Cartesian Coordinate System Right-Hand Rule
The Cartesian Coordinate System herein refers to the standard coordinate system of an CNC Machine Tool's six primary axes A,B,C,X,Y, and Z Axes. The Right-Hand Rule is used to determine both the Axis Designation and Axis Direction.
Cartesian coordinate system right-hand rule - x axis, y axis, z axis
Axis Designation
This first part uses the cartesian coordinate system right-hand rule to show the axis designations of the three primary linear axes, X axis, Y axis, and Z axis. The axes letter names are designated by the relationship shown on the image to the right. The thumb, index finger, and middle finger of the right hand are held so that they form three angles positioned 90 degrees from each other. The thumb represents X axis, the index finger Y axis, and the middle finger Z axis.

Understanding the right-hand rule along with some machine tool builder guidelines makes it possible to determine axis designations on a machine that one is not familiar with. The first machine tool builder guideline is, the linear axis that moves parallel to the main spindle's centerline is designated Z axis.

The second machine tool builder guideline that pertains to a milling type machine is, the longest travel axis is designated X axis. The only axis left since this article refers to machines with the three primary linear axes is Y axis. The object now is to rotate your hand until your thumb is parallel to X and your middle finger is parallel to Z axes then, your index finger will be parallel Y axis.

Cartesian coordinate system right-hand rule - a axis, b axis, c axis The three primary rotary axes are A, B, and C. Each one is designated by indentifying the primary linear axis that its rotary plane is perpendicular to. That plane could be said to rotate around the linear axis it is relative to. A axis rotates around X axis, B axis rotates around Y axis, and C axis rotates around Z axis. A B C is in order respectively with X Y Z. One way to remember that is with this little rhyme. A B C, X Y Z. The small image to the right demonstrates A axis relative to X axis.
Axis Direction
Notice the image of the cartesian coordinate system right-hand rule again. At the end of each arrow next to the axis letters X, Y, and Z there is a + sign. In the right-hand rule, the direction that each finger points to is the positive direction of motion for that axis.

If you look at the image again you will notice that all the arrows representing the axes positive directions originate at a common zero. That zero represents the known Zero Location by which the cartesian coordinate system defines other locations at a distance away in either the positive or negative direction in 3D space.

Programmers need to know which direction the machine is going to move relative to the Zero Location. They know that by the viewpoint of the machine. The viewpoint is most likely viewed from the front of the machine. However, it can be setup to be viewed from the back, or somewhere else depending upon the type of machine.

Now for the cartesian coordinate system right-hand rule as it applies to a rotary axis direction. Imagine wrapping your right hand around a linear axis with your thumb pointing toward the positive direction. The direction that your fingers are wrapped represents the positive direction for the rotary axis that rotates around that linear axis. The arrow for A axis on the above right small image shows its positive direction.
Left-Handed Coordinate Systems
The right-handed coordinate system is the standard but there are some CNCs that use the left-handed coordinate system. As the name implies, the left hand is used to designate the axes directions instead of the right. The thumb still represents X-Axis and so on. If your new to this subject, you might be surprised at how easy it is to think, that a CNC is using the left-hand system, because an axis direction doesn't match up to the right-hand rule, and be absolutely wrong! The next section explains how that happens.
Key Point
The coordinate system is viewed from the programmer's perspective. The programmer calculates tool movements relative to a "stationary work surface". Because of that, an axes + direction can appear to be backwards when the tool is stationary and the work surface moves to machine the part. The key is to always view the coordinate system as if the tool is moving and the work surface is stationary, even if it's not! Then the axis + direction by the right-hand rule should make sense. One consistancy for a mill is, the Z-Axis + direction always points from the tool into the spindle behind it.
Who invented the Cartesian Coordinate System?
René Descartes (March 31, 1596 – February 11, 1650), also known as Renatus Cartesius (latin), a highly influential French philosopher, mathematician, scientist, and writer. Much of subsequent western philosophy is a reaction to his writings. His most famous statement is " Cogito ergo sum " ("I am thinking therefore I exist." ). As the inventor of the Cartesian Coordinate System, he formulated the basis of modern analytic geometry, which in turn influenced the development of modern calculus.