Quaternions and Rotations *

Up until now we have learned that a rotation in R3 about an axis through the origin can be represented by a 3×3 orthogonal matrix with determinant 1. However, the matrix representation seems redundant because only four of its nine elements are independent. Also the geometric interpretation of such a matrix is not clear until we carry out several steps of calculation to extract the rotation axis and angle. Furthermore, to compose two rotations, we need to compute the product of the two corresponding matrices, which requires twenty-seven multiplications and eighteen additions. Quaternions are very efficient for analyzing situations where rotations in R3 are involved. A quaternion is a 4-tuple, which is a more concise representation than a rotation matrix. Its geometric meaning is also more obvious as the rotation axis and angle can be trivially recovered. The quaternion algebra to be introduced will also allow us to easily compose rotations. This is because quaternion composition takes merely sixteen multiplications and twelve additions. The development of quaternions is attributed to W. R. Hamilton [5] in 1843. Legend has it that Hamilton was walking with his wife Helen at the Royal Irish Academy when he was suddenly struck by the idea of adding a fourth dimension in order to multiply triples. Excited by this breakthrough, as the couple passed the Broome Bridge of the Royal Canal, he carved the newfound quaternion equations