Game Development Reference
In-Depth Information
2. 4x + 7y = 42
7y = −4x + 42
y = −(4/7)x + 6
The slope is −4/7 and the y-intercept is 6.
3.
(a) p
min
= (−5,−7,−5), p
max
= (7, 11, 8)
(b) (x
min
, y
min
, z
min
) = (−5,−7,−5)
(x
min
, y
min
, z
max
) = (−5,−7, 8)
(x
min
, y
max
, z
min
) = (−5, 11,−5)
(x
min
, y
max
, z
max
) = (−5, 11, 8)
(x
max
, y
min
, z
min
) = (7,−7,−5)
(x
max
, y
min
, z
max
) = (7,−7, 8)
(x
max
, y
max
, z
min
) = (7, 11,−5)
(x
max
, y
max
, z
max
) = (7, 11, 8)
(c) c = (p
min
+ p
max
)/2 = (1, 2, 1.5)
s = (p
max
−p
min
) = (12, 18, 13)
(d) v
′
v
′
1
= (−2.828, 12.728,−5.000)
2
= (−0.707, 3.5355, 8.000)
v
′
v
′
3
= (−4.243, 0.000, 1.000)
4
= (1.414,−8.485, 0.000)
v
′
5
= (2.121, 6.364, 4.000)
(e) p
min
= (−4.243,−8.485,−5), p
max
= (2.121, 12.728, 8)
(f) First, we determine which products to take by using the technique from Listing 9.4:
x
′
x
′
min
= m
11
x
min
max
= m
11
x
max
(m
11
> 0)
+ m
21
y
max
+ m
21
y
min
(m
21
< 0)
+ 0
+ 0
(m
31
= 0)
y
′
y
′
min
= m
12
x
min
max
= m
12
x
max
(m
12
> 0)
+ m
22
y
min
+ m
22
y
max
(m
22
> 0)
+ 0
+ 0
(m
32
= 0)
z
′
z
′
min
= 0
max
= 0
(m
13
= 0)
+ 0
+ 0
(m
23
= 0)
+ z
min
+ z
max
(m
33
= 1)
Summing the appropriate products, we have
x
′
min
= m
11
x
min
+ m
21
y
max
+ 0 = 0.707−5 + (−0.707)11 + 0 = −11.312,
y
′
min
= m
12
x
min
+ m
22
y
min
+ 0 = 0.707−5 + 0.707−7 + 0 = −8.484,
z
′
min
= z
min
= −5,
x
′
max
= m
11
x
max
+ m
21
y
min
+ 0 = 0.7077 + (−0.707)−7 + 0 = 9.898,
y
′
max
= m
12
x
max
+ m
22
y
max
+ 0 = 0.7077 + 0.70711 + 0 = 12.726,
z
′
max
= z
max
= 8.
Notice how much larger this box is than the one of the transformed points!
Search WWH ::
Custom Search