August 2014
Intermediate to advanced
469 pages
16h 30m
English
Content preview from GPGPU Programming for Games and Science
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292 GPGPU Programming for Games and Science
Secondly, Equations (6.57), (6.58), and (6.60) imply
SV = U
2
× V
= a
0
U
1
− a
1
U
0
=(U
0
· V)U
1
− (U
1
· V)U
0
= U
1
(U
T
0
V) − U
0
(U
T
1
V)
=(U
1
U
T
0
− U
0
U
T
1
)V
(6.64)
This equation is true for all vectors V,so
S = U
1
U
T
0
− U
0
U
T
1
(6.65)
Thirdly, Equations (6.57), (6.59), and (6.61) imply the relationship
S
2
V = U
2
× (U
2
× V)
= −a
0
U
0
− a
1
U
1
=(U
2
· V)U
2
− V
= U
2
(U
T
2
V) −V
=(U
2
U
T
2
− I)V
(6.66)
This equation is true for all vectors V,so
S
2
= U
2
U
T
2
− I (6.67)
Combining these relationships with Equation (6.56),
R
1
= γ(U
0
U
T
0
+ U
1
U
T
1
)+σ(U
1
U
T
0
− U
0
U
T
1
)+U
2
U
T
2
Equation (6.56)
= γ(I − U
2
U
T
2
)+σ(U
1
U
T
0
− U
0
U
T
1
)+U
2
U
T
2
by Equation (6.63)
= γ(I − U
2
U
T
2
)+σS + U
2
U
T
2
by Equation (6.65)
= I + σS +(1− γ)(U
2
U
T
2
− I)
= I +(sinθ)S +(1− cos θ)S
2
by Equation (6.67)
(6.68)
This equation provides ...