A surface is proved flat by comparison, never by looking at it: blue contact against a companion plate shows high spots, a straight edge held against a lamp shows light gaps, and a glass flat under single-colour light shows interference fringes each worth about 0.3 micrometre of height. Because two plates can fit each other while both are curved, only a result that survives all three pairings of three plates proves flatness, and the same test shows later which plate has drifted.
Part of Precision Tool Making
Methods for testing a plate or edge for flatness, the arithmetic of what each can resolve, and how a settlement detects that a reference has lost its accuracy.
Two Plates Can Agree And Both Be Wrong
Describe each surface along one line by its curvature k, the reciprocal of its radius, taken as positive where convex and negative where concave. Two surfaces fit when their curvatures cancel, so a pair A and B fits when kA + kB = 0. That is satisfied by a convex plate on a concave one, and by two flat plates, so a good fit between two plates proves nothing about flatness.
Add a third. If all three pairs fit: kA + kB = 0, kB + kC = 0 and kA + kC = 0. Add the first two and subtract the third: (kA + kB) + (kB + kC) - (kA + kC) = 2 kB = 0. So kB = 0, and then kA = kC = 0. All three are flat along the line tested. No reference was used; the proof is closed algebra. This is the three-plate method stated as arithmetic (Three-Plate Method), and the same result applies to a second direction across the plate, which is why twist is tested separately.
Each Test Has A Floor Below Which It Sees Nothing
Blue contact shows where two surfaces touch. A film of blue has thickness of its own, so a gap smaller than that film is invisible; the wiki gives no figure for the film, so measure the limit of your own blue by testing a plate against a feeler-leaf shim (Precision Measurement).
A straight edge against a lamp. Straightedge Making says the light-gap test detects errors of 0.002-0.005 mm (Straightedge Making). That figure is that article’s and has not been verified here.
The reversed edge. Turning an edge over doubles its bend, so a bend that is half the resolvable gap becomes visible (Straight Edges and Surface Plates).
Twist. Support a plate on three points, lay the second on top, and press opposite corners alternately; a plate that rocks on one diagonal and not the other is twisted (Three-Plate Method).
Optical flat. A flat piece of glass laid on the surface under single-colour light, such as sodium light (the wiki mentions a sodium flame through salt), shows light and dark fringes. Each fringe marks a gap change of half a wavelength. For sodium light at 589 nm that is 294.5 nm, about 0.3 micrometre or 0.0003 mm. This is the one test whose scale is set by nature rather than by a plate you own. The catch is that the glass flat is itself a reference that must be made flat; the standard way is again to lap three together (recalled, not in the wiki).
What You Need
- Two companion plates and engineer’s blue
- A straight edge and a lamp with a dark backing
- A feeler-leaf set to calibrate the limit of the blue
- For the optical test: a glass flat, single-colour light, and a dark room
- A log in which each test, date and result is written down
Method: A Flatness Audit
Step 1: Contact test the plate against its two companions in all three pairs.
Step 2: Straight-edge test along both directions and both diagonals, edge reversed each time.
Step 3: Twist test by the diagonal rocking method.
Step 4: Optical test, if a glass flat is available, counting fringes across the plate.
Step 5: Write down the result and the date, and the temperature of the workshop.
Step 6: Re-test after any knock, after heavy use, and at an interval you set from the log of how fast the plate has actually drifted.
Worked Example
Sag and radius: a surface that is 0.01 mm lower at the middle than at the ends of a 300 mm span has radius R = L squared / (8 x s) = 300 x 300 / (8 x 0.01) = 90,000 / 0.08 = 1,125,000 mm, which is 1,125 m. At the carrier’s master reference level of 0.002 mm the radius is 90,000 / 0.016 = 5,625,000 mm, 5,625 m. These are very gentle curves, which is why they cannot be seen.
Fringes: 0.002 mm = 2,000 nm, and 2,000 / 294.5 = 6.8 fringes; 0.01 mm is 10,000 / 294.5 = 34 fringes. The light-gap range of 0.002-0.005 mm is 6.8 to 17 fringes wide. An optical test reads a fraction of one fringe, so it can resolve finer than the lamp test, if the sodium wavelength and the half-wavelength rule are taken as given.
Temperature: steel grows 0.012 mm per metre per degree, so a 300 mm plate is 0.0036 mm longer for each degree. A plate warmed by a hand on one side bends by an unknown fraction of that; let a plate reach uniform temperature before any test finer than about 0.005 mm.
Common Mistakes
- Passing a plate on a single pair. Two plates can fit while both are curved.
- Testing a warm plate. Heat bends it.
- Trusting the glass flat without asking how it was made. It is another owned reference.
- Treating a stated flatness as measured. This wiki’s articles give different figures for the same number of scraping cycles; no figure here is a substitute for the audit.
Safety
Heavy plates, glass edges and a flame
Plates are heavy: lift with help. Glass flats chip at the edge and the chips are sharp. A sodium flame is an open flame: keep it away from fuel, oil-soaked rags and solvents, and do not use it where flammable vapour could be present. Cast-iron scraping dust is a respiratory matter; wipe it with a damp cloth (Dust and Respiratory Protection).
What’s Next
- Learn next: Gauge Blocks - the flat surfaces that wring together
- Combine with: Surface Plate Construction - what you are auditing