HEIDENHAIN MANUALplus 620 Technical Manual page 750

Example:
750
Below is a sketch of a simple lathe with the linear axes X and Z and a C axis.
Starting from the machine bed, the X axis is on the Z axis on the tool side.
The coordinate system of the machine bed is determined by the position of
the main spindle and the arrangement of the tool carrier (here: "behind
workpiece".) As the kinematics chains is evaluated three-dimensionally, an
(imaginary) Y axis on the X axis is taken into account.
For the transformation sequence on the workpiece side, only the "machine
table" must be taken into account. For kinematics with a C axis, the "machine
table" and the C axis must be taken into account.
HEIDENHAIN Technical Manual MANUALplus 620