Game Development Reference
In-Depth Information
i
r
(
w
c
where
i
,
j
=
1
3
) is the element of
R
at the
i
th
row and
j
th
column, and
[
]
()
T
T
t
=
t
t
t
=
−
R
w
c
t
w
c
.
x
y
z
Then, substituting equation 40 into equation 41 yields (assuming
t
y
≠
0):
[
]
w
w
w
w
w
w
y
ˆ
ˆ
y
y
ˆ
y
ˆ
−
x
ˆ
−
x
ˆ
−
x
ˆ
⋅
a
−
x
ˆ
=
0
where
a
contains the seven unknowns:
[
]
T
1
2
3
4
5
6
7
a
=
a
a
a
a
a
a
a
T
s
s
s
s
.
y
y
y
y
=
t
−
1
r
1
1
t
−
1
r
2
t
−
1
r
3
t
−
1
t
t
−
1
r
1
2
t
−
1
r
2
2
t
−
1
r
3
2
y
y
1
y
1
y
x
y
y
y
s
s
s
s
x
x
x
x
With more than seven 3-D world points at general positions, one can estimate the
seven unknowns from an over-determined linear system (by stacking multiple
equation 42). After that,
R
w
c
,
t
x
, and
t
y
can be calculated as:
s
s
() () ()
() () ()
2
2
2
y
x
4
1
2
3
t
=
a
t
2
2
2
=
t
⋅
a
+
a
+
a
5
6
7
t
y
=
a
+
a
+
a
,
,
,
x
y
y
s
s
y
x
[
]
s
[
]
T
T
[
]
[
]
1
1
2
3
1
1
2
3
r
r
r
=
x
t
a
a
a
T
T
,
1
2
2
2
3
2
5
6
7
,
r
r
r
=
t
a
a
a
1
y
s
y
y
[
]
[
]
[
]
T
T
T
1
3
2
3
3
1
1
2
3
1
2
2
2
3
r
r
r
=
r
r
r
×
r
r
r
.
3
1
1
2
The sign of
t
y
can be determined by requiring that
ˆ and
x
(respectively ˆ and
c
y
) have opposite signs.
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