4.3. Matrices 157
z the upper 3 × 3 matrix U, which represents the rotation and/or scale,
z a 1 × 3 translation vector t,
z a 3 × 1 vector of zeros 0 = [ 0 0 0 ]T, and
z a scalar 1 in the bott om-right corner of the matrix.When a point is multiplied by a matrix that has been partitioned like this, the
result is as follows:
4.3.7.1. Translation
The following matrix translates a point by the vector t:
or in partitioned shorthand:
To invert a pure translation matrix, simply negate the vector t (i.e., negate tx ,
ty , and tz).
4.3.7.2. Rotation
All 4 × 4 pure rotation matrices have the form:
The t vector is zero and the upper 3 × 3 matrix R contains cosines and sines of
the rotation angle, measured in radians.
The following matrix represents rotation about the x-axis by an angle φ:
33 31
13 13
13[ 1] [ 1] [( ) 1].
1
××
××
×⎡⎤
′ = ⎢⎥=+
⎣⎦
U0
rr rU t
t1000
0 1 00
[ 1] 0010
1
[( )( )( )1],
xyzxyz
xxyyzzrrrt
rt rt rt⎡⎤
⎢⎥
+= ⎢⎥
⎢⎥
⎢⎥
⎣⎦
=+ + +
rttt[ 1] [( 1 ) 1].
⎡⎤
⎢⎥=+
⎣⎦
I0
rrt t[ 1] [ 1].
1
⎡⎤
⎢⎥=
⎣⎦
R0
rrR
01 0 00
0 cos sin 0
rotate ( , ) [ 1]. 0 sin cos 00 0 01x rrrxyz⎡⎤
⎢⎥φφ
φ= ⎢⎥
⎢⎥−φ φ
⎢⎥
⎣⎦r