Concentricity measurement of features at different heights

A drawing hands you a 1.05 mm pin that must run concentric to its 20 mm cylindrical housing, and the pin tip stands 8 mm closer to the camera than the datum face. One camera, one shot, two feature planes. The calibration question is real: a scale proven at one height is not automatically true at the other. This guide gives the decision for telecentric and ordinary lenses, the depth-of-field and scale arithmetic behind it, and a bench check that proves the calibration before parts run. The same logic covers any two-plane stack: flanges, steps, rings or marks at different heights on one part.

The decision: focus midway, calibrate once — or twice, never halfway

With an object-side telecentric lens, focus so the sharp depth range brackets both planes and calibrate once, with the grid anywhere inside that range. Put the focus plane midway, 4 mm in front of the housing face here, so both planes share the defocus margin. Telecentric magnification does not depend on object distance, so one scale serves both heights, and chief rays parallel to the axis add no parallax shift.

With an ordinary lens, no single calibration is right. Calibrate twice, once with the grid at each height, and stop the iris down until both planes are sharp. A mid-plane calibration is the worst of the three choices: the next section prices its error at both heights.

For a 20 mm part this usually lands on telecentric lenses; the arithmetic below shows when they are not needed. If the gauging task itself is in question, start with when you need a telecentric lens for gauging.

What the height step costs an ordinary lens

Take an ordinary C-mount lens at 0.3x magnification with the housing plane at 110 mm working distance. From m = f/(WD − f), the focal length is f = m × WD / (1 + m) = 0.3 × 110 / 1.3 ≈ 25 mm. The pin plane sits at 102 mm, so its magnification is 25/(102 − 25) = 0.325, against 25/(110 − 25) = 0.294 at the housing.

Calibrated at the housing plane, the 1.05 mm pin face reads 1.05 × 0.325/0.294 = 1.16 mm — a 0.11 mm error from scale alone. A pin offset of 0.05 mm reads 0.055 mm. Calibrated halfway, at 106 mm (m = 25/81 = 0.309), the 20 mm housing reads 20 × 0.294/0.309 = 19.03 mm and the pin reads 1.05 × 0.325/0.309 = 1.10 mm: wrong at both heights, with the datum circle itself off by almost a millimetre.

That settles it for ordinary lenses: reject every single-calibration variant, including "somewhere in between". The entocentric versus telecentric comparison shows what a telecentric buys at this magnification.

Why telecentric scale ignores the height step

An object-side telecentric lens images every point at one scale wherever it sits inside the depth range, because chief rays run parallel to the optical axis. Move a feature closer by the height step and its image does not grow; shift it sideways and the image position does not lean with height. For a cylinder pointing at the camera this matters twice: the silhouette width equals the true diameter, and the silhouette center stays on the cylinder axis.

What a telecentric does not buy is extra depth. Depth of field follows from magnification, aperture and the blur you accept — the same physics as any lens, as the working f-number and magnification guide works through. So the plan is to choose a magnification low enough that the depth range swallows the step, then exploit constant scale across it. The telecentric depth of field validation guide shows how to prove that range on your part.

Worked example: a 1.05 mm pin standing proud of a 20 mm housing

Inputs, with their sources: pin Ø 1.05 mm, protrusion 8 mm, housing Ø 20 mm (the drawing, as asked); an illustrative 2/3" camera with 2448 × 2048 pixels of 3.45 µm; a ±0.02 mm concentricity tolerance, typical for a gauged pin; collimated backlight for silhouette edges; and the lens below from the catalog.

The example uses the LS-TCL-1390, a 0.179x object-side telecentric lens for 2/3" sensors, C-mount, with 260 mm working distance.

Three numbers decide the set-up:

Field of view: 8.8 mm / 0.179 = 49.2 mm wide, so the 20 mm housing sits with 14.6 mm of margin per side and placement tolerance costs nothing. Object pixel: 3.45 µm / 0.179 = 19.3 µm, and sub-pixel edge fitting at about 0.1 pixel on backlit edges puts single-edge noise near 2 µm, an order below the ±0.02 mm tolerance. Depth of field at F8: 2 × 8 × 19.3 / 0.179² = 308.8 / 0.03204 = 9,638 µm ≈ 9.6 mm, against the step — it fits with 1.6 mm of margin. Diffraction stays hidden: the Airy disk at F8 is 2.44 × 0.55 µm × 8 = 10.7 µm, well under the 19.3 µm pixel.

Decision: focus at the mid-plane — 4 mm in front of the housing face, so the focused plane sits at 260 mm and the housing face at 264 mm — calibrate once, and run both planes on one scale at F8. If the bench check below shows drift beyond one pixel, move to F11: the same estimate gives 13.3 mm and the Airy disk 14.8 µm, still under a pixel.

Size your own set-up from the height step

For your own camera, invert the depth estimate to get the largest magnification that spans a step Δh at working aperture F#:

With 3.45 µm pixels, F8 and Δh = 8,000 µm: (2 × 8 × 3.45 / 8,000)^(1/3) = (0.0069)^(1/3) ≈ 0.19. That is why 0.179x fits and 0.3x does not. At 0.3x the object pixel is 3.45/0.3 = 11.5 µm and the estimate is 2 × 8 × 11.5 / 0.09 = 2,044 µm ≈ 2.0 mm at F8; even F22 reaches only 506/0.09 = 5.6 mm while the Airy disk grows to 2.44 × 0.55 × 22 = 29.5 µm — nearly three pixels — so edges soften exactly where they must stay crisp.

Check the pixel budget the other way with the object space pixel size guide. For a ±0.02 mm tolerance, 19-23 µm object pixels with 0.1-pixel edges hold; for ±5 µm, take a second station at higher magnification for the close plane rather than stretching one calibration.

For the ordinary-lens route, browse machine vision lenses in the lens finder and turn field of view and working distance into a focal length here:

Three telecentrics that fit a 20 mm housing

The depth rule for this part lands near 0.18x, and the catalog holds models from under 0.1x up to 0.3x on C-mount. Three of them bracket the sensible choices:

All 3 share: Mount C-mount. Values as printed in each product's data sheet; open a part for its drawing and full specifications.

For this step, the 0.179x pick from the worked example stands: 9.6 mm of estimated depth at F8 and 19.3 µm object pixels.

The 0.5x model suits a different job: a height step well under a millimetre plus face features lit coaxially. Its depth estimate at F8 is 2 × 8 × 6.9 / 0.25 = 441.6 µm ≈ 0.44 mm, its field on a 1" camera is 12.8/0.5 = 25.6 mm wide, and the 110 mm working distance keeps the fixture compact.

The LS-TCL-2194 is that 0.5x option: 1" format, C-mount, 110 mm working distance, with built-in coaxial light.

The 0.166x bi-telecentric model covers the outliers: a 12.8/0.166 = 77 mm field on a 1" camera when the part runs well off-center or a variant grows, and about 12.1 mm of estimated depth at F8. For backlit silhouette gauging none of the three needs its coaxial port; pair whichever you choose with a collimated backlight.

The ordinary-lens fallback: two calibrations, one per plane

If a telecentric is out of the question, drop the magnification until one iris setting holds both planes sharp. At 0.15x, a 2/3" camera sees 8.8/0.15 = 58.7 mm, object pixels are 3.45/0.15 = 23 µm, and the depth estimate at F8 is 2 × 8 × 23 / 0.15² = 368 / 0.0225 = 16,356 µm ≈ 16.4 mm — the step fits. Stopping down from F2.8 to F8 costs three stops, so add light or exposure, roughly 8× of it.

1. Fixture the part so both planes sit square to the camera axis.
2. Stop down to F8-F11 and confirm a test grid is sharp at both heights.
3. Calibrate with the grid on the fixture, at the housing plane, and save that calibration.
4. Move the grid onto a spacer matching the protrusion, at the pin plane, calibrate again, and save.
5. Convert the housing tools with the first calibration and the pin tools with the second; concentricity is the pin-center-to-housing-center distance in datum coordinates.

Silhouette edges need light behind the part; the collimated versus diffuse backlight guide covers which backlight holds an edge position stable for gauging.

Bench checks before you run parts

Calibrate first, prove second. Measure the 20 mm housing diameter with the part seated, then again with the part raised by the height offset on a spacer: if the reading moves by more than one object pixel — 19 µm in the worked example — the depth range is not holding, so stop down one more F-stop and repeat. Zoom to the far-plane pin edge next: if the grey transition spans more than about three pixels, sub-pixel fitting degrades. Last, budget distortion: residual geometric distortion of 0.05% across a 10 mm radius is up to 5 µm, the same order as a tight concentricity tolerance, so calibrate it out with a grid — the geometric versus TV distortion guide explains which datasheet number applies.

When the one-calibration answer is wrong

• The protrusion varies. If pin height swings to 12 mm between lots, the 1.6 mm margin at F8 is gone: hold the height in the fixture, move to F11 (13.3 mm), or drop magnification.
• The tolerance tightens. Below about ±10 µm, 19-23 µm object pixels dominate the error budget; split the task across two stations at higher magnification.
• The pin is not square. One degree of tilt over the step height moves the tip 8 × tan 1° ≈ 0.14 mm laterally — a real offset that no calibration removes, so fixture squareness is part of the measurement.
• Someone proposes the mid-plane calibration on an ordinary lens. Section two already priced it: nearly a millimetre of error on the datum. It is the one option that is always wrong.

Set-up checklist

• Field of view covers the housing plus the placement margin you need: 49.2 mm against a 20 mm part leaves 14.6 mm per side.
• Object pixel small enough for the tolerance: 19.3 µm with 0.1-pixel edges gives about 2 µm of noise against a ±0.02 mm tolerance.
• Depth estimate clears the step with margin: 9.6 mm against 8 mm at F8.
• Focus plane midway; defocus margin shared by both planes.
• One calibration inside a telecentric depth range; two calibrations, one per plane, on an ordinary lens.
• Bench proof done: housing diameter repeats at both heights within one pixel.
• Backlight bright enough for the working aperture: three stops cost about 8× the light.
• Next step: spec the lens and camera together in the configurator, starting from the magnification you just derived.