Need Math to Rotate XYZ Point Around A-B Rotary Axes

I need help with some CNC Machine Tool logic that has to do with Coordinate Rotation. I have a known XYZ Point that I need to rotate around A(First Rotary) and B(Second Rotary) angles. The A-Axis is Parallel to "Z" and the B-Axis is Parallel to the "Y" Axis.

My logic for "Z" shown here works correctly to give me the pre-rotated Z-Axis value.

Z=(Z*(cos(Rotary B Angle)))-(X*(sin(Rotary B Angle))))

I can't come up with the correct formula for the same X-Y values.

Example:

This is a known value after Coordinate Rotate: X-8.8103 Y6.473 Z4.6117 A= -90 B= -45

I need a formula for the XYZ value BEFORE Coordinate Rotation which is: X6.473 Y9.4908 Z-2.96887

The Z Formula above works to properly convert Z4.6117 to Z-2.96887. I need help with a formula to do the same conversion for X-Y.

The software I am using to write the formula is a proprietary CAM Software with limited Math Operators such as ACS, ATN, COS, SIN, EXP, SQR, SQRT, TAN

[Matt J]

The software I am using to write the formula is a proprietary CAM Software with limited Math Operators such as ACS, ATN, COS, SIN, EXP, SQR, SQRT, TAN

Does that mean that you are not working in Matlab? If so, this is probably not the right forum.

[Matt J]

Randy's comment moved here:

I am new to Matlab so I appologize for my question which was a plea for general Math help and not something that pertains only to working in Matlab. If you are a Moderator you can delete my non Matlab related question.

Answer 1

[Matt J]

See Rodrigues' formula,

https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula

Also, though you are apparently not working in Matlab, you could use Matlab to verify your calculations using this FEX submission,

https://www.mathworks.com/matlabcentral/fileexchange/30864-3d-rotation-about-shifted-axis?focused=5191309&tab=function