Rotation in Three Dimensions - 3D Math Primer for Graphics and Game Development - page 263

}

/ / I n t e r p o l a t e

w = w0 ∗ k0 + w1 ∗ k1 ;

x = x0 ∗ k0 + x1 ∗ k1 ;

y = y0 ∗ k0 + y1 ∗ k1 ;

z = z0 ∗ k0 + z1 ∗ k1 ;

Listing 8.3
Quaternion slerp

8.5.13 Advantages and Disadvantages of Quaternions
Quaternions offer a number of advantages over other methods of represent-
ing angular displacement:

• Smooth interpolation. The interpolation provided by slerp provides

smooth interpolation between orientations. No other representation
method provides for smooth interpolation.

• Fast concatenation and inversion of angular displacements. We can

concatenate a sequence of angular displacements into a single angular
displacement by using the quaternion cross product. The same oper-
ation using matrices involves more scalar operations, although which
one is actually faster on a given architectures is not so clean-cut:
single instruction multiple data (SIMD) vector operations can make
very quick work of matrix multiplication. The quaternion conjugate
provides a way to compute the opposite angular displacement very
e ciently. This can be done by transposing a rotation matrix, but is
not easy with Euler angles.

• Fast conversion to and from matrix form. As we see in Section 8.7,

quaternions can be converted to and from matrix form a bit faster
than Euler angles.

• Only four numbers. Since a quaternion contains four scalar values,

it is considerably more economical than a matrix, which uses nine
numbers. (However, it still is 33% larger than Euler angles.)

These advantages do come at some cost, however. Quaternions suffer
from a few of the problems that affect matrices, only to a lesser degree:

• Slightly bigger than Euler angles. That one additional number may

not seem like much, but an extra 33% can make a difference when
large amounts of angular displacements are needed, for example, when
storing animation data. And the values inside a quaternion are not
“evenly spaced” along the [−1,+1] interval; the component values do
not interpolate smoothly, even if the orientation does. This makes