Many things in nature are governed by one simple goal: Take as much space as economically possible. Before cities were intelligently designed, they would just grow following this principle. Streets wandered around the terrain forming a network that covered all available space. Similar networks are found in living things like blood vessels, nervous systems.
Trees develop in the same way. Their growth is determined by how much sunlight they can get. The better they expand in space, the bigger they become. You can argue that once you encounter an adult tree, it is there thanks to its successful colonization of the space around it. So somehow this principle is built into the tree.
But hold this thought for a moment. There is another approach to modeling things that grow. If you are into procedural things you surely heard of it: L-Systems. At their heart, they are just a way to describe how one thing can become a series of things over time.
For instance, a L-System may have a single rule that says a dot will become a dot and a line. If you start with one dot:
After one unit of time has passed, this dot is replaced by a dot and a line:
After another unit of time:
If we let this thing run for ten iterations, we will end up with nine lines and one dot. Even this very simple replacement rule has the ability to grow over time.
You could easily have a rule that produces branching, pretty much like a tree. If you say a stick will be replaced by three sticks:
It is not hard to imagine that with richer rules and better end elements that just lines and dots, you could grow something that is very close to a real-life tree. This is how many commercial-grade products generate vegetation, and they are very good at it.
The problem with L-systems is that it is their nature to blindly replace things. If you want them to become aware of external factors, like sunlight, presence of other objects or even be aware of themselves so a branch will not intersect other branches, you need to start tweaking them. Then they stop being so fun.
An algorithm that will use L-Systems to create a realistic looking tree is not trivial. While the basic branching idea of the tree is easily conveyed by the L-System, the system is not aware of the main forces that make a tree look like a tree.
Some folks at the University of Calgary saw this. They asked, what if you do it the opposite way. Instead of growing the tree from scratch and making sure it will grow the way you want, what if we start from the space we know the tree is going to take and just fill that volume.
The problem becomes about space colonization. This can be solved by an algorithm that is much simpler than extended L-Systems. You can see their paper here, but I will describe briefly how it works.
It starts by defining the volume the crown of the tree will take. The simplest volume is a sphere, just a point and a radius. The volume is then filled with random points. You can think of these points like targets the colonization algorithm will try to reach.
Then we add one segment at the base of the tree. From this point the tree will grow.
A segment has two ends and some length. Soon you will find that the average segment length will be in part responsible of the overall appearance of the tree. Smaller segments will result in curvaceous and intricate trees while larger segments will make for straight trunks and branches.
The two ends of the segment are of great importance too. For each segment end the algorithm will compute an attraction vector towards the cloud of target points. If there are target points close enough to the segment end, a new segment is added. The new segment will follow the same direction as the attraction vector at that point.
Whenever a segment end is too close to a target point, the target point is removed. As new segments are added in the direction of the target points, they end up eating all the points. Once there are no more points left, or they are too few of them, the algorithm is finished.
The results are very realistic. Branches naturally avoid each other, each one appears to have developed as the result of seeking sunlight. The same method can be used to create roots. Roots also expand in some form of space colonization.
The following image shows a tree generated with this technique. The ground is removed so the roots can be seen.
How many different trees can be achieved with this technique? Well there are many factors you can play with, like the size of the segments, the attractor vector cut-off zone and the distance where segments remove target points. On top of that you can introduce space warps that will mess up with attraction vectors. This can be used to simulate gravity for some heavy branches.
As you can see, the algorithm is pretty simple, still the results are quite good. I think this beats L-Systems for large trees. Next, I plan to use it for generating the chaotic layouts of old cities. When I get there I will surely post about that.
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