Calculation Method of the CNC Machine Tool Arc

An arc is the part of a circle between two points. In a CNC block you never describe the whole circle. You give the controller a start point, an end point, a direction, and enough data to fix one unique circle through them. Everything below follows from that.

The straight line joining the two points is the chord. The distance from the chord midpoint to the arc is the sagitta. The angle between the two radii drawn to the endpoints is the central angle, usually written as θ. These three values are linked, so knowing any two of them yields the third.

Two of those values come straight off the drawing. The chord is the distance between two features you can measure with a caliper. The sagitta is the height of the arc above that chord, which a height gauge or an optical comparator can pick up. The radius is often the unknown, because a large radius on a small feature is hard to measure directly.

Worked example. A drawing gives a chord of 5,285 mm and a radius of 3,359 mm. Halve the chord to get 2,642.5 mm. The midpoint of the chord, the circle center, and one endpoint form a right triangle, so the sagitta follows from the Pythagorean relation: s = R − √(R² − (c/2)²). Plugging in the numbers gives s ≈ 1,096.7 mm. That is a shallow arc, about 1,096 mm of rise over 5,285 mm of span.

Once the sagitta is known, the central angle comes from sin(θ/2) = (c/2) / R. Here θ/2 ≈ 51.85°, so θ ≈ 103.7°. The arc length is then L = R × θ in radians, which gives roughly 6,079 mm. Checking the arc length against the chord tells you how much of the circle the move actually covers.

The R format is the short form. You write the endpoint and a radius value, and the control solves the center. It is compact and easy to read on the screen. The catch is that two circles of the same radius pass through any two points, and the sign of R picks which one you get. A positive R takes the arc of 180° or less; a negative R takes the arc longer than 180°.

The I and J format is the long form. Instead of a radius you give the vector from the start point to the center, measured along X and Z. I is the X component, J is the Z component, and K is used for the Y axis on a mill. Because the center is stated outright, there is no ambiguity about which side of the chord the arc sits on.

That ambiguity is why R blocks fail more often than I/J blocks. A rounding error in R can put the center on the wrong side of the chord, and the control rejects the block or, worse, cuts a shallow arc in the wrong direction. On a full circle the start and end points coincide, and R alone cannot define the center at all.

Choose R when the arc is short, the geometry is simple, and a human has to read the program. Choose I and J when the arc runs past 180°, when it is a full circle, or when the same profile is posted from CAM thousands of times. The incremental center vector survives posting and re-posting without drifting.

The calculation method cnc machine operators actually use on the floor is not the full derivation. It is a quick check: does the arc length match the chord and radius the print shows? A mismatch of more than a few tenths usually means a transposed digit somewhere in the block.

Tolerance matters here. On a part held to ±0.005 mm, an arc that is geometrically correct but posted with truncated coordinates can still miss. Truncating I and J to three decimals on a 3,000 mm radius moves the center by more than the tolerance. Keep full precision in the center vector and let the post handle rounding.

Controller behavior differs too. Some controls alarm out when the programmed endpoint does not sit on the circle defined by I and J. Others quietly move the endpoint to the nearest point on that circle. The second behavior is more forgiving in a prove-out and more dangerous in production, because the error never shows up as an alarm.

For very shallow arcs, another option is to skip the arc entirely and use a series of G01 moves. The chord error of each short segment has to stay under the tolerance, which means the segment length drops as the radius grows. On a 4,000 mm rail this can mean a lot of blocks, but the toolpath becomes predictable and easy to verify.

Small radius, short arc, manual programming: R is fine. The two candidate centers are far apart, and the sign convention is easy to remember. Most operators read an R block at a glance and know what the tool will do.

Long arc, full circle, or CAM output: use I and J. Full circles in particular cannot be written with R, because the start and end points are identical. Some controls accept a full circle only when the center vector is explicit.

When the arc is part of a mating surface, check the chord and sagitta after the first article comes off the machine. A comparator overlay on a shallow arc shows the error faster than any CMM routine. If the arc is off, the cause is usually in the block, not in the machine.

GreatLight machines arc-heavy profiles on 16 simultaneous 5-axis centers, with 127 high-precision CNC machines in total and a maximum processing size of 4,000 mm. Arc segments are verified in process and again before shipment, and inspection reports are available on request.