    \
                 *      *        FG     *
                               -/  \+
                               /    \
                              /      *
                           -/  \+
                           /    \
                          *      *
   Now, we would like some way to eliminate the segments JC and CL, so
       that we will be left with the boundary segments of the union.
       Examine the segment BC in the tree. What we would like to do is
       split BC with the hyperplane JF. Conveniently, we can do this by
       pushing the BC segment through the node for JF. The resulting
       segments can be classified with the rest of the JF subtree. Notice
       that the segment BJ lands in an out node, and that JC lands in an
       in node. Remembering that we can discard interior nodes, we can
       eliminate JC. The segment BJ replaces BC in the original tree.
       This process is repeated for segment CD, yielding the segments CL
       and LD. CL is discarded as landing in an interior node, and LD
       replaces CD in the original tree. The result looks like this:

A               B
 |             |
 |             |
 |             |J       F
 |             +-------+
 |                     |
 |                     |
 |        L            |
 +-------+             |
D        |             |
         |             |
         |             |
        H      K        G

                      -/  \+
                      /    \
                     /      *
                  -/  \+
                  /    \
                 /      \
                LD       \
              -/  \+      \
              /    \       \
             /      \       \
            DA       \       \
          -/  \+      \       \
          /    \       \       \
         *      *       \       \
                        KH       \
                      -/  \+      \
                      /    \       \
                     /      *       \
                    HL              JF
                  -/  \+          -/  \+
                  /    \          /    \
                 *      *        FG     *
                               -/  \+
                               /    \
                              /      *
                           -/  \+
                           /    \
                          *      *
   As you can see, the result is the union of the polygons ABCD and EFGH.
       To perform other boolean operations, the process is similar. For
       intersection, you discard segments which land in exterior nodes
       instead of internal ones. The difference operation is special. It
       requires that you invert the polytope before insertion. For simple
       objects, this can be achieved by scaling with a factor of -1. The
       insertion process is then cinducted as an intersection operation,
       where segments landing in external nodes are discarded.
       Tree merging
       Last Update: 04/30/95 15:45:20
   How do you perform collision detection with a BSP Tree?
       Detecting whether or not a point moving along a line intersects
       some object in space is essentially a ray tracing problem.
       Detecting whether or not two complex objects intersect is
       something of a tree merging problem.
       Typically, motion is computed in a series of Euler steps. This
       just means that the motion is computed at discrete time intervals
       using some description of the speed of motion. For any given point
       P moving from point A with a velocity V, it's location can be
       computed at time T as P = A + (T * V).
       Consider the case where T = 1, and we are computing the motion in
       one second steps. To find out if the point P has collided with any
       part of the scene, we will first compute the endpoints of the
       motion for this time step. P1 = A + V, and P2 = A + (2 * V). These
       two endpoints will be classified with respect to the BSP tree. If
       P1 is outside of all objects, and P2 is inside some object, then
       an intersection has clearly occurred. However, if P2 is also
       outside, we still have to check for a collision in between.
       Two approaches are possible. The first is commonly used in
       applications like games, where speed is critical, and accuracy is
       not. This approach is to recursively divide the motion segment in
       half, and check the midpoint for containment by some object.
       Typically, it is good enough to say that an intersection occurred,
       and not be very accurate about where it occurred.
       The second approach, which is more accurate, but also more time
       consuming, is to treat the motion segment as a ray, and intersect
       the ray with the BSP Tree. This also has the advantage that the
       motion resulting from the impact can be computed more accurately.
       Last Update: 04/30/95 15:45:20
   How do you handle dynamic scenes with a BSP Tree?
       So far the discussion of BSP tree structures has been limited to
       handling objects that don't move. However, because the hidden
       surface removal algorithm is so simple and efficient, it would be
       nice if it could be used with dynamic scenes too. Faster animation
       is the goal for many applications, most especially games.
       The BSP tree hidden surface removal algorithm can easily be
       extended to allow for dynamic objects. For each frame, start with
       a BSP tree containing all the static objects in the scene, and
       reinsert the dynamic objects. While this is straightforward to
       implement, it can involve substantial computation.
       If a dynamic object is separated from each static object by a
       plane, the dynamic object can be represented as a single point
       regardless of its complexity. This can dramatically reduce the
       computation per frame because only one node per dynamic object is
       inserted into the BSP tree. Compare that to one node for every
       polygon in the object, and the reason for the savings is obvious.
       During tree traversal, each point is expanded into the original
       Implementation notes
       Inserting a point into the BSP tree is very cheap, because there
       is only one front/back test at each node. Points are never split,
       which explains the requirement of separation by a plane. The
       dynamic object will always be drawn completely in front of the
       static objects behind it.
       A dynamic object inserted into the tree as a point can become a
       child of either a static or dynamic node. If the parent is a
       static node, perform a front/back test and insert the new node
       appropriately. If it is a dynamic node, a different front/back
       test is necessary, because a point doesn't partition three
       dimesnional space. The correct front/back test is to simply
       compare distances to the eye. Once computed, this distance can be
       cached at the node until the frame is drawn.
       An alternative when inserting a dynamic node is to construct a
       plane whose normal is the vector from the point to the eye. This
       plane is used in front/back tests just like the partition plane in
       a static node. The plane should be computed lazily and it is not
       necessary to normalize the vector.
       Cleanup at the end of each frame is easy. A static node can never
       be a child of a dynamic node, since all dynamic nodes are inserted
       after the static tree is completed. This implies that all subtrees
       of dynamic nodes can be removed at the same time as the dynamic
       parent node.
       Advanced methods
       Tree merging, "ghosts", real dynamic trees... MORE TO COME
       Last Update: 04/29/95 03:14:22
   How do you compute shadows with a BSP Tree?
       Last Update: 04/30/95 15:45:20
   How do you extract connectivity information from BSP Trees?
       Last Update: 04/30/95 15:45:20
   How are BSP Trees useful for robot motion planning?
       Last Update: 04/30/95 15:45:20
   How are BSP Trees used in DOOM?
       Before you can understand how DOOM uses a BSP tree to accelerate
       its rendering process, you have to understand how the world is
       represented in DOOM. When someone creates a DOOM level in a level
       editor they draw linedefs in a 2d space. Yes, that's right, DOOM
       is only 2d. These linedefs (ignoring the special effects linedefs)
       must be arranged so that they form closed polygons. One linedef
       may be used to form the outline of two polygons (in which case it
       is known as a two-sided linedef) and one polygon may be contained
       within another, but no linedefs may cross. Each enclosed area of
       the world (i.e. polygon) is assigned a floor height, ceiling
       height, floor and ceiling textures, a lower texture and an upper
       texture. The lower texture is visible when a linedef is viewed
       from a direction where the floor is lower in the adjoining area.
       An equivalent thing is true for the upper texture. A set of these
       enclosed areas that all have the same attributes is known as a
       When the level is saved by the editor some new information is
       created including the BSP tree for that level. Before the BSP tree
       can be created, all the sectors have to be split into convex
       polygons known as sub-sectors. If you had a sector that was a
       square area, then that would translate exactly into a sub-sector.
       Whereas if that sector was contained inside another larger square
       sector, the larger one would have to be split into four, four
       sided sub-sectors to make all the sub-sectors convex. When more
       complex sectors are split into sub-sectors the linedefs that bound
       that sector may need to be broken into smaller lengths. These
       linedef sections are called segs.
       Given a point on the 2d map, the renderer (which isn't discussed
       here) wants a list of all the segs that are visible from that
       viewpoint in closest first order. Because of the restrictions
       placed on the DOOM world, the renderer can easily tell when the
       screen has been filled so it can stop looking for segs at this
       time. This is quicker than rendering all the segs from back to
       front and using a method like painters algorithm.
       Each node in the BSP tree defines a partition line (this does not
       have be a linedef in the world but usually is) which is the
       equivalent to the partition plane of a 3d BSP tree. It then has
       left and right pointers which are either another node for further
       sub-division or a leaf, the leaf being a sub-sector in DOOM. The
       BSP tree in DOOM is effectively being used to sort whole
       sub-sectors rather than individual lines front to back. Each node
       also defines an orthogonal bounding box for each side of the
       partition. All segs on a particular side of the partition must be
       within that box. This speeds up the searching process by allowing
       whole branches of the tree to be discarded if that bounding box
       isn't visible. The test for visibility is simply if the bounding
       box lies wholly or partly within the cone defined by the left and
       right edges of the screen.
       During the display update process the BSP tree is searched
       starting from the node containing the sub-sector that the player
       is currently in. The search moves outwards through the tree
       (searching the other half of the current node before moving onto
       the other half of the parents node). When a partition test is
       performed the branch chosen is the one on the same side as the
       player. This facilitates the front to back searching. Each time a
       leaf is encountered the segs in that sub-sector are passed to the
       renderer. If the renderer has returned that the screen is filled
       then the process stops, otherwise it continues until the tree has
       been fully searched (in which case there is an error in the level
       In case you're thinking that it is inefficient to dump a whole
       sub-sectors worth of segs into the renderer at once, the segs in a
       sub-sector can be back-face culled very quickly. DOOM stores the
       angle of linedefs (of which segs are part). When the angle of the
       players view is calculated this allows segs to be culled in a
       single instruction! Angles are stored as a 16 bit number where 0
       is east an 65535 is 1/63336 south of east.
       Last Update: 04/30/95 15:45:20
   How can you make a BSP Tree more robust?
       Last Update: 04/30/95 15:45:20
   How efficient is a BSP Tree?
       Space complexity
       For hidden surface removal and ray tracing accelleration, the
       upper bound is O(n ^ 2) for n polygons. The expected case is O(n)
       for most models. MORE LATER
       Time complexity
       For hidden surface removal and ray tracing accelleration, the
       upper bound is O(n ^ 2) for n polygons. The expected case is O(n)
       for most models. MORE LATER
       Last Update: 04/30/95 15:45:20
   How can you make a BSP Tree more efficient?
       Bounding volumes
       Bounding spheres are simple to implement, take only a single plane
       comparison, using the center of the sphere.
       Optimal trees
       Construction of an optimal tree is an NP-complete problem. The
       problem is one of splitting versus tree balancing. These are
       mutually exclusive requirements. You should choose your strategy
       for building a good tree based on how you intend to use the tree.
       Minimizing splitting
       An obvious problem with BSP trees is that polygons get split
       during the construction phase, which results in a larger number of
       polygons. Larger numbers of polygons translate into larger storage
       requirements and longer tree traversal times. This is undesirable
       in all applications of BSP trees, so some scheme for minimizing
       splitting will improve tree performance.
       Bear in mind that minimization of splitting requires pre-existing
       knowledge about all of the polygons that will be inserted into the
       tree. This knowledge may not exist for interactive uses such as
       solid modelling.
       Tree balancing
       Tree balancing is important for uses which perform spatial
       classification of points, lines, and surfaces. This includes ray
       tracing and solid modelling. Tree balancing is important for these
       applications because the time complexity for classification is
       based on the depth of the tree. Unbalanced trees have deeper
       subtrees, and therefore have a worse worst case.
       For the hidden surface problem, balancing doesn't significantly
       affect runtime. This is because the expected time complexity for
       tree traversal is linear on the number of polygons in the tree,
       rather than the depth of the tree.
       Balancing vs. splitting
       If balancing is an important concern for your application, it will
       be necessary to trade off some balance for reduced splitting. If
       you are choosing your hyperplanes from the polygon candidates,
       then one way to optimize these two factors is to randomly select a
       small number of candidates. These new candidates are tested
       against the full list for splitting and balancing efficiency. A
       linear combination of the two efficiencies is used to rank the
       candidates, and the best one is chosen.
       Reference Counting
       Other Optimizations
       Last Update: 05/16/95 01:16:38
   How can you avoid recursion?
       standard binary tree search/sort techniques apply.
       Last Update: 03/02/95 23:40:07
   What is the history of BSP Trees?
       Last Update: 04/30/95 15:45:20
   Where can you find sample code and related online resources?
       BSP tree FAQ companion code
       The companion source code to this document is available via FTP
          + file://
   or, you can also request that the source be mailed to you by sending
       e-mail to with a subject line of
       "SEND BSP TREE SOURCE". This will return to you a UU encoded copy
       of the sample C++ source code.
       Other BSP tree resources
       Pat Fleckenstein and Rob Reay have put together a FAQ on 3D
       graphics, which includes a blurb on BSP Trees, and an ftp site
       with some sample code. They seem to have an unusual affinity for
       ftp sites, and therefore won't link the BSP tree FAQ from their
          + 3D FAQ
          + file://
       Implementing and Using BSP Trees
    1. Accompanying C++ source
       Michael Abrash's columns in the '95 DDJ Sourcebook are an
       excellent introduction to the concept of BSP trees, especially in
       two dimensions. The source code for these is available as part of
       a package.
          + Abrash BSP tree source, and other C++ stuff
       Ekkehard Beier has made available a generic 3D graphics kernel
       intended to assist development of graphics application interfaces.
       One of the classes in the library is a BSP tree, and full source
       is provided. The focus seems to be on ray tracing, with the code
       being based on Jim Arvo's Linear Time Voxel Walking article in the
       ray tracing news.
          + Generic 3d kernel
       Eddie Edwards wrote a commonly referenced text which describes 2D
       BSP trees in some detail for use in games like DOOM. It includes a
       bit of sample code, too.
       Mel Slater has made available his C source code for computing
       shadow volumes based on BSP trees:
          + A Comparison of Three Shadow Volume Algorithms
       Graphics Gems
       Peter Shirley and Kelvin Sung have C sample code for ray tracing
       with BSP trees in Graphics Gems III
       Norman Chin has provided a wonderful resource for BSP trees in
       Graphics Gems V. He provides C sample code for a wide variety of
       More sources for sample BSP tree code
          + file://
       General resources for computer graphics programming
       Algorithm, Incorporated, an Atlanta-based Scientific and
       Engineering Research and Development Company specializing in
       Computer Graphics Programming and Business Internet
       Communications, has lots of good pointers and useful offerings.
       If you are interested in game programming, check out the FAQ.
       Last Update: 08/04/95 12:16:09
       A partial listing of textual info on BSP trees.
    2. Abrash, M., BSP Trees, Dr. Dobbs Sourcebook, 20(14), 49-52,
       may/jun 1995.
    3. Dadoun, N., Kirkpatrick, D., and Walsh, J., The Geometry of Beam
       Tracing, Proceedings of the ACM Symposium on Computational
       Geometry, 55--61, jun 1985.
    4. Chin, N., and Feiner, S., Near Real-Time Shadow Generation Using
       BSP Trees, Computer Graphics (SIGGRAPH '89 Proceedings), 23(3),
       99--106, jul 1989.
    5. Chin, N., and Feiner, S., Fast object-precision shadow generation
       for area light sources using BSP trees, Computer Graphics (1992
       Symposium on Interactive 3D Graphics), 25(2), 21--30, mar 1992.
    6. Chrysanthou, Y., and Slater, M., Computing dynamic changes to BSP
       trees, Computer Graphics Forum (EUROGRAPHICS '92 Proceedings),
       11(3), 321--332, sep 1992.
    7. Naylor, B., Amanatides, J., and Thibault, W., Merging BSP Trees
       Yields Polyhedral Set Operations, Computer Graphics (SIGGRAPH '90
       Proceedings), 24(4), 115--124, aug 1990.
    8. Chin, N., and Feiner, S., Fast object-precision shadow generation
       for areal light sources using BSP trees, Computer Graphics (1992
       Symposium on Interactive 3D Graphics), 25(2), 21--30, mar 1992.
    9. Naylor, B., Interactive solid geometry via partitioning trees,
       Proceedings of Graphics Interface '92, 11--18, may 1992.
   10. Naylor, B., Partitioning tree image representation and generation
       from 3D geometric models, Proceedings of Graphics Interface '92,
       201--212, may 1992.
   11. Naylor, B., {SCULPT} An Interactive Solid Modeling Tool,
       Proceedings of Graphics Interface '90, 138--148, may 1990.
   12. Gordon, D., and Chen, S., Front-to-back display of BSP trees, IEEE
       Computer Graphics and Applications, 11(5), 79--85, sep 1991.
   13. Ihm, I., and Naylor, B., Piecewise linear approximations of
       digitized space curves with applications, Scientific Visualization
       of Physical Phenomena (Proceedings of CG International '91),
       545--569, 1991.
   14. Vanecek, G., Brep-index: a multidimensional space partitioning
       tree, Internat. J. Comput. Geom. Appl., 1(3), 243--261, 1991.
   15. Arvo, J., Linear Time Voxel Walking for Octrees, Ray Tracing News,
       feb 1988.
   16. Jansen, F., Data Structures for Ray Tracing, Data Structures for
       Raster Graphics, 57--73, 1986.
   17. MacDonald, J., and Booth, K., Heuristics for Ray Tracing Using
       Space Subdivision, Proceedings of Graphics Interface '89, 152--63,
       jun 1989.
   18. Naylor, B., and Thibault, W., Application of BSP Trees to Ray
       Tracing and CSG Evaluation, Tech. Rep. GIT-ICS 86/03, feb 1986.
   19. Sung, K., and Shirley, P., Ray Tracing with the BSP Tree, Graphics
       Gems III, 271--274, 1992.
   20. Fuchs, H., Kedem, Z., and Naylor, B., On Visible Surface
       Generation by A Priori Tree Structures, Conf. Proc. of SIGGRAPH
       '80, 14(3), 124--133, jul 1980.
   21. Paterson, M., and Yao, F., Efficient Binary Space Partitions for
       Hidden-Surface Removal and Solid Modeling, Discrete and
       Computational Geometry, 5(5), 485--503, 1990.
       Last Update: 06/19/95 09:59:42
   This document was last updated on
   Bretton Wade (