How to Calculate I and J for a G02 Arc (Worked Examples)

Q: How do you calculate I and J for a G02 arc?

Subtract the start point from the center point. I equals the center's X coordinate minus the start point's X coordinate, and J equals the center's Y minus the start point's Y, with the signs kept as they fall. An arc starting at X20 Y10 around a center at X15 Y10 gets I-5. J0, because the center sits five units in the negative X direction from where the arc begins.

Q: Are I and J measured from the start point or the end point?

From the start point, on standard incremental-center controls. Computing the offsets from the endpoint is the most common calculation mistake and produces an arc with a misplaced center. The start point is wherever the tool sits when the G02 block begins executing, which is the endpoint of the previous move.

Q: How do you check that I and J are correct?

Use the radius twice. The square root of I squared plus J squared must equal the arc radius, since I and J span from the start point to the center. Then confirm the distance from that center to the arc's endpoint is the same radius. If the two distances disagree, the center, the endpoint, or the offsets are wrong, and the control will alarm or cut a wrong arc.

Q: What is the best way to learn arc calculations like I and J?

Drill the concepts with active recall until the subtraction rule is automatic, then practice on paper arcs. A free app like G-Code Sprint quizzes G02, G03, and the arc words as quick timed questions and repeats whichever ones you miss, which makes the calculation rule stick alongside the codes themselves.

Working out I and J by hand intimidates beginners, but the entire calculation is one rule: subtract the start point from the center, coordinate by coordinate. Every wrong arc traced back to this calculation comes from subtracting the wrong pair of points or dropping a sign.

What are I and J measuring?

I and J describe the straight-line offset from the arc’s start point to the arc’s center: I along X, J along Y. They are a small vector on the ordinary Cartesian grid, and on standard controls they are incremental, measured from the start point, as the LinuxCNC arc format specifies. (If you are still deciding whether to use this format or the simpler radius word at all, the R versus I/J overview covers that choice first.) The start point is simply wherever the previous move left the tool.

So the rule is:

I = center X - start X
J = center Y - start Y

Keep the signs exactly as the subtraction produces them. A center left of the start point gives a negative I; a center below gives a negative J.

A worked example, step by step

The tool sits at X20. Y10. and the drawing shows a 5 mm radius arc around a center at X15. Y10., ending at X10. Y10.:

Step	Calculation	Result
Start point	from the previous move	X20, Y10
Center	from the drawing	X15, Y10
`I`	15 - 20	-5.0
`J`	10 - 10	0
Radius check	sqrt((-5)^2 + 0^2)	5.0 ✓

The block comes out as:

G02 X10. Y10. I-5. J0 F150

The negative I is doing real work: it says the center lies five units in the negative X direction from where the arc begins.

A second example with both offsets

Start point X0 Y0, center at X3. Y4., full circle back to the start. The subtraction gives I3. J4., and the radius check gives the hypotenuse: the square root of 9 plus 16 is 5, so this is a 5 mm radius circle:

G02 X0 Y0 I3. J4. F150   (full circle, center up and to the right)

Full circles are the case where this calculation is unavoidable, because the radius word cannot define them, the limitation explained in I, J, K vs R in G02. The same start-to-center subtraction, generalized to many points, is how a bolt hole circle’s coordinates get computed with sine and cosine.

How do you verify the numbers before running?

The radius gives you a two-step self-check that catches nearly every slip. First, I and J span start-to-center, so the square root of I squared plus J squared must equal the arc radius. Second, the center must sit at that same radius from the endpoint, because both ends of an arc lie on the circle. Run both checks; if they disagree, one of your three inputs (start, end, center) is wrong. Controls enforce this geometry, and an inconsistent arc either alarms or cuts a shape you did not intend, one of the failure paths covered in why G02 cuts a straight line.

Where do the calculations go wrong?

Three mistakes account for almost all bad I J values:

Mistake	What happens	Guard
Using the center’s absolute coordinates as `I J`	Center lands far from the arc	Always subtract the start point
Measuring from the endpoint	Mirrored or shifted center	The start point is the reference
Dropped minus sign	Arc bulges the wrong way	Keep signs from the subtraction

The direction of travel around that center is the separate clockwise question settled by G02 vs G03, and the standard code references list both forms of the arc words for quick lookup.

Bottom line

I and J are the center minus the start point, sign included: one subtraction for X, one for Y. Verify with the radius twice, start-to-center and center-to-end, and the arc is proven before the machine ever sees it. Making that subtraction rule reflex is a recall job, and a routine on the G-code practice hub handles it alongside the codes themselves.

Sources

Frequently asked questions

How do you calculate I and J for a G02 arc?

Subtract the start point from the center: I equals center X minus start X, J equals center Y minus start Y, signs kept. An arc starting at X20 Y10 around a center at X15 Y10 gets I-5. J0.

Are I and J measured from the start point or the end point?

From the start point on standard controls. Computing from the endpoint is the most common mistake and misplaces the center.

How do you check that I and J are correct?

The square root of I squared plus J squared must equal the radius, and the center must sit at the same radius from the endpoint. If the two distances disagree, one of the inputs is wrong.

What is the best way to learn arc calculations like I and J?

Drill the rule with active recall and practice on paper arcs. A free app like G-Code Sprint quizzes G02, G03, and the arc words and repeats whichever ones you miss.

G-Code Sprint is a study and practice tool only. Always follow your instructor, employer, machine manual, and shop safety procedures.