Game Development Reference
In-Depth Information
9. q 1 q 2 =
w 1
x 1
y 1
z 1
w 2
x 2
y 2
z 2
2
4 w 1 w 2 −x 1 x 2 −y 1 y 2 −z 1 z 2
3
5
0
@
1
A
w 1 x 2 + x 1 w 2 + y 1 z 2 −z 1 y 2
w 1 y 2 + y 1 w 2 + z 1 x 2 −x 1 z 2
w 1 z 2 + z 1 w 2 + x 1 y 2 −y 1 x 2
=
(w 1 w 2 −x 1 x 2 −y 1 y 2 −z 1 z 2 ) 2
+ (w 1 x 2 + x 1 w 2 + y 1 z 2 −z 1 y 2 ) 2
+ (w 1 y 2 + y 1 w 2 + z 1 x 2 −x 1 z 2 ) 2
+ (w 1 z 2 + z 1 w 2 + x 1 y 2 −y 1 x 2 ) 2
=
After expanding these products and then canceling terms (a step that we have omitted
because it is very messy), we then factor:
w 1 2 w 2 2 + x 1 2 x 2 2 + y 1 2 y 2 2 + z 1 2 z 2 2
+ w 1 2 x 2 2 + x 1 2 w 2 2 + y 1 2 z 2 2 + z 1 2 y 2 2
+ w 1 2 y 2 2 + y 1 2 w 2 2 + z 1 2 x 2 2 + x 1 2 z 2 2
+ w 1 2 z 2 2 + z 1 2 w 2 2 + x 1 2 y 2 2 + y 1 2 x 2 2
q 1 q 2 =
w 1 2 (w 2 2 + x 2 2 + y 2 2 + z 2 2 )
+ x 1 2 (w 2 2 + x 2 2 + y 2 2 + z 2 2 )
+ y 1 2 (w 2 2 + x 2 2 + y 2 2 + z 2 2 )
+ z 1 2 (w 2 2 + x 2 2 + y 2 2 + z 2 2 )
=
=
(w 1 2 + x 1 2 + y 1 2 + z 1 2 )(w 2 2 + x 2 2 + y 2 2 + z 2 2 )
q 1 2 q 2 2
=
= q 1 q 2
B.9
Chapter 9
(Page 339.)
1. First, we convert the ray to implicit form by using Equation (9.5):
a = d y = 5,
b = −d x = 7,
d = x org d y −y org d x = 55−3(−7) = 46.
Then, we convert this to slope-intercept form according to Equation (9.6):
m = −a/b = −5/7,
y 0 = d/b = 46/7.
So the equation of the line is y = −(5/7)x + 46/7.
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