八叉树(octree)简介(zz)

Introduction To Octrees

by Jaap Suter (13 April 1999)


Introduction


Hidden surface removal is among the biggest problemswhen writing a 3D engine.

I struggled with it since the very beginning of writing 3D engines and stillhave no satisfactory solution to it. The ideal visibility detection scheme would allowunlimited data, extremely dynamic worlds and would have zero overdraw. The first 3dengine which implements these three demands still has to be written.

The Z buffer for example allows dynamical worlds and even crossing faces, but itsuffers from immense overdraw. The BSP-tree on the other hand, if well implemented,has no overdraw at all but needs that much pre-processing that dynamical worlds are adefinite nono.

It wasn't until recently i first heard of the octree, and I must admit i was struckby it's simplicity. I never actually implemented this structure and therefore I willpresent no pseudo code at all. This explanation is merely an attempt to make it more clearto people who have never heard of it. Besides, if you really understand the structure,then implementation is a piece of cake.

The Octree Structure


Our virtual world or level is actually nothing more then a soup ofpolygons. Some of you people might have thrown in a couple of curves and voxels but mostof it will be polygons. Here it is:



Fig 1. Our little level.

In the picture I just built a simple level containing no more than 250 polys. Nowimagine a cube surrounding the world like in the next image:



Fig. 2. Our little level surrounded by a cube.

Now it isn't hard to see that this cube can be divided into eight smaller cubes,hence the name octree. Take a look at this picture:



Fig 3. Our little level with the surrounding cube subdivided.

Call the larger cube the parent and the smaller cubes the children. On their turn subdivideeach children into eight smaller cubes, and you will notice we are creating a tree where eachnode has eight children.

There is still one little problem left. When should we stop dividing cubes into smaller ones?There are two possible solutions. The first one is to stop when a cube has some size smaller thena fixed number. The second one is more common. You might have noticed that every child has lesspolygons then it's parent. The trick is to stop subdividing when the number of polygons in a cubeis smaller then some fixed number.

Creating The Octree


Trees are recursion, recursion is trees. It is as simple as that.If you have a correct definition of you cubeNode it is very easy to create an octreerecursively. First of all you check all polygons against the boundarys of the cube. Thisis very simple cause these boundaries are all axis aligned. This means that the cube hassix plane equations, which are:

1. X = Q.X

2. Y = Q.Y

3. Z = Q.Z

4. X = Q.X + cubeSize

5. Y = Q.Y + cubeSize

6. Z = Q.Z + cubeSize

Where Q is the position of one corner of the cube. This are very easy equations and the allparent polygons can very easily be checked against them.

It could occur that a polygon crosses a cube boundary. Again two possible solution are athand. First of all we could clip the polygon against the cube, which is simple, because ofthe axis aligned boundarys. On the other hand we could put the polygon in all cubes it is in.This means that some cubes can contain the same polygons. In order to prevent us from drawingone poly more than one time we should have a flag on each polygon which will be set if the polyis drawn for the first time.

The implementation of an octree is very straight forward. I haven't done it myself yet,but I will soon. It is all matter of recursion. In order to construct a tree, the first thingyou should think of is recursion. Whether we are talking about binary trees, quad trees oroctrees, it doesnt matter, just build the darn thing recursively. So have a class definitionof one cubeNode and put the creation of the tree in it's constructor. In this constructor youwill call the constructor itself for smaller cubes.

The Purpose Of The Octree


An octree is actually nothing more then a data structure. Itcan be used for very different things. It is not only handy for visibility detection butalso for collision detection, realtime shadows and many more things. The most importantthing to understand about octrees is that if a parent is not important then it's childrenaren't either. Let's makes this a little bit more clear with an example.

We will do this in 2d, which therefore resembles a quadtree, but with some imaginationit can very easily be extended to 3d. Here we test the cubes (squares) against the viewingfrustrum. Take a look at the next picture:



Fig 4. An octree from the top and a viewing frustrum.

In this picture a colored square that has one color was dumped without checking it’s children. Asyou can see some squares had to be checked all the way to the final node, but some large squarescould be dumped at once. The colored squares are the ones that are outside the viewing frustrum andthe greyscale ones are the one inside the viewing frustrum. As you can see this is actually a worstcase scenario because the viewing frustrum crosses the middle of the surrounding square and thereforeall the four children have to be checked.

You could also apply the octree to many other things. Collision detection for example. Just checkin which cube your bounding sphere or box is and you will only have to check against those faces.There are many more examples.

Conclusion


There is already a lot written about octrees. I tried togive my view on them in the hope somebody might read this. As you can see octrees areway easier to implement than BSP-trees (although some disagree) while offering a betterstructure. The main point about an octrees is that when a parent is discarded so areit's children. Actually that is all there is to it.

Code clean, play Goldeneye and go vegetarian.

Jaap Suter a.k.a .........
原文链接http://www.flipcode.com/archives/Introduction_To_Octrees.shtml