## General

Orthogonal transformations a, b and c of the group of orthogonal transformations in 3 dimensions O(3) must uphold the following laws: completeness. assocativity. unit. inverse.

## Specific

The page tries to calculate what happens to a point after rotation around an arbitrary axis through the origin.

The following variables are used:

R The distance of the point to the origin (the origin is also assumed to be where the axis goes through). The axis (direction and angular velocity). The location of the point relative to the origin.

Given that the point will only move on a plane perpendicular to the axis, we get: Given that the distance of the point to the origin will stay the same, we get: Solve the latter equation for z: Put into the first equation:    For the positive branch, that is:    FIXME: solved for x, assumed :  For the negative branch, that is:    FIXME: solved for x, assumed :  ## Cross Product Solving  Ooookay...    Better do it with a matrix...     Abbreviate: :   