All truths are easy to understand once they are discovered; the challenge is to discover them… - Galileo

The first thing I like to explain is that Cartesian deals with the X axis, Y axis, and Z axis. I’m sure you know the “left hand rule” where you hold your hand out and point your index finger forward, your middle finger to the right, and your thumb up, and say “X, Y, and Z… and your fingers are pointing in the positive direction.” Now would be a good time to move your CMM probe to different areas (after a PCS is developed) and show where the probe is along each axis. From here I jump way ahead (briefly) to explain that Polar works in Radius (from XY origin), Angle (from X axis 0), and Height, and I tell people they can just cross “Height” off their list because wherever the probe is on the Z axis IS THE SAME AS THE HEIGHT. Just like 100ml is the same as 100cc, 100mm (positive) on the Z axis (Cartesian) is also 100mm “Height” in Polar. AWESOME, we’re 1/3 of the way there.
What we need to explain now is how coordinates on the X axis (left to right) and the Y axis (forward and backward) “translate” to a Radius and an Angle. Since we’ve taken the Z axis (or “Height’ in Polar coordinates) away we can view everything as only 2 dimensions and the best perspective is the “top view” of a part. The X and Y coordinates are just like the grid (city block) system invented by Alexander the Great. After an X,Y,Z origin is established on a part, another feature might be located at X 1.501” Y 3.502” which means if your probe moved (from zero) to the right (X positive direction) 1.501” and stopped, and then moved 3.502” forward (Y positive direction) you’d be at the center of the feature.

IMAGE 1 – top view of a part with the left, front corner as the origin and a circle located at X 1.501” Y 3.502”

It’s just like walking to a friend’s house… 1.5 blocks East, turn left and walk 3.5 blocks North and you’re there. If you can view Cartesian as walking (North, South, East, or West) then think of Polar as “the direction (and distance) a crow would fly” to get to the same destination. Birds don’t fly to the right and then make a left turn, birds head in a direction and fly until they get there. So let’s look at the same “top view” of the same part and think of the X axis as 0 degrees and the degree increase in a counter clockwise manner. X positive is 0 degrees, Y positive is 90 degrees, X negative is 180 degrees, and Y minus is 270 degrees (or minus 90 degrees).

So all we have to decide is, what direction (in degrees) would we need to go, and how far would we (or our probe) have to travel (in a straight line) to get to our feature that we know is located at X 1.501” Y 3.502” ? Your CMM software will make this calculation (or translation) for you but you must be able to visualize it. A point located at X 1.501, Y 3.502 is also 3.8101 from the origin (the Radius), at a direction (Angle) of 66.7994 degrees.

IMAGE 2 – same as image 1 but with Polar added

That’s all there is to it:

Cartesian Polar
X 1.501      R 3.8181

Y 3.502     A 66.7994

Richard Clark is a Metrologist, CMM programmer, and CMM trainer from Indiana. Feel free to e-mail him at This email address is being protected from spambots. You need JavaScript enabled to view it.