# Randomly dividing a 2d complex enclosed region

First I will define:

• Region: big stuff manually created I want to divide.

• Zone: small stuff I want to generate.

I have a map. The world map in fact. And I want to divide it into small zones. The size of the zones will be dependent on what region the zone is in. For instance very small for Europe (maybe Europe will have like 200 zones) but only a couple of huge ones for the Atlantic Ocean.

I can manually create points to enclose a region. I will create regions for each big space I want it to have different size than other spaces. For instance I will create an enclosed region for Europe. So I got a butch of (latitude, longitude) points defining the limits of the Europe region. The shape is of course not regular and there are holes in the middle of it (I don't want to create small zones over the Mediterranean sea but a big one). So what we got is a huge 2D shape to be filled up with zones.

Zones themselves are n-sized polygons, number of sizes can be randomly chosen or subject to other constraints. The area of each zone is also limited random (like 50 plus/minus 40%) although this constraint again can be relaxed (as exception, not as rule). Zones can not overlap and the whole region must be divided.

My obvious question is if someone has any idea of how this can be done. I even have a hard time to know if a given point is inside my enclosed region or not. In case this problem is not solved or not generally solved, I would like to know if you think it can be significant enough to get a publication about the solution of it.

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When you ask mathematicians a question, you really need to be much more precise with your definitions and your usage of terms. As it stands, I have no idea what exactly the problem is. It would help a lot if you gave us an example of an instance of your problem and an example of a solution to it. –  Qiaochu Yuan Jul 23 '10 at 7:26
Do you have any constraints on the shapes of the zones other than their area and the fact that they are polygons? In the absence of such constraints, this doesn't seem like a mathematically interesting question. I encourage votes to close. –  S. Carnahan Jul 23 '10 at 14:27