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Path: senator-bedfellow.mit.edu!bloom-beacon.mit.edu!newsfeed.internetmci.com!news.mathworks.com!news.kei.com!travelers.mail.cornell.edu!willow-farm.cit.cornell.edu!news.graphics.cornell.edu!uranus.graphics.cornell.edu!not-for-mail From: BSP-FAQ <bsp-faq@graphics.cornell.edu> Newsgroups: comp.graphics.algorithms,comp.answers,news.answers Subject: Binary Space Partitioning Trees FAQ Followup-To: comp.graphics.algorithms Date: 26 Sep 1995 11:28:16 -0400 Organization: Cornell University Program of Computer Graphics Lines: 1365 Sender: bwade@Graphics.Cornell.edu Approved: news-answers-request@MIT.EDU Message-ID: <44966g$dmo@uranus.graphics.cornell.edu> NNTP-Posting-Host: uranus.graphics.cornell.edu Summary: Frequently asked questions about Binary Space Partitioning trees. Xref: senator-bedfellow.mit.edu comp.graphics.algorithms:21402 comp.answers:14484 news.answers:53762 Archive-name: graphics/bsptree-faq Posting-Frequency: monthly URL: http://www.graphics.cornell.edu/bspfaq/ BSP TREE FREQUENTLY ASKED QUESTIONS (FAQ) _________________________________________________________________ Questions 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 Tree? 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 _________________________________________________________________ Answers About this document General 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 comp.graphics.algorithms. It is also available via WWW at the URL: http://www.graphics.cornell.edu/bspfaq/ The most recent newsgroup posting of this document is available via ftp at the URL: ftp://rtfm.mit.edu:/pub/usenet/news.answers/graphics/bsptree-faq 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 bsp-faq@graphics.cornell.edu 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 © 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 bsp-faq@graphics.cornell.edu. 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 Graphics. -- Last Update: 07/05/95 03:46:05 Acknowledgements 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 (bwade@graphics.cornell.edu or bsp-faq@graphics.cornell.edu) before trying to contact anyone else on this list. Contributors + Bruce Naylor (naylor@research.att.com) + Richard Lobb (richard@cs.auckland.ac.nz) + Dani Lischinski (danix@cs.washington.edu) + Chris Schoeneman (crs@lightscape.com) + Philip Hubbard (pmh@graphics.cornell.edu) + Jim Arvo (arvo@graphics.cornell.edu) + Kevin Ryan (kryan@access.digex.net) + Joseph Fiore (fiore@cs.buffalo.edu) + Lukas Rosenthaler (rosenth@foto.chemie.unibas.ch) + Anson Tsao (ansont@hookup.net) + Robert Zawarski (zawarski@chaph.usc.edu) + Ron Capelli (capelli@vnet.ibm.com) + Eric A. Haines (erich@eye.com) + Ian CR Mapleson (mapleson@cee.hw.ac.uk) + Richard Dorman (richard@cs.wits.ac.za) + Steve Larsen (larsen@sunset.cs.utah.edu) + Timothy Miller (tsm@cs.brown.edu) + Ben Trumbore (wbt@graphics.cornell.edu) + Richard Matthias (richardm@cogs.susx.ac.uk) + Ken Shoemake (shoemake@graphics.cis.upenn.edu) + Seth Teller (seth@theory.lcs.mit.edu) + Peter Shirley (shirley@graphics.cornell.edu) + Michael Abrash (mikeab@idece2.idsoftware.com) + Robert Schmidt (robert@idt.unit.no) 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 bsp-faq@graphics.cornell.edu. -- Last Update: 03/29/95 14:12:10 About the pseudo C++ code Overview 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. Example 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? Overview 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 used. 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 splits. 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, *back; }; 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, back_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); break; case IN_BACK_OF: backlist.Add_To_List (poly); break; case IN_FRONT_OF: frontlist.Add_To_List (poly); break; 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); break; } } 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 efficient?". -- Last Update: 05/08/95 13:10:25 How do you partition a polygon with a plane? Overview 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 plane: void Split_Polygon (polygon *poly, plane *part, polygon *&front, polygon *&back ) { int count = poly->NumVertices (), out_c = 0, in_c = 0; point ptA, ptB, outpts[MAXPTS], inpts[MAXPTS]; 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; } else 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? Overview 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 algorithm: +------+ | | +---------------| |--+ | | | | | | | | | | | | | +--------| |--+ | | | | +--| |--------+ | | | | | | | | | | | | | +--| |---------------+ | | +------+ 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 accordingly. 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 effect. 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? Overview -- Last Update: 05/20/95 22:56:51 How do you accelerate ray tracing with a BSP Tree? Overview 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. MORE TO COME -- Last Update: 04/30/95 15:45:19 How do you perform boolean operations on polytopes with a BSP Tree? Overview 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: AB -/ \+ / \ / * BC -/ \+ / \ / * CD -/ \+ / \ / * DA -/ \+ / \ * * 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 AB -/ \+ / \ / * BC -/ \+ / \ / \ CD \ -/ \+ \ / \ \ / \ \ DA \ \ -/ \+ \ \ / \ \ \ / * \ \ EJ KH \ -/ \+ -/ \+ \ / \ / \ \ / * / * \ LE HL JF -/ \+ -/ \+ -/ \+ / \ / \ / \ * * * * FG * -/ \+ / \ / * GK -/ \+ / \ * * 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 AB -/ \+ / \ / * BC -/ \+ / \ / \ CD \ -/ \+ \ / \ \ / \ \ DA \ \ -/ \+ \ \ / \ \ \ * * \ \ KH \ -/ \+ \ / \ \ / * \ HL JF -/ \+ -/ \+ / \ / \ * * FG * -/ \+ / \ / * GK -/ \+ / \ * * 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 AB -/ \+ / \ / * BJ -/ \+ / \ / \ LD \ -/ \+ \ / \ \ / \ \ DA \ \ -/ \+ \ \ / \ \ \ * * \ \ KH \ -/ \+ \ / \ \ / * \ HL JF -/ \+ -/ \+ / \ / \ * * FG * -/ \+ / \ / * GK -/ \+ / \ * * 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? Overview 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? Overview 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 object. 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? Overview -- Last Update: 04/30/95 15:45:20 How do you extract connectivity information from BSP Trees? Overview -- Last Update: 04/30/95 15:45:20 How are BSP Trees useful for robot motion planning? Overview -- Last Update: 04/30/95 15:45:20 How are BSP Trees used in DOOM? Overview 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 sector. 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 design). 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? Overview -- 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? Overview -- 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 at: + file://ftp.graphics.cornell.edu/pub/bsptree/ or, you can also request that the source be mailed to you by sending e-mail to bsp-faq@graphics.cornell.edu 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 document: + http://www.csh.rit.edu/~pat/misc/3dFaq.html + file://ftp.csh.rit.edu/pub/3dfaq/ 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. + http://www.ddj.com/ddj/issues/j9507a.htm + http://www.ddj.com/ddj/issues/j9507b.htm 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. + ftp://ftp.mv.com/pub/ddj/1995/1995.cpp/asc.zip 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. + ftp://metallica.prakinf.tu-ilmenau.de/pub/PROJECTS/GENERIC/gen eric1.1.tar.gz 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. + file://x2ftp.oulu.fi/pub/msdos/programming/theory/bsp_tree.zip Mel Slater has made available his C source code for computing shadow volumes based on BSP trees: + http://www.dcs.qmw.ac.uk/~mel/BSP.html Graphics Gems The Graphics Gems archive at file://ftp.princeton.edu/pub/Graphics/GraphicsGems/ 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 uses. More sources for sample BSP tree code + file://ftp.idsoftware.com/tonsmore/utils/level_edit/node_build ers/ + file://ftp.cs.brown.edu/pub/sphigs.tar.Z 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 rec.games.programmer.faq: http://www.ee.ucl.ac.uk/~phart/FAQ/rgp_FAQ.html. -- Last Update: 08/23/95 10:16:23 References 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 _________________________________________________________________ This document was last updated on Andrew Kunz (ank@graphics.cornell.edu)