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Billboarding example from "OpenGL Game Programming" Book

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Ok thanks to Sneftel , I finally understood the use of modelview matrix . But could someone explain to me how the Billboarding example from "OpenGL Game Programming" book works because I have problems in understanding it? I can''t understand why we transpose the matrix and why we don''t use directly the up and right vectors of the modelview matrix(which I assume are the fisrt two columns of the 3x3 submatrix). Thanks

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