Quadtrees Question submitted by (15 March 2000) | Return to The Archives

| I'm just getting into 3D graphics as a whole and I thought a good place to start would be a world represented by a quad tree. I've read all the info I can find on the Internet and set up the tree itself. The problem is that I can't figure out a speed efficient routine that determines whether a particular square lies (either completely or partially) inside the view triangle. All I've come up with is a whole load of point-in-polygon tests and a mess of if statements and still it doesn't work properly.
Am I missing something obvious?

| Your question, in itself, is a bit confusing. I'm going to assume a few things. First, that when you say "world" you're referring to a landscape or a terrain, rather than an environment that has a lot of detail along the Y (height) axis. Second, that when you wrote, "view triangle" you're referring to the view frustum, as projected onto the ground.
As a side-note, I don't mean to insult you with these clarifications. Anybody who has ever learned anything would know that, until you've learned something, you're not always aware of all the variables involved in solving a problem. Translated, this means that when asking questions about something you're in the process of learning, you're not necessarily able to ask complete questions (i.e. with all the variables listed.) That's my job, figuring out what you're trying to accomplish, then helping you get there. Anyway, on to the solution!
Quadtrees are a great tool for terrain engines. Provided that your world is a terrain, you're certainly headed in the right direction. Aside from the indirect advantages, quadtrees provide two important things directly: a way to store your data, and a hierarchy. It appears as though you've got your data into the quadtree, now you just need to know how to take advantage of the hierarchy for performance.
As far as I can tell, it's not possible to determine if a square (node) and a [view] triangle intersect with just point-in-polygon tests. Consider the following example:
Figure 1: A triangle intersects a square, with none of the vertices of either inside the other
You'll notice that even if you were clever and tried a few extra points (the center point of each) in this example, you would still find that the center point of the square is outside the triangle, and the center of the triangle is outside the square. How do you solve the problem? Clipping.
Clipping may sound like a lot of (slow) work, but hear me out. You need accurate results and clipping can provide those results. Also, understand that you are always clipping to the same three lines (your triangle). This means that you can pre-calculate a few things to speed up your clipper by at least 25%. And finally slowness incurred by clipping is far outweighed by the speed benefits you get from the hierarchy of the quad-tree. Here's how:
Don't think of it as clipping a 'node' to a triangle, think of it as clipping a 'tree' to the triangle. Consider writing a routine that traverses your tree, with the inputs: (1) a list of pointers and (2) a triangle. As you traverse your tree, you want to determine if the current node (to start with, the root node) intersects the triangle completely, partially, or not at all.
If a node intersects completely, then add the node to the list, and do not recurse any deeper into this branch of the tree. The rule here is that if a node is completely inside the triangle, then so will its children.
If the node doesn't intersect at all, then you don't add the node to the list, and you also don't traverse any deeper along this branch.
If a node intersects partially, you need to do one of two things. You either need to traverse deeper, or add the node to the list. If this is a leaf node (i.e. no children) then you want to add it to the list. If this is a node (i.e. it has children) then you don't need to add this node to the list, since it is the children you're after, so just traverse deeper along this branch without adding to the list. The intersecting children will eventually end up in the list. This assumes that your nodes contain no data, and your leaves contain no children.
When your traversal is complete, you've got a list of the nodes and/or leaves that intersect the triangle. This means that to get all the data, you may have to traverse into some of the nodes in the resulting list (the ones that aren't leaves.)
Notice, that of those three tests, two of them are quick-outs ("quick-exits" or "early terminators"). By this, I mean that two of the three tests will result in no more traversing and no more clipping. Only the partial-intersection test results in more clipping. And, this doesn't mean that a partial intersection will result in constant clipping as the tree is traversed deeper. It just means that there are potentially other nodes deeper along that branch that might or might not intersect. Very possibly, there will be more quick-outs along that branch of the tree, reducing the tree to fewer and fewer nodes of a "known state". This is the beauty of the hierarchy.
And speaking of quick-outs, your clipper itself can have its own set of quick-outs. Of course, this is another topic altogether.
If you're comfortable with your understanding of what I've described here, and with your ability to code it, you should have all the key elements to make a top-notch hierarchical frustum culling tool for a terrain system.
Response provided by Paul Nettle

This article was originally an entry in flipCode's Fountain of Knowledge, an open Question and Answer column that no longer exists.

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