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How do Tangents and UVs relate?

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Hello, I wish to understand how to compute tangents for a given mesh to render normal mapped objects, however, in most references to tangent computation, there is math that involves grabbing texture coordinate information. I believe that I understand how to calculate a tangent but I do not understand why some calculations involve UV's in their calculations. Perhaps someone can clear this up for me.

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You can't calculate tangents and binormals without UV information, since the tangent points in the direction of increasing U and the binormal points in the direction of increasing V.

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