rtr

$$begin{array}{l|lll}
OpenGL & right-handed & column-major & CBA vec v newline
DirectX & left-handed & row-major & vec v ABC
end{array}$$

OpenGL : column-major VS DirectX : row-major

$$
left[
begin{array}{cccc}
1 & 0 & 0 & t_x newline
0 & 1 & 0 & t_y newline
0 & 0 & 1 & t_z newline
0 & 0 & 0 & 1
end{array}
right]
*
left[
begin{array}{c}
x newline
y newline
z newline
1
end{array}
right]
or
left[
begin{array}{cccc}
x & y & z & 1
end{array}
right]
*
left[
begin{array}{cccc}
1 & 0 & 0 & 0 newline
0 & 1 & 0 & 0 newline
0 & 0 & 1 & 0 newline
t_x & t_y & t_z & 1
end{array}
right]
$$
inverse matrix 逆矩阵: $ T^{-1}(t) = T(-t)$

2 dimensions ( OpenGL ) :
$$begin{bmatrix}
costheta & -sintheta newline
sintheta & costheta newline
end{bmatrix}$$

3 dimensions ( OpenGL ) :

$$R_x(theta) =
begin{bmatrix}
1 & 0 & 0 & 0 newline
0 & costheta & -sintheta & 0 newline
0 & sintheta & costheta & 0 newline
0 & 0 & 0 & 1
end{bmatrix}$$

$$R_y(theta) =
begin{bmatrix}
costheta & 0 & sintheta & 0 newline
0 & 1 & 0 & 0 newline
-sintheta & 0 & costheta & 0 newline
0 & 0 & 0 & 1
end{bmatrix}$$

$$R_z(theta) =
begin{bmatrix}
costheta & -sintheta & 0 & 0 newline
sintheta & costheta & 0 & 0 newline
0 & 0 & 1 & 0 newline
0 & 0 & 0 & 1
end{bmatrix}$$

逆矩阵: $ R_i^{-1}(theta) = R_i(-theta)$

for 3x3 rotation mateix, the trace ( the sum of the diagonal elements in a matrix ) is constant :
$ tr(R) = 1+2costheta $

$$begin{bmatrix}
S_x & 0 & 0 & 0 newline
0 & S_y & 0 & 0 newline
0 & 0 & S_z & 0 newline
0 & 0 & 0 & 1
end{bmatrix}$$

逆矩阵: $ S^{-1}(s) = S(frac{1}{S_x},frac{1}{S_y},frac{1}{S_z}) $

• 如果有两项negative = > rotate $ pi $ radians
• 如果有一项或三项negative => reflection matrix
  可能导致incorrect lighting或backface culling，需要先计算行列式determinant是否$<0$
  $$
  begin{array}{|lll|}
  a_1 & b1 & c1 newline
  a_2 & b2 & c2 newline
  a_3 & b3 & c3 newline
  end{array}
  = a_1b_2c_3 + b_1c_2a_3 + c_1a_2b_3 - a_3b_2c_1 - b_3c_2a_1 - c_3a_2b_1
  $$

TRS is the order commonly used( OpenGL ),so S is applied first

其余等用到再看