Deriving the Formula

The model illustrates the smooth function that establishes the relationship between two directions (orientated 45º to one another in the example), in three dimensions, simultaneously. It involves synchronous quantification of direction, from any single event, which can be understood as simultaneous finite rotations which commute in much the same manner as infinitesimal rotations commute in conventional plane geometry. This differs (mathematically and conceptually) from the usual (two dimensional) discrete finite rotations or infinitesimal rotations which are used to manipulate objects in Euclidean 3-space.

The methods and solutions presented in this section have not been verified.