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problems with polygonal tessellation

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Alrighty, I can't figure out polygonal tessellation. I do not want to use GLU. The polygons I am trying to tessellate can be concave, but aren't always. They can NOT have holes, and they can't have over lapping segments. They are defined by a list of 2D vertecies (X, Y), which define the outline of the polygon in a counter-clockwise direction. I need to break each polygon (with any amount of vertecies, and any posible shape that follows the above rules) into triangles. If anyone knows of a good, inexspensive way of doing this, then I would appreciate you sharing the knowlage you have.

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