Frame of reference conversion: 3 phase voltage vector to 3D co-ordinate system

Hi,

PROBLEM: I wish to define the rotating vector defined using a 3 axis co-ordinate system with axis spaced at 120degrees in two dimensions, as a rotating vector in an 3 orthogonal axis co-ordinate system.

DESCIPTION: In the most straight forward case the vector maps a circle in the original frame of reference. In the general case the three phases may have different magnitudes resulting in an ellipical mapping, where the centre of the 120degree spaced frame of reference can be off (ellipse) centre.

OBSERVATIONS:
You can think of the circle or ellipse as the end of a cone centred at (0,0,0) of the orthogonal co-ordiniate system. then the rotating vector is centre at (0,0,0) and follows around the perimeter of the circle or ellipse forming the outside of the cone.

HELP REQUIRED:
I want a set of equations to convert the vector frame of reference from the 2D 3 axis coordinate system to the 3D 3 axis orthogonal co-ordinate system. How do I get these?
 

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