Eduard Study - Euclidean Space Group and Biquaternions

Euclidean Space Group and Biquaternions

In 1891 Eduard Study published “Of Motions and Translations, in two parts”. It treats Euclidean space through the space group. The second part of his article constructs a seven-dimensional space out of “dual biquaternions”, that is numbers

where a, b, c, and d are dual numbers and {1, i, j, k} multiply as in the quaternion group. He uses these conventions:

The multiplication table is found on page 520 of volume 39 (1891) in Mathematische Annalen under the title “Von Bewegungen und Umlegungen, I. und II. Abhandlungen”. Eduard Study cites William Kingdon Clifford as an earlier source on these biquaternions. In 1901 Study published Geometrie der Dynamen to highlight the applications of this algebra. Due to Eduard Study’s profound and early exploitation of this eight-dimensional associative algebra, it is frequently referred to as Study Biquaternions. Study’s achievement is celebrated, for example, in A History of Algebra (1985) by B. L. van der Waerden, who also cites Clifford’s earlier note.

Since the space group is important in robotics, the Study biquaternions are a technical tool, now sometimes referred to as dual quaternions. For example, Joe Rooney has profiled the use of this algebra by several modelers of mechanics (see external link).