normal transform

conclusion:

if T is orthogonal

n'  = Tn

else

       n' = inverse(transpose(T))n

Proof:

 normal n, arbitray point p1, p2  on plane.

 dot(n, (p1- p2)) = 0          ==>             transpose(n)(p1 - p2) = 0  ①     (matrix version of dot)

so transpose(n')(p1' - p2') = 0

==>  transpose(n')(Tp1  - Tp2) = 0

==> transpose(n')T(p1 - p2) = 0  ②

① ②  ==>  transpose(n) =  transpose(n')T

           ==>  n = transpose(T)n'

           ==>  n' = inverse(transpose(T))n