A computer numerical control (CNC) machine moves controlled axes according to a programmed geometric path.

The machine has a fixed coordinate system tied to its physical travel. A work coordinate system places the programmed part origin at a convenient location on the setup. Tool geometry and length offsets then relate the commanded tool path to the actual cutting edge.

For a three-axis machine, positions are commonly described by coordinates $(X,Y,Z)$. A move from

$$A=(10,20,5)$$

to

$$B=(40,20,5)$$

changes only $X$ and therefore follows a straight line parallel to the $X$ axis.

When several axes move simultaneously, the controller must coordinate them. Linear interpolation advances the axes so that the tool follows a straight segment between programmed points. Circular interpolation coordinates two axes to follow an arc in a selected plane.

Coordinate mistakes can create large physical errors even when the part geometry in the program is correct. Important distinctions include:

• machine coordinates versus work coordinates;
• absolute versus incremental positions;
• tool centre path versus finished-part geometry;
• commanded coordinates versus offsets applied by the controller.

CNC therefore separates geometric intent from machine realization: the program specifies a path in a reference frame, while offsets and interpolation make that path correspond to the actual setup and tool.