Game Development Reference
In-Depth Information
A3
×
3matrix
abc
def
ghi
M
=
has its inverse given by
ei
fh
ch
bi bf
ce
1
det M
M 1
=
fg
di
ai
cg cd
af
dh
eg bg
ah ae
bd
where det M is the determinant of the matrix:
det M
=
aei
+
dhc
+
gbf
ahf
gec
dbi
A quaternion
w
x
y
z
θ
ˆ
=
represents the same rotation as the matrix
( 2 y 2
2 z 2 )
1
+
2 xy
+
2 zw
2 xz
2 yw
( 2 x 2
2 z 2 )
Θ
=
2 xy
2 zw
1
+
2 yz
+
2 xw
( 2 x 2
2 y 2 )
2 xz
+
2 yw
2 yz
2 xw
1
+
A transformation matrix M t can be changed into a new coordinate system, using
a transform M b to the new coordinate system according to
M t =
M b M t M 1
b
D.4
I NTEGRATION
To update an object's position
1
2 ¨
p =
pt 2
p
+ ˙
pt
+
is normally replaced by the less accurate
p =
+ ˙
p
pt
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