Mathematical Structures for Computer Graphics

5.2.4 Quaternion Algebra

In the middle of the nineteenth century, the Irish mathematician William Hamilton invented a new algebraic system by starting with a collection of imaginary quantities called quaternions. Although it might seem reasonable to start with complex numbers and add another imaginary quantity, quaternions require two new imaginary quantities giving a set of three denoted by . (Again, these are not vectors although the letters conventionally used are also used for orthonormal basis vectors.) The square of each of these quantities is , that is, . A quaternion is a generalized complex number: . Historically, the tricky part was to define the algebra properly so that we get all the appropriate properties associated with addition, multiplication, and a norm. Hamilton hit on a set of rules for the imaginary quantities that led to a useful algebraic system.

Definition 5.2 (Quaternions)

A quaternion is a mathematical object of the form , where the imaginary quantities