Rotation in Three Dimensions - 3D Math Primer for Graphics and Game Development - page 272

• Conceptually, a quaternion expresses angular displacement by using

an axis of rotation and an amount of rotation about that axis.

• A quaternion contains a scalar component w and a vector component

v . They are related to the angle of rotation θ and the axis of rotation
n by

v = n sin(θ/2).

w = cos(θ/2),

• Every angular displacement in 3D has exactly two different represen-

tations in quaternion space, and they are negatives of each other.

• The identity quaternion, which represents “no angular displacement,”

is [1, 0 ].

• All quaternions that represent angular displacement are “unit quater-

nions” with magnitude equal to 1.

• The conjugate of a quaternion expresses the opposite angular dis-

placement and is computed by negating the vector portion v . The
inverse of a quaternion is the conjugate divided by the magnitude. If
you use quaternions only to describe angular displacement (as we do
in this topic), then the conjugate and inverse are equivalent.

• Quaternion multiplication can be used to concatenate multiple rota-

tions into a single angular displacement. In theory, quaternion mul-
tiplication can also be used to perform 3D vector rotations, but this
is of little practical value.

• Quaternion exponentiation can be used to calculate a multiple of an

angular displacement. This always captures the correct end result;
however, since quaternions always take the shortest arc, multiple rev-
olutions cannot be represented.

• Quaternions can be interpreted as 4D complex numbers, which creates

interesting and elegant parallels between mathematics and geometry.

A lot more has been written about quaternions than we have had the
space to discuss here. The technical report by Dam et al [11] is a good
mathematical summary. Kuiper's book [41] is written from an aerospace
perspective and also does a good job of connecting quaternions and Eu-