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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