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I have been searching for an answer to this for a really long time and I have not found any definitive answers as of yet. What I am trying to do is determine if and when two bodies collided between the times t_{0} and t_{1}. Calculating this is much more straight forward if each body is only either translating or rotating (about center or any point), but when adding multiple transformations on top of each other, things become a bit more messy. I made this animation to illustrate the problem.
As you can see, at times t_{0} and t_{1} there are no collisions, but between there are multiple collisions occurring.
One way I thought of for tackling this problem was to reduce the size of the time intervals. So, between t_{0} and t_{1} there would be a total of n updates to check for collisions. This works, but the only way I found that I could guarantee no false negatives, i.e. not finding a collision that happened, was to integrate over extremely small timesteps. As you can imagine, this is very costly because it resulted in the tens to hundreds of timesteps per body per update cycle. I am not saying this idea has no merit, but I would need to find a way to calculate the minimum number of timesteps rather than uniformly moving each body forward one unit of distance/rotation until they reach the desired position and orientation.
So, my question is two parts:
As you can see, at times t_{0} and t_{1} there are no collisions, but between there are multiple collisions occurring.
One way I thought of for tackling this problem was to reduce the size of the time intervals. So, between t_{0} and t_{1} there would be a total of n updates to check for collisions. This works, but the only way I found that I could guarantee no false negatives, i.e. not finding a collision that happened, was to integrate over extremely small timesteps. As you can imagine, this is very costly because it resulted in the tens to hundreds of timesteps per body per update cycle. I am not saying this idea has no merit, but I would need to find a way to calculate the minimum number of timesteps rather than uniformly moving each body forward one unit of distance/rotation until they reach the desired position and orientation.
So, my question is two parts:
 Is there a better way of determining if and when a collision occurs?
 If not, is it possible to calculate the minimum number of timesteps?
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