Background Information and Techniques - Computer Animation: Algorithms and Techniques - page 499

2
4

3
5

2
4

3
5

0
0

W

W 0
W 3
W 6
W 9
W 10
W 11

0
3

W

1
kW 0 6 k

0
6

W

W ¼ ¼ (B.180)

0 9

W

0 10
1

W

c 0 ¼ W 0 W 6
r 0 ¼ W 1 W 6
f u ¼kW 0 c 0 W 6 k
t 0 ¼ðw 9 c 0 Þ=f u
t 1 ¼ðw 10 r 0 Þ=f v
t 2 ¼ w 11
R 0 ¼ðW 0 c 0 W 6 Þ=f u
R 1 ¼ðW 3 r 0 W 6 Þ=f v
R 2 ¼ W 6

(B.181)

0
0 c 0 W

0
6 Þ=f u

R 0 ¼ðW

0
3 r 0 W

0
6 Þ=f v

R 1 ¼ðW (B.182)

0
6

R 2 ¼ W

c 0 ¼ W 0 R 2
r 0 ¼ W 1 R 2
f u ¼kW 0 c 0 R 2 k
f v ¼kW 3 r 0 R 2 k
t 0 ¼ðW 9 c 0 Þ=f u
t 1 ¼ðW 10 r 0 Þ=f v
t 2 ¼ w 11

(B.183)

Given an approximation to a rotation matrix, R, the objective is to find the closest valid rotation
matrix to the given matrix ( Eq. B.184 ) . This is of the form shown in Equation B.185 , where the matrices
C and D are notated as shown in Equation B.186 . To solve this, define a matrix B as in Equation B.187 .
If q ¼ ( q 0 , q 1 , q 2 , q 3 ) T is the eigenvector of B associated with the smallest eigenvalue, R is defined by

Equation B.188 [ 26 ].

kR Rk min (B.184)

kR C Dk (B.185)

C ¼½C 1 ; C 2 ; C 3
D ¼½D 1 ; D 2 ; D 3

(B.186)