Binary Space Partitioning Trees FAQ

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Sep 26, 1995, 3:00:00 AM9/26/95
Archive-name: graphics/bsptree-faq
Posting-Frequency: monthly




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
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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
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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 (
+ Michael Abrash (
+ 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)
tree->polygons.Add_To_List (poly);
case IN_BACK_OF:
backlist.Add_To_List (poly);
frontlist.Add_To_List (poly);
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:

| |
| |
| E | F
| +-----+-------+
| | | |
| | | |
| | | |
+-------+-----+ |
D | C |
| |
| |
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:

| |
| |
| E |J F
| +-----+-------+
| | | |
| | | |
| | | |
+-------+-----+ |
D |L :C |
| : |
| : |

-/ \+
/ \
/ *
-/ \+
/ \
/ \
CD \
-/ \+ \
/ \ \
/ \ \
DA \ \
-/ \+ \ \
/ \ \ \
/ * \ \
-/ \+ -/ \+ \
/ \ / \ \
/ * / * \
-/ \+ -/ \+ -/ \+
/ \ / \ / \
* * * * 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:

| |
| |
| |J F
| +-------+
| | |
| | |
| | |
+-------+-----+ |
D |L :C |
| : |
| : |

-/ \+
/ \
/ *
-/ \+
/ \
/ \
CD \
-/ \+ \
/ \ \
/ \ \
DA \ \
-/ \+ \ \
/ \ \ \
* * \ \
KH \
-/ \+ \
/ \ \
/ * \
-/ \+ -/ \+
/ \ / \
* * 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:

| |
| |
| |J F
| +-------+
| |
| |
| L |
+-------+ |
D | |
| |
| |

-/ \+
/ \
/ *
-/ \+
/ \
/ \
LD \
-/ \+ \
/ \ \
/ \ \
DA \ \
-/ \+ \ \
/ \ \ \
* * \ \
KH \
-/ \+ \
/ \ \
/ * \
-/ \+ -/ \+
/ \ / \
* * 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

+ file://

Dr. Dobbs Journal has an article in their July '95 issue about BSP
trees, By Nathan Dwyer. It describes the construction of BSP trees
for visible surface processing, how to split polygons with planes,
and how to dump the tree to a file. There is C++ source code to
accompany the article.


Michael Abrash's columns in the '95 DDJ Sourcebooks 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.


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.


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:


Graphics Gems
The Graphics Gems archive at
file:// is an
invaluable resource for all things graphical. In particular, there
are some BSP tree references worth looking over.

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
Last Update: 08/23/95 10:16:23


A partial listing of textual info on BSP trees.

1. Abrash, M., BSP Trees, Dr. Dobbs Sourcebook, 20(14), 49-52,
may/jun 1995.

2. Dadoun, N., Kirkpatrick, D., and Walsh, J., The Geometry of Beam
Tracing, Proceedings of the ACM Symposium on Computational
Geometry, 55--61, jun 1985.

3. Chin, N., and Feiner, S., Near Real-Time Shadow Generation Using
BSP Trees, Computer Graphics (SIGGRAPH '89 Proceedings), 23(3),
99--106, jul 1989.

4. 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.

5. Chrysanthou, Y., and Slater, M., Computing dynamic changes to BSP
trees, Computer Graphics Forum (EUROGRAPHICS '92 Proceedings),
11(3), 321--332, sep 1992.

6. 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.

7. 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.

8. Naylor, B., Interactive solid geometry via partitioning trees,
Proceedings of Graphics Interface '92, 11--18, may 1992.

9. Naylor, B., Partitioning tree image representation and generation
from 3D geometric models, Proceedings of Graphics Interface '92,
201--212, may 1992.

10. Naylor, B., {SCULPT} An Interactive Solid Modeling Tool,
Proceedings of Graphics Interface '90, 138--148, may 1990.

11. Gordon, D., and Chen, S., Front-to-back display of BSP trees, IEEE
Computer Graphics and Applications, 11(5), 79--85, sep 1991.

12. 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.

13. Vanecek, G., Brep-index: a multidimensional space partitioning
tree, Internat. J. Comput. Geom. Appl., 1(3), 243--261, 1991.

14. Arvo, J., Linear Time Voxel Walking for Octrees, Ray Tracing News,
feb 1988.

15. Jansen, F., Data Structures for Ray Tracing, Data Structures for
Raster Graphics, 57--73, 1986.

16. MacDonald, J., and Booth, K., Heuristics for Ray Tracing Using
Space Subdivision, Proceedings of Graphics Interface '89, 152--63,
jun 1989.

17. Naylor, B., and Thibault, W., Application of BSP Trees to Ray
Tracing and CSG Evaluation, Tech. Rep. GIT-ICS 86/03, feb 1986.

18. Sung, K., and Shirley, P., Ray Tracing with the BSP Tree, Graphics
Gems III, 271--274, 1992.

19. 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.

20. 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


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