Rodrigues' rotation formula

Rodrigues the formula, named after O. Rodrigues, is a formula for the exponential function of an anti- symmetrical 3 × 3 matrix, which describes a cross-product in a matrix form. It reads:

Its main application lies in the fact that the result describes a rotation around the axis with angle as the matrix.

Derivation

The exponential function can be divided into an infinite series that converges absolutely for all values ​​from view as:

The equation can also be applied for any square matrices. One that is suitable for their particular properties for the matrix of the cross product. It reads for the three-dimensional real space:

Multiplying out we obtain the following formula:

Wherein the length of the vector. This means that you can reduce to powers with exponents less than 3 powers of the matrix always. Therefore, these matrices are suitable for insertion into a power series.

For sine and cosine, there are also Taylor expansions. They are:

These equations can be combined: straight exponent terms can be replaced with the development and cosine terms with odd exponents by the sinusoidal development. After some simplifications we obtain the Rodrigues equation.

Properties

Be. Then:

Application

Especially in the robotics and computer graphics, the Rodrigues formula plays a role. There is always by a coordinate system, defined in which is valid for a vector:

This means that the matrix represents a rotation about the axis. The rotational angle is, that is the length of the vector.