Game Development Reference
In-Depth Information
represents an orientation of θ about the axis:
x
y
z
Two quaternions can be multiplied together:
=
w 1
x 1
y 1
z 1
w 2
x 2
y 2
z 2
w 1 w 2
x 1 x 2
y 1 y 2
z 1 z 2
w 1 x 2 +
x 1 w 2
y 1 z 2
z 1 y 2
w 1 y 2
x 1 z 2
+
y 1 w 2
z 1 x 2
w 1 z 2 +
x 1 y 2
y 1 x 2 +
z 1 w 2
A quaternion representing orientation can be adjusted by a vector representing
amount of rotation according to
1
2 θ
θ
ˆ
=
θ
ˆ
+
θ
ˆ
ˆ
where the rotation is converted into a quaternion according to
[ θ x θ y θ z ]
[0 θ x θ y θ z ]
D.3
M ATRICES
An n
m matrix has n rows and m columns.
Matrices can be post-multiplied (we don't use pre-multiplication in this topic) by
vectors with the same number of elements as the matrix has columns:
×
=
abc
def
ghi
x
y
z
ax
+
by
+
cz
dx
+
ey
+
fz
gx
+
hy
+
iz
Matrices can be multiplied together, providing that the number of columns in the
first matrix is the same as the number of rows in the second:
C (i,j) =
A (i,k) B (k,j)
k
where C (i,j) is the entry in matrix C at the i th row and j th column; and where k ranges
up to the number of columns in the first matrix.