December 2013
Intermediate to advanced
508 pages
16h 50m
English
Content preview from Practical Algorithms for 3D Computer Graphics, 2nd Edition
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2.9. Reflection in a plane 21
L
ine
1
Line 2
P
2
P
1
x
b
A
A´
d
r
x
y
z
O
^
^
Figure 2.7: Closest distance of approach between two lines.
Once λ and µ are determined the length of b is readily found:
l =
P
1
+ λ
ˆ
r − (P
2
+ µ
ˆ
d)
(2.6)
Taking the dot product of 2.5 with
ˆ
r will eliminate b and give an express ion for
λ:
λ = µ(
ˆ
d ·
ˆ
r) − (P
1
− P
2
) ·
ˆ
r (2.7)
The dot product of 2.5 with
ˆ
d eliminates b and substituting λ us in g 2.7 gives:
µ =
((P
1
− P
2
) ·
ˆ
d) − ((P
1
− P
2
) ·
ˆ
r)(
ˆ
d ·
ˆ
r)
1 − (
ˆ
d ·
ˆ
r)
2
(2.8)
With µ known 2.7 gives λ and then the distance of closest approach follows
from 2.6.
Note: If the lines are parallel
(
ˆ
d·
ˆ
r)
< ǫ (where ǫ is the machine tolerance of
zero, approximately ...
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