Page 137 - Computer Graphics Handout

3.14 QUATERNIONS

Quaternions are an extension of complex numbers that provide an alternative method for describing and manipulating rotations.
Although less intuitive than our original approach, quaternions provide advantages for animation and hardware implementation
of rotation.

3.14.1 Complex Numbers and Quaternions
In two dimensions, the use of complex numbers to represent operations such as rotation is well known to most students of
engineering and science. For example, suppose that we let i denote the pure imaginary number such that i2=−1. Recalling Euler’s
identity,
iθ
e = cos θ + i sin θ ,
we can write the polar representation of a complex number c as
iθ
c = a + ib = re ,

However, we are really interested in rotations in a three-dimensional space. In three dimensions, the problem is more difficult
because to specify a rotation about the origin we need to specify both a direction (a vector) and the amount of rotation about it (a
scalar). One solution is to use a representation that consists of both a vector and a scalar. Usually, this representation is written as
the quaternion
a = (q0, q1, q2, q3) = (q0, q),
where q = (q1, q2, q3). The operations among quaternions are based on the use of three “complex” numbers i, j, and k with the
properties
i2 = j2 = k2 = ijk =−1.
These numbers are analogous to the unit vectors in three dimensions, and we can write q as
q = q1i + q2j+ q3k.
Now we can use the relationships among i, j, and k to derive quaternion addition and multiplication. If the quaternion b is given by

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