Eric Bainville - Mar 2007

This page is an introduction to quaternions, oriented towards their application to geometry. Properties and definitions used in other articles are given here in a condensed form. Interested readers should refer to the numerous articles found on the Internet on quaternions. Good starting points are the Wikipedia and MathWorld articles.


There are several different ways to define a quaternion. For applications to geometry, the most convenient definition is the following:

Sum and product are compatible with the identification with reals. Sum, product, and dot product are compatible with the identification with vectors (product of vector quaternions corresponds to cross product). The set Q of quaternions can be seen as a normed vector space of dimension 4, and a non-commutative R algebra. We have the following properties: