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bump mapping theory question

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I was reading up on bump mapping and was wondering why you needed to construct the two temp vectors (A and B) from the cross products of the surface normal and the Pu and Pv surface partial derivatives. As I understand it you create these two vectors in order to create a tangent plane with respect to the surface at point P. Then you take these two vectors and multiply them respectively by the s and t partial derivatives from the bump map to find the offset vector D for N. Since the surface normal was created via the cross product of the partial derivatives in the first place, couldn't you save time and multiply the P/u, P/v partial derivative vectors directly with the s, t partial derivative vectors from the texture map to find D? A = N X P , B = N X P , and D = B A - B B v u s t

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