172 4. 3D Math for Games
This is analogous to the reversal that occurs when transposing or inverting
matrix products.4.4.3. Rotating Vectors with Quaternions
How can we apply a quaternion rotation to a vector? The fi rst step is to rewrite
the vector in quaternion form. A vector is a sum involving the unit basis vectors
i, j, and k. A quaternion is a sum involving i, j, and k, but with a fourth scalar
term as well. So it makes sense that a vector can be writt en as a quaternion
with its scalar term qS equal to zero. Given the vector v, we can write a cor-
responding quaternion v = [ v 0 ] = [ vx vy vz 0 ].
In order to rotate a vector v by a quaternion q, we pre-multiply the vec-
tor (writt en in its quaternion form v) by q and then post-multiply it by the
inverse quaternion, q–1. Therefore, the rotated vector v’ can be found as fol-
lows:This is equivalent to using the quaternion conjugate, because our quaternions
are always unit length:
(4.4)
The rotated vector v’ is obtained by simply extracting it from its quaternion
form v’.
Quaternion multiplication can be useful in all sorts of situations in real
games. For example, let’s say that we want to fi nd a unit vector describing the
direction in which an aircraft is fl ying. We’ll further assume that in our game,
the positive z-axis always points toward the front of an object by convention.
So the forward unit vector of any object in model space is always FM ≡ [ 0 0 1 ]
by defi nition. To transform this vector into world space, we can simply take
our aircraft ’s orientation quaternion q and use it with Equation (4.4) to rotate
our model-space vector FM into its world space equivalent FW (aft er converting
these vectors into quaternion form, of course):4.4.3.1. Quaternion Concatenation
Rotations can be concatenated in exactly the same way that matrix-based trans-
formations can, by multiplying the quaternions together. For example, consid-
er three distinct rotations, represented by the quaternions q1 , q 2 , and q3 , with
matrix equivalents R1 , R 2 , and R 3. We want to apply rotation 1 fi rst, followed
by rotation 2 and fi nally rotation 3. The composite rotation matrix Rnet can be
found and applied to a vector v as follows:v′==rotate(q, )v qvq*.F WM==qF q−−^11 q [0 0 1 0] q.v′==rotate(q, )v qvq .−^1