正交矩阵
正交矩阵不用计算逆矩阵，计算转置矩阵就是它的逆矩阵
MM^T = I
因为MM^-1 = I
所以M^T = M^-1
检查正交矩阵
MM^T = I
M = [
m11 m22 m33
m21 m22 m23
m31 m32 m33
]
// 将每一行用向量表示
r1 = [m11,m12,m13]
r2 = [m21,m22,m23]
r3 = [m31,m32,m33]
M = [
r1,
r2,
r3
]
M^T = [r1,r2,r3]
// 检查正交矩阵
r1 * r1 = 1
r1 * r2 = 0
r1 * r3 = 0
r2 * r1 = 0
r2 * r2 = 1
r2 & r3 = 0
r3 * r1 = 0
r3 * r2 = 0
r3 * r3 = 1
矩阵正交化
修正矩阵计算误差，正交化修正
// 规范化
r1' = r1
r2' = r2 - (r1'r2)
r3' = r3 - (r1'r3)r1' - (r2'r3)r2'
r1' = r1-k(r1*r2)r2-k(r1*r3)r3
r2' = r2-k(r1*r2)r1-k(r2*r3)r3
r3' = r3-k(r1*r3)r1-k(r2*r3)r2
齐次矩阵
利用齐次矩阵将线性变换矩阵与平移矩阵结合到一个矩阵中
[
m11 m12 m13
m21 m22 m23
m31 m32 m33
]
[
m11 m12 m13 0
m21 m22 m23 0
m31 m32 m33 0
0 0 0 1
]
[x y z 1] * [
m11 m12 m13 0
m21 m22 m23 0
m31 m32 m33 0
0 0 0 1
]
= [
xm11+y+m21+zm31
xm12+y+m22+zm32
xm13+y+m23+zm33
1
]^T
平移矩阵
[x y z 1] * [
1 0 0 0
0 1 0 0
0 0 1 0
deltax deltay deltaz 1
]
[x+deltax y+deltay z+deltaz 1]
// 线性变换 + 平移
R = [
r11 r12 r13 0
r21 r22 r23 0
r31 r32 r33 0
0 0. 0. 1
]
T = [
1 0 0 0
0 1 0 0
0 0 1 0
deltax deltay deltaz 1
]
M = RT = [
r11 r12 r13 0
r21 r22 r23 0
r31 r32 r33 0
deltax deltay deltaz 1
]