Game Engine Architecture

158 4. 3D Math for Games

The matrix below represents rotation about the y-axis by an angle θ. Notice

that this one is transposed relative to the other two—the positive and negative

sine terms have been refl ected across the diagonal:

This matrix represents rotation about the z-axis by an angle γ:

Here are a few observations about these matrices:

z The 1 within the upper 3 × 3 always appears on the axis we’re rotating

about, while the sine and cosine terms are off -axis.

z Positive rotations go from x to y (about z), from y to z (about x), and from

z to x (about y). The z to x rotation “wraps around,” which is why the

rotation matrix about the y-axis is transposed relative to the other two.

(Use the right-hand or left -hand rule to remember this.)

z The inverse of a pure rotation is just its transpose. This works because

inverting a rotation is equivalent to rotating by the negative angle. You

may recall that cos(–θ) = cos(θ) while sin(–θ) = –sin(θ), so negating the

angle causes the two sine terms to eff ectively switch places, while the

cosine terms stay put.

4.3.7.3. Scale

The following matrix scales the point r by a factor of sx along the x-axis, sy

along the y-axis, and sz along the z-axis:

cos 0 sin 0

01 0 0

rotate ( , ) [ 1] .sin 0 cos 0

00 0 1

y rrrxyz

⎡⎤θ −θ

⎢⎥

θ= ⎢⎥

⎢⎥θθ

⎢⎥

⎣⎦

r

cos sin 0 0

sin cos 0 0

rotate ( , ) [ 1]. 0 0 10

0 0 01

z rrrxyz

⎡⎤γγ

⎢⎥−γ γ

γ= ⎢⎥

⎢⎥

⎢⎥

⎣⎦

r

0 00

0 00

[1]

00 0

0 0 01

[ 1].

x

y

xyz z

xxyyzz

s

s

rrr

s

sr sr sr

⎡⎤

⎢⎥

= ⎢⎥

⎢⎥

⎢⎥

⎣⎦

=

rS