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swept solids

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I am not sure exactly what I am looking for, but I imagine it as a swept solid. I have been searching for some time to find out how to take for example a box or cylinder at any orientation, move it through space in any direction, and have the simplest polygon result set of the area it 'swept'. I have seen some voxel oriented things, but I would prefer more accurate polygons. I also ran across something called 'swept differential equations - SDE' but I never saw it explained simple enough I could write code from. I do see some post regarding collision detection, but that is not quite what I am looking for. I am currently using samples to get the volume, but that produces too many point depending on the detail rate. So anyhow, any advice on what I should be searching for (what this is really called) would be helpful. Thanks.

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