In mathematics and mechanics, the Euler–Rodrigues formula describes the rotation of a vector in three dimensions. It is based on Rodrigues' rotation formula, but uses a different parametrization.
Contents
- Definition
- Vector formulation
- Symmetry
- Composition of rotations
- Rotation angle and rotation axis
- Connection with quaternions
- Connection with SU2 spin matrices
- References
The rotation is described by four Euler parameters due to Leonhard Euler. The Rodrigues formula (named after Olinde Rodrigues), a method of calculating the position of a rotated point, is used in some software applications, such as flight simulators and computer games.
Definition
A rotation about the origin is represented by four real numbers, a, b, c, d such that
When the rotation is applied, a point at position
Vector formulation
The parameter a may be called the scalar parameter, while ω→ = (b, c, d) the vector parameter. In standard vector notation, the Rodrigues rotation formula takes the compact form
Symmetry
The parameters (a, b, c, d) and (−a, −b, −c, −d) describe the same rotation. Apart from this symmetry, every set of four parameters describes a unique rotation in three-dimensional space.
Composition of rotations
The composition of two rotations is itself a rotation. Let
It is straightforward, though tedious, to check that
Rotation angle and rotation axis
Any central rotation in three dimensions is uniquely determined by its axis of rotation (represented by a unit vector
Note that if
In particular, the identity transformation (
Connection with quaternions
The Euler parameters can be viewed as the coefficients of a quaternion; the scalar parameter
which is a quaternion of unit length (or versor) since
Most importantly, the above equations for composition of rotation are precisely the equations for multiplication of quaternions. In other words, the group of unit quaternions with multiplication, modulo the negative sign, is isomorphic to the group of rotations with composition.
Connection with SU(2) spin matrices
The Lie group SU(2) can be used to represent three-dimensional rotations in 2×2-matrices. The SU(2)-matrix corresponding to a rotation, in terms of its Euler parameters, is
Alternatively, this can be written as the sum
where the