Gear
Thick rulers read from above
A thick ruler lifts its scale away from the surface being measured, adding a small parallax-like offset.
Guides on Gear: Every instrument, and what each costs you, Everything you can wrap, and what each reads, Verifying the thing you measure with
A ruler has thickness, and its printed scale sits on the top face, a few millimetres above whatever the ruler is resting against. If your eye is not directly above the tip, you read the tip against a mark that is not the one directly beside it. The error is the gap multiplied by the tangent of the viewing angle, and it grows with both.
The geometry
Picture a ruler laid beside the shaft, with its scale on the upper face. The tip is at one height and the scale is at another, separated by the ruler's thickness plus any air gap. Look straight down and the two line up, so the mark you read is the true one. Tilt your head and your line of sight crosses the gap at an angle, landing on the scale displaced from the true position.
The displacement is gap x tan(angle), where the angle is measured from straight above. This is parallax, and it is the same mechanism as the ordinary kind covered in viewing angle when reading your own ruler. The NIST report on forensic photo scales, Ferrucci et al. at the National Institute of Standards and Technology, makes the same point about comparing two scales: care should be taken to reduce parallax caused by the viewing angle, because oblique viewing introduces error. What is added here is the dependence on the ruler's own thickness.
How big the offset gets
These offsets are computed from that formula, not measured, and the gaps are illustrative:
| Scale-to-surface gap | At 10 degrees off | At 20 degrees off | At 30 degrees off |
|---|---|---|---|
| 0.5 mm (thin) | 0.09 mm | 0.18 mm | 0.29 mm |
| 1 mm | 0.18 mm | 0.36 mm | 0.58 mm |
| 2 mm | 0.35 mm | 0.73 mm | 1.15 mm |
| 3 mm (thick) | 0.53 mm | 1.09 mm | 1.73 mm |
A thin ruler at a modest angle is below anything you can read off a millimetre scale. A thick one at a casual angle is already a millimetre or more, which is an appreciable part of the error budget for a bone-pressed length.
Which way it errs
The sign depends on which side your eye is on. If you look from behind the ruler toward the tip, the apparent mark shifts one way; from the other side it shifts the other. Because most people read their own ruler from the same side and from the same posture each time, the offset is consistent and does not average out over repeats. That makes it a systematic error, of the kind systematic and random error distinguishes, and three readings will agree with each other while all being off.
When it matters, and the plain fix
It matters most when the ruler is thick, the scale is on the top face rather than the edge, and you must view from an angle, which describes self-measurement from standing. It matters least with a thin ruler, with the eye directly above the tip, or with a partner reading it square-on.
The fixes are in order of effort. Sight straight down onto the tip rather than across it. Choose the thinnest rigid ruler you have. Use a ruler designed to remove the gap, since bevelled-edge rulers and transparent rulers each attack it by design, and are described in their own posts. Or have someone else read the scale, accepting the trade covered in having someone else measure.
Edge lift is not zero offset
A thick ruler can also be a problem at the zero end, but that is a separate matter: zero offset is a printed-scale-versus-end issue, not a view-angle one. The two can stack.
None of this has anything to do with an image or a score. Rate Cock works on photographs, where perspective plays a related but separate role. If you are measuring to see where you sit among men who have been measured, the ruler's thickness matters; if you are measuring because you want to start selling on a platform, a scale you can defend matters for a different reason. A cleaned-up photo scored against others never touches a ruler at all, and neither does a reviewer's written opinion.
The takeaway is a single number to carry: the offset is the gap times the tangent of your viewing angle, so halve either and you halve the error.