Graphics Reference
In-Depth Information
This symmetry was discovered by the Swiss mathematician Gabriel Cramer [1704-1752] and is
now known as Cramer's rule. However, it only works when the determinant in the denominator
is non-zero.
Similarly, for three simultaneous equations:
d
1
=
a
1
x
+
b
1
y
+
c
1
z
d
2
=
a
2
x
+
b
2
y
+
c
2
z
d
3
=
a
3
x
+
b
3
y
+
c
3
z
we can state
⎡
⎤
⎡
⎤
⎡
⎤
d
1
d
2
d
3
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
x
y
z
⎣
⎦
=
⎣
⎦
⎣
⎦
and
d
1
b
1
c
1
d
2
b
2
c
2
d
3
b
3
c
3
a
1
d
1
c
1
a
2
d
2
c
2
a
3
d
3
c
3
a
1
b
1
d
1
a
2
b
2
d
2
a
3
b
3
d
3
x
=
y
=
z
=
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
a
1
b
1
c
1
a
2
b
2
c
2
a
3
b
3
c
3
Before applying Cramer's rule, let's tidy up Eq. (6.28) as follows:
k
=
t
−
p
1
p
21
=
p
2
−
p
1
p
31
=
p
3
−
p
1
Therefore,
⎡
⎤
r
s
=
−
vp
21
p
31
⎣
⎦
k
(6.29)
and
x
k
x
21
x
31
y
k
y
21
y
31
z
k
z
21
z
31
−
x
v
x
k
x
31
−
−
x
v
x
21
x
k
y
v
y
k
y
31
−
−
y
v
y
21
y
k
z
v
z
k
z
31
−
z
v
z
21
z
k
=
=
=
r
s
(6.30)
DET
DET
DET
where
−
x
v
x
21
x
31
−
DET
=
y
v
y
21
y
31
−
z
v
z
21
z
31
In their topic
Real-Time Rendering
, Tomas Akenine-Möller and Eric Haines [Akenine-Möller,
2002] take the above solution one stage further by exploiting the fact that the determinant
abc
=−
acb
=−
a
×
c
·
b
=−
c
×
b
·
a
You may recall that we covered this in Section 2.9.1 and proved that the box product
