Rotations and quaternions: The easy and convenient way: Rotaciones y cuaterniones (sin secretos) - Mora Flores, Walter (2025)
There is a lot of scattered literature on quaternions and rotations that is oriented to practical applications but not so much to develop the intuition and mathematics behind the formulas. In this paper we start from the common basic knowledge of Linear Algebra courses2 and introduce quaternions and their application in rotations, following a natural, theoretical, practical and intuitive flow. The set of quaternions, denoted H, is a vector space isomorphic to R4 and a multiplication is defined which gives it a non-commutative field structure. Multiplication by a unitary quaternion applies a rotation in two planes, in a simultaneous manner, in a similar way as multiplication by a unitary complex number applies a rotation. To use this fact in rotations in R3, we choose a suitable orthonormal basis of H (this gives us two planes), such that in one plane the axis of rotation is fixed (i.e., no rotation) and in the other plane the desired rotation is applied.