3D Game Engine Design

Chapter 2  basic mathematics

1. matrix

　 1)base

      symmetric matrix: M = M';

      skew-symmetric matrix: M=-M'

    2)scaling

　    scaling matrix: diagonal matrix D = diag{d1,...,dn} for all di > 0

    3)rotation

       rotation maxtrix: R if RR' = I

       then R has a corresponding unit length axis of rotation U and angle of rotation θ

       if U=(u0,u1,u2), define the skew-symmetric matrix S = []

       also, the rotation corresponding to axis U and the angle θ is R = I + S sinθ + S² (1-cosθ)

    4)translation

       translation of vectors by a fixed vector T:  Y=X+T

    5)homogeneous transformations

       vector map:   (x,y,z) -> (x,y,z,1)

                           (x,y,z,w) -> (x/w,y/w,z/w,1) when set w=1

      transformations can be applied to homogeneous coordinates to other homogeneous coordinates