Nutating head kinematics

I think that the head joint of this type is called cardanic head
www.google.gr/url?sa=i&rct=j&q=&esrc=s&s...ust=1575006942281400
To find the offsets is relative easy procedure with simple moves. When you find them you have to put in the post some trigonometric calculations
That part is the difficult part for me

I have this question witch is more trigonometry than linuxcnc
any help is highly appreciated
In a 5 axis head -head cnc with C,B and Z axis on the head of the machine the
Z axis is depended by the B axis rotation
the calculation formula to find correct Z value is =offset of B by the Z axis multiply by the sin of b angle
in my case Z[-(412*SIN(ANGLE_B))]
When i try a test program a p/2 of a cycle i have correct Z position in the beginning and the end of the path
but the trajectory of the tool is linear (not circular as expected)
Suggestions?