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points "cut" in an aabb by a moving sphere

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Please, can someone help me with some hints on this problem:

I have a cloud of points with the same distance between them, align along the 3axes (it can be assimilated with an aabb) and a sphere that is moving toward it (linear moving can be assumed). I want to determine the points hit by the sphere.
I postpone this check until the sphere touch the aabb (I use a collision test based on Arvo's algorithm). My problem is that I don't want to use a brute force approach, that is, while the sphere verify the collision test check all the points of the cloud after each move and see if they are touched and mark them if they are. Is there a faster way to solve this problem?


P.S. After some reading I was thinking to try to divide that aabb in smaller aabb's but I need some help in finding algorithm for this aproach (if it is the right one).

[Edited by - iust79 on August 20, 2010 12:13:24 PM]

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