Euler–Rodrigues parameters

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.

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.

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 ${\displaystyle {\vec {x}}}$ rotates to its new position

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

${\displaystyle {\vec {x}}'={\vec {x}}+2a({\vec {\omega }}\times {\vec {x}})+2\left({\vec {\omega }}\times ({\vec {\omega }}\times {\vec {x}})\right)~~.}$

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