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八叉树入门 出 处:  中国游戏开发者
2003-06-08 ] 作 者:Jaap Suter

  目 录
  1 引言
  2 八叉树的结构
  3 创建八叉树
  4 八叉树的用途
  5 结论

   原 文Introduction To Octrees
   译 者Jerrywang
   版 本the first editionVer 1.0

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

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

  Z buffer允许动态世界以及即使是交叉的面,但是得忍受大量的重画。另一方面,BSP树如果能够被很好的实现,可以避免重画,但是它需要大量的预处理计算而且不适合动态世界。
   The Z buffer for example allows dynamical worlds and even crossing faces, but it suffers 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 a definite nono.

  直到最近我第一次听说了八叉树( Octree),我必须承认我被它的简易性“狠狠地打了一下儿”。我从来没有真正地去实现这个结构,因此我不会去实现一些代码。这个解释只是想让从没有听说过八叉树的人感到更清晰易懂。如果你真的了解了这个结构( Octree),实现它只是一个小意思。
   It wasn't until recently i first heard of the octree, and I must admit i was struck by it's simplicity. I never actually implemented this structure and therefore I will present no pseudo code at all. This explanation is merely an attempt to make it more clear to people who have never heard of it. Besides, if you really understand the structure, then implementation is a piece of cake.

2、八叉树的结构The Octree Structure
  我们的实际‘世界’或者说是关卡只不过是一些多边形。你们或许在其中加入了一些曲面( Curves)和Voxels(地形实现方法的一种),但是大多数也都是多边形。
   Our virtual world or level is actually nothing more then a soup of polygons. Some of you people might have thrown in a couple of curves and voxels but most of 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. Now imagine 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 subdivide each children into eight smaller cubes, and you will notice we are creating a tree where each node 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 then a fixed number. The second one is more common. You might have noticed that every child has less polygons then it's parent. The trick is to stop subdividing when the number of polygons in a cube is smaller then some fixed number.

3、创建八叉树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 octree recursively. First of all you check all polygons against the boundarys of the cube. This is very simple cause these boundaries are all axis aligned. This means that the cube has six 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
   WhereQ is the position of one corner of the cube. This are very easy equations and the all parent polygons can very easily be checked against them.

   It could occur that a polygon crosses a cube boundary. Again two possible solution are at hand. First of all we could clip the polygon against the cube, which is simple, because of the 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 drawing one poly more than one time we should have a flag on each polygon which will be set if the poly is 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 thing you should think of is recursion. Whether we are talking about binary trees, quad trees or octrees, it doesnt matter, just build the darn thing recursively. So have a class definition of one cubeNode and put the creation of the tree in it's constructor. In this constructor you will call the constructor itself for smaller cubes.

4、八叉树的用途The Purpose Of The Octree
   An octree is actually nothing more then a data structure. It can be used for very different things. It is not only handy for visibility detection but also for collision detection, realtime shadows and many more things. The most important thing to understand about octrees is that if a parent is not important then it's children aren't either. Let's makes this a little bit more clear with an example.

  我们用2D来表示它,这样就很像一个四叉树,但是想象一下它也可以很容易的扩展到3D上。在这里我们沿着视锥( viewing frustrum)检测立方体(正方形)。
   We will do this in 2d, which therefore resembles a quadtree, but with some imagination it can very easily be extended to 3d. Here we test the cubes (squares) against the viewing frustrum. 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抯 children. As you can see some squares had to be checked all the way to the final node, but some large squares could be dumped at once. The colored squares are the ones that are outside the viewing frustrum and the greyscale ones are the one inside the viewing frustrum. As you can see this is actually a worst case scenario because the viewing frustrum crosses the middle of the surrounding square and therefore all the four children have to be checked.

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

   There is already a lot written about octrees. I tried to give my view on them in the hope somebody might read this. As you can see octrees are way easier to implement than BSP-trees (although some disagree) while offering a better structure. The main point about an octrees is that when a parent is discarded so are it's children. Actually that is all there is to it.
  Code clean, play Goldeneye and go vegetarian. Jaap Suter a.k.a .........