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How do complex collisions work?


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I tried to play around with some complex polygons and run into some issues. I first fooled with code&web physics editor and that work like supposed. Then, i took a look on JSON format and tried create similar one by hand. After some trial and error i got it right. It collided properly with world borders and body debug looked right. However, objects didn't collide properly with other objects and now I'm wondering why..

Also, one weird error said something like "Convex polys must be in conter-clockwise order" So can anyone open this mystery, how does convex polygons collision work behind the scenes?

I also noticed that shapes were divided into multiple smaller shapes in JSON. Is this simply to prettify JSON or is there actually a reason behind this?

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