sequence - Mathematical technique to check intersection -

Imagine that the size of the hollow cube has a very large room in the fixed discrete posts of the room There are magic balls in the air. No magic ball is exactly above each other. If we take a imaginary horizontal plane of the infinite area and pass through the cube, how can we ensure that the plane does not cut any magic balls?

The height of a magic ball is given as a function of its position (X and Y). Distribution is such a way that some balls are at the same height, while others are at different altitudes to do this function - z = axy + bx + cy
where a, b, c positive The integer constants are position (x-axis and y-axis value) and height (z) are discrete values ​​(for simplicity, we can consider them positive integer).

If ball delivery function is Z = 10xy + 8x + 4y, then it is impossible to assume 15 or 21. So z = 15 or z = 21 will not cut any of the balls in a plane! In fact, in this case, any aircraft is not cut through balls with height (z = any strange number), it is worth noting that there are some aircraft with height that do not cut balls.

We do not want to find the height of all the magic balls and compare it to the height of the horizontal plane, as if trying all possible combinations, and the computer will take too long .

Our goal is to find a quick way through which we can tell whether any combination of a given value (height) can be prepared by (x, y) (position). If a given z can not be produced, an aircraft does not cut with any balls at that height! This question is also similar to the search whether a given number exists in the sequence produced by a function of two variables.

If u can give me some suggestions to solve this problem then it will be very helpful. Thank you. (I have already tried evolutionary computing like GA, PSO, DE, SA etc.) The method should be deterministic.

It seems that there are many balls in the room, the height of the room is z = A to z = B . What are you interested in is that if any height is on z, without the need for all the balls to do this, you have to start by assuming that A to B vacancy is empty and walk through the balls again, Mark the parts of this category as complete, unless it is completely complete or there is no ball. In the former case, no aircraft will be satisfied, but you will not be able to walk through all the balls again. In the latter case, you have a range of z in which there are no balls and you can use them to check more than one plane easily, although at the initial cost of walking through all the balls.