BSP Trees

BSP

Binary Space Partition Trees use Half-Spaces to cut the space in half. This is applied recursively on the new branches. Each half space further sub-dividing the region in two.

In two dimensions the cutter is a line. Here is an example and the associated bsp tree.

If a line is visible to its parent line then the line appears on the rhs branch, else its on the left hand side branch.

The leafs are the regions.

Here I use points to construct the lines. An arbitrary point of the remaining points was chosen to create the next partitions. The remaining points were divided according to which side of the line the points was on. Quick Hull uses a similar partitioning strategy.

It is easy to test if a point is inside the region. Just follow the branch and if all the tests are satisfied then the point is in the region. The test is a dot product of the point with the line (translating both to the origin first) > 0.

Visible Surface Generation

Consider a space with geometric objects made up of polygons. A BSP can partition or tessellate this space. For a static scene where the scene geometry does not change the tessellation can be used in the rendering process because the tessellation orders the polygons - literally a tessellation is an ordering.

Since the painters algorithm paints the objects furthest away first the tessellation is ideal. The tessellation is transversed from furthest partitions to the nearest partitions. Since this is represented in the BSP tree a fancy tree transversing algorithm is used.

Since the geometry is static then the tree is static. So the tree transverse algorithm works on a static tree from any partition within the tree. Since the tree/geometry is constructed once it is efficient.

"On Visible Surface Generation by a Priori Tree Structure" by H Fuchs, Z. Kedem and B. Naylor (1980 ACM 0-89791-021-4/80/0700) paper apparently introduced BSP trees for this.

This paper harps on about minimizing the number of polygons generated. When a polygon crosses a partition boundary it is split so that it lies distinctly in one of the partitions. Such a minimization is extremely silly by to days standards but academics often waffle their papers and at the time there was justification because the hardware cost millions of dollars, but now we would not care about this. Frankly I would not care about it then either.

So when you read their paper you can see that their drama over "a large increase from the original number of polygons..." is a drama because the half space cuts maybe 20-30%. This is insignificant when you look at the tools they bring out to deal with it. So this large increase is a fiction.

The paper is good and has other things I do not understand.

From what I gather BSP trees can be used in may ways. The paper separates the idea of fixed areas and replaces them with the geometric objects. And while polygons (and hence simplexes) make up a BSP trees objects, any geometric object can do. If memory is not an issue the whole surface as simplexes could be in the tree.

Given N geometric objects to be rendered transversing is O(N). ie we do not have to decide if object x is in front of y, so processing is linear.

Other uses of the BSP tree could be used in conjunction with LOD to render the scene.

No culling was done. <TODO> - read up on BSP trees and culling