What are I and J in G-code arcs?

I and J are the incremental offsets from the arc start point to the arc center along the X and Y axes. Most controllers use incremental I and J by default in the XY plane, so I equals center X minus start X and J equals center Y minus start Y. This tool outputs them in that incremental form.

Why are there two possible arcs?

A start point, end point, and radius do not define a unique arc, because the center can sit on either side of the chord connecting the points. That gives two centers and two arcs, one shorter and one longer. The tool shows both and marks the shorter solution.

What is the difference between G02 and G03?

G02 cuts a clockwise arc and G03 cuts a counter-clockwise arc, both viewed looking down the Z axis in the XY plane. The tool reports the correct direction for the minor arc to each center; to take the major arc the long way around, you reverse the G02 or G03 code.

Why does it say my radius is too small?

An arc must reach both endpoints, so its radius cannot be smaller than half the straight-line distance between them. If you see that warning, increase the radius to at least the half-chord value the tool reports, or move the endpoints closer together.

Does this work for the YZ or XZ plane?

This calculator targets the default XY plane used with G17, which covers the large majority of arc programming. For YZ work with G19 or XZ work with G18 the same geometry applies but the offset letters change to J and K or I and K respectively.

G-Code Arc (I/J/K) Calculator

CNC controllers cut arcs with circular interpolation, and the trickiest part of hand-coding one is the I and J center offsets. This tool takes a start point, an end point, and a radius and computes the exact I and J values plus the correct G02 or G03 direction so you can paste a working arc move straight into your program.

How it works

The arc center must be equidistant from both endpoints, so it lies on the perpendicular bisector of the chord. Using the chord length and the radius, the distance from the chord midpoint to the center is found by Pythagoras, giving two centers — one on each side:

chord = distance(start, end)
half  = chord / 2
a     = sqrt(radius² − half²)
center = midpoint ± a × (unit perpendicular to chord)
I = center X − start X
J = center Y − start Y

For each center the tool measures the swept angle in both directions to decide whether the minor arc is clockwise (G02) or counter-clockwise (G03), and flags the solution with the smaller sweep as the shorter arc.

Example and notes

An arc from (0, 0) to (10, 10) with radius 10 produces two centers, at (10, 0) and (0, 10), each giving a quarter-circle but in opposite directions. Pick the one whose direction and bulge match your toolpath. The output uses incremental I and J, which is the default on Fanuc, Haas, LinuxCNC, and most other controllers; if your machine is set to absolute arc centers with G90.1, use the printed center coordinates instead of the I and J values.