Any nonzero quaternion has a corresponding unitary (length one) quaternion in the same direction as . Unitary quaternions are an elegant and efficient way to formalize 3D rotations.

A quaternion can be interpreted in terms of a geometric transformation. Consider the quaternion with magnitude ; then the unitary quaternion of is .

The unitary quaternion can be expressed in the form , where is real and is a 3D vector.

Consider the rotation around the axis with direction through the angle . The image of a vector can be calculated from the quaternion using the formula , where is the conjugate of .

Reference

[1] R. T. Farouki, Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable, Berlin: Springer, 2008.