why normal transform use inverse-transpose matrix

Contents

1. 1. 证明

定义$T$为切线,$N$为法线,$M$为切线的变换矩阵,$Q$为法线的变换矩阵

证明

$$ T\cdot N^T = 0\\\\[2ex] T\cdot M = T'\\\\[2ex] N\cdot Q = N'\\\\[2ex] T'\cdot (N')^T = 0\\\\[2ex] T\cdot M \cdot (N\cdot Q)^T = 0\\\\[2ex] T\cdot M \cdot Q^T \cdot N^T = 0\\\\[2ex] \therefore M\cdot Q^T = I\\\\[2ex] \therefore Q = (M^{-1})^T $$