Roundness error measurement from a 3D scan: four fitting methods

Roundness looks like the simplest number on the drawing and hides the most decisions. A 3D scan of a bore, a bearing seat or a drawn shell returns a cloud of edge points, and the software condenses them into one value in micrometres. That value changes with the circle-fitting method behind it, and four methods are in common use. This guide is for quality and process engineers who have to read the value, defend it in an audit, or set it up on a new part. It defines roundness from scanned data, works a 40 mm bore end to end, compares the four fits on the same points, and ends with the method to pick for each mating function.

What does a 3D scan report as roundness error?

A 3D scan reports roundness as one number in micrometres: the radial gap between two concentric circles that just contain every measured point of the feature, so a 12 µm result means the whole edge profile stays inside a ring-shaped zone 12 µm wide.

ISO 1101 draws the same zone on the drawing: two concentric circles, with every point of the real surface required to fall between them. The scan supplies the points; the fitting method decides where the two circles sit. Centre position and diameter come out of the same fit, which is why one circular-edge tool reports all three at once.

The points come from triangulation: a laser line swept across the bore, or a projected pattern snapped over the surface. The geometry is the same one explained in how to create a 2D image from laser triangulation, and once the points exist, every fitting method consumes them identically.

Which four fitting methods are used for roundness?

Four standardized methods dominate: the least squares circle (LSC), the minimum zone circle (MZC), the maximum inscribed circle (MIC) and the maximum circumscribed circle (MCC); each places the two concentric circles differently, and each returns a different micrometre value from the same points.

The least squares circle minimizes the sum of squared radial residuals, and its roundness is the spread of those residuals between a concentric circumscribed and inscribed circle. The minimum zone circle finds the tightest concentric pair that contains every point, which is exactly the zone ISO 1101 describes. The maximum inscribed circle is the largest circle that fits inside a bore, a virtual pin gauge. The maximum circumscribed circle is the smallest circle that contains a shaft, a virtual ring gauge. The ISO GPS roundness standard, ISO 12181-1, formalizes them as LSCI, MZCI, MICI and MCCI.

Least squares is the most commonly used reference circle in CMM and vision software because the fit is linear, fast and stable in the presence of noise. Minimum zone needs an iterative optimizer but reads closest to the tolerance zone on the drawing. MIC and MCC care about a handful of extreme contact points, which makes them honest for fit prediction yet sensitive to outliers.

How do you calculate roundness from scanned points?

Scan a 40 mm bore at 720 points, fit the least-squares circle, and if the radial deviations span from −5 µm to +7 µm the roundness is 7 − (−5) = 12 µm. The whole method is a subtraction once the centre is known.

The full pass, with the arithmetic shown:

1. Sweep the bore: one profile every 0.5° of rotation, 720 profiles in total, D = 40 mm.
2. Circumference = π × 40 mm = 125.7 mm, so consecutive points sit 125.7 / 720 = 0.17 mm apart.
3. Least-squares fit of the 720 points returns the centre and a mean radius of 20.000 mm.
4. Radial deviations from that circle: maximum +7 µm, minimum −5 µm.
5. RON_LSC = 7 − (−5) = 12 µm.
6. Largest diameter = 2 × (20.000 mm + 0.007 mm) = 40.014 mm; smallest = 2 × (20.000 mm − 0.005 mm) = 39.990 mm.
7. The minimum-zone optimizer on the same points returns 11.2 µm.

The bore is a worked example with chosen numbers; the arithmetic is the part to copy. Two habits keep the value honest: state radial or diametral explicitly, because the diametral reading here is 2 × 12 µm = 24 µm, and keep the sweep symmetric, since the optimizer assumes full-circle coverage with no gaps.

How do the four methods differ on the same scan?

On that same 720-point scan the four methods return four different numbers: LSC 12.0 µm, MZC 11.2 µm, MIC 12.0 µm and MCC 12.8 µm, and the minimum-zone value can never exceed the least-squares value on identical points.

Minimum zone trades an optimizer for up to about a micrometre of credit on lobed parts. MIC and MCC slide the centre toward the functional extreme, so their roundness follows the few contact points that set the fit. Against a 12 µm drawing tolerance the method becomes the verdict: LSC at 12.0 µm sits on the limit, MZC at 11.2 µm passes with 0.8 µm to spare. Name the method in the inspection plan and both numbers tell the truth.

Which roundness method matches which mating function?

Match the method to what the circle does on the part: verify a drawing tolerance with the minimum zone circle, gauge a bore with the maximum inscribed circle, gauge a shaft with the maximum circumscribed circle, and track process drift with the least squares circle.

• Drawing verification and disputes: MZC. The tolerance zone on the drawing is a pair of concentric circles, so the minimum-zone fit mirrors it; this is the number that holds up in an audit.
• Bore receiving a pin, plug gauge or bearing race: MIC. It answers the only question that matters — will the largest pin go in?
• Shaft entering a housing or ring gauge: MCC. It answers whether the smallest ring will pass over.
• SPC and tool-wear trending: LSC. It uses all 720 points, moves smoothly as the tool wears, and most software reports it by default.

Roundness earns its place on the drawing because it keeps the lubricating film alive in a cylinder bore and spreads the load on a bearing seat. Pick the method that models the mating contact, not the one that prints the friendlier number. Where a drawing calls out only a roundness value of 0.012 mm, write the chosen method and the filter band into the inspection plan before the first report goes out.

How many scan points does a roundness fit need?

A circle fit needs more than two sample points per undulation it must see: a three-lobe part therefore needs over 6 points, and a 720-point scan spaced 0.17 mm apart on a 125.7 mm circumference resolves undulations down to about 0.35 mm of arc.

The rule is Nyquist applied around a circle: to detect n lobes you need more than 2n points, and metrology practice keeps four or more per lobe. Turning in a three-jaw chuck leaves 3 lobes; grinding can leave 10 or more. A few hundred points has margin to spare, and decimating afterwards is the wrong way to get there — point cloud x y z resolution settings explains how thinning after capture shifts points rather than removing them.

Standoff comes first: give the scanner room so the whole circle stays inside the field, following the 3D camera working distance and positioning guide. The workhorse capture is a laser profiler swept by a rotary axis: it projects a line across the bore and records one profile per trigger, the same sweep that serves surface defect detection with a 3D laser line profiler, so one capture does both jobs. The profiler line in the shop covers the common standoffs and point densities:

If the part can sit still, a projected structured-light 3D camera such as the MV-DPS200P-02 captures the whole surface in a single shot, so the fit sees the complete circle with no stitching and no motion error folded in.

On a tube or extrusion line, a line-scan 2.5D system such as the MV-HLVS-16K100-L60 pairs 16K resolution with a 120 kHz sampling rate and a 600 mm emitting area, capturing every cross-section as the part passes the light sheet. For transparent or thin-wall tube, place line scan lights behind the part and measure the silhouette instead; a backlit edge gives the fit a far cleaner transition than reflection does.

Sampling and filtering travel together. Roundness filters cut the profile by undulations per revolution (upr), with common bands at 1–15, 1–50 and 1–150 upr, and a narrower band reads lower because it drops short-wave content. Lobes from turning sit mostly in the low harmonics, which is why removing the first harmonic from an FFT spectrum is the standard repair when residual eccentricity contaminates the profile. Fix the band once, print it on the report, and never compare two values measured with different bands.

How does scan quality limit the roundness number you can trust?

The reported value is only as good as the edge points: a single outlier sitting 10 µm off the surface inflates a least-squares roundness reading by up to 10 µm, which is most of a typical 12 µm tolerance.

Three failure paths dominate:

• Laser line noise. Sub-pixel extraction jitter lands directly in the deviations. Get a clean height line first — laser line extraction noise: get a clean height line shows the test.
• Edge artifacts. Stereo and structured-light depth smears across surface boundaries, and those flying pixels between object and background sit exactly on the circle being fitted, biasing its extremes.
• Dropouts. A shiny bore can steer the laser away and leave gaps; missing points in a 3D camera point cloud covers the diagnosis. MIC and MCC feel gaps hardest because they key on extreme contact points.

Two controls set the trust limit. Clean before fitting: reject spikes, clip flying pixels, bridge short gaps, then refit; least squares degrades smoothly while MIC and MCC can jump when one contact point changes. Then budget the noise: keep measured edge-point noise at or below one fifth of the tolerance, so 2 µm of noise supports a 10 µm tolerance or looser. Tighter than that, improve the optics or move the number to a spindle-type roundness tester.

When is the usual roundness answer wrong?

Three cases change the number: an arc shorter than about 90° destabilizes the fit, diametral reporting doubles 12 µm to 24 µm, and a thin-wall bore clamped in a three-jaw chuck gains real three-lobe distortion the scan measures faithfully.

Partial arcs: CMM practice treats least-squares fits as sound from about 90° of span upward. Below that, centre and radius couple badly, and MIC and MCC lose meaning because their contact points must sit on opposed sides of the circle. For a short arc, report radial deviation against the nominal radius instead, or re-fixture so the scanner sees the whole circumference.

Diametral doubling: radial roundness of 12 µm and diametral roundness of 24 µm describe the same points, and tools and drawings disagree about which one they mean. Compare a diametral value against a radial tolerance and you reject good parts or ship bad ones.

Clamping: the part can be honest while the fixture is not. A thin-wall bore squeezed in a three-jaw chuck distorts into a three-lobe shape, and the scan reports that lobing whether it was machined in or squeezed in. Where free-state measurement applies, unclamp before scanning or move to collets and soft jaws, and record the clamping state on the report.

What should you check before trusting a roundness report?

Before signing a roundness report, check five things: full-circle coverage, point spacing finer than the shortest undulation, outlier-cleaned points, a stated filter band, and the fitting method named beside the value — an 11.2 µm MZC number and a 12.0 µm LSC number are not interchangeable.

The list, with the reason each item earns its place:

• Coverage: the sweep samples every sector of the circle; a gap destabilizes MIC and MCC, which need contact points spread around the circumference.
• Spacing: points closer than half the shortest undulation to be caught; 0.17 mm spacing catches lobes down to about 0.35 mm of arc.
• Cleaning: spikes and flying pixels rejected and the fit rerun; uncorrected outliers inflate the value by their own size.
• Filter band: printed in upr on the report, the same band on both sides of any comparison.
• Method and basis: LSC, MZC, MIC or MCC named, with micrometres, radial or diametral, and free-state or clamped stated.

Pick among the shop's 3D sensors by standoff, field and point density with the camera and 3D sensor finder, or start a build in the system configurator at the camera step. The 3D vision topic page collects the neighbouring point-cloud guides. If a bore on your line needs a number you can defend, bring the drawing and the tolerance to the contact page and we will size the scan to it.