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Discretize 2d shapes

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Given a shape, how could you process it to make it a good starting point for copying those various elements at its points ?

At first it looked like something related to the Bresenham algorithm, but neighbor pieces have  to have one common edge.. The anti aliasing algorithm avoids such situation, so it won't work

I wonder what other mathematical problem this is related to.Could it be as simple as a snap (fuse), 

to a grid, then take care of the 45 degree edges ?




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