A spherometer is a precision instrument that measures the radius of curvature of a spherical surface by reading the sagitta — the height between a curved surface and the plane of three reference legs. Optical workshops grinding telescope mirrors and ophthalmic lens blanks rely on it daily. The instrument places a fine-pitch micrometer screw at the centre of an equilateral leg triangle and reads the screw's vertical displacement against the curved surface. From that single sagitta reading you back-calculate the radius to within ±0.05 mm on a 250 mm focal-length mirror — accuracy a calliper cannot touch.
Spherometer Interactive Calculator
Vary the leg spacing, sagitta reading, and micrometer drum details to see the calculated surface radius, mirror focal length, curvature, and reading resolution.
Equation Used
The spherometer radius equation uses the known triangular leg spacing l and the measured sagitta h from the central micrometer screw. Smaller sagitta readings produce larger radii, so dust, zero error, or poor touch pressure can strongly affect long-radius optical surfaces.
- The three outer legs form an equilateral triangle.
- Sagitta is the corrected vertical micrometer displacement from a flat zero reference.
- The measured surface is locally spherical.
- Mirror focal length is approximated as f = R/2.
Inside the Spherometer
The spherometer works on a geometric trick. Three fixed legs sit on the surface in an equilateral triangle of known leg-spacing, and a fourth central screw — the micrometer screw gauge — moves up or down until its tip just touches the surface. The vertical displacement of that central screw, the sagitta, plus the known leg-spacing, gives you the radius of curvature through a single closed-form equation. No fixturing, no reference standard, no comparator block. You zero the screw on a flat optical surface — a known plano reference — then transfer the instrument to the curved part and read the offset.
The geometry is what makes the reading fragile. The three legs must be coplanar to within roughly 2 µm or you bias the sagitta reading. The screw pitch is normally 0.5 mm with a graduated drum giving 100 divisions, so each division is 5 µm — and the vernier scale on better instruments resolves a further 0.5 µm. If you over-tighten the screw against the glass, you flex the frame, lift the legs off contact, and read short. If you under-touch, you read long. Operators learn the touch by sound and feel — a faint tick as the screw kisses the surface is the cue to stop.
Failure modes are usually contamination and wear. A speck of dust under any leg shows up as 5-15 µm of phantom sagitta, which on a long-radius mirror can throw the calculated radius off by hundreds of millimetres. Worn leg tips go from sharp points to small flats, and the contact point shifts inward, shortening the effective leg-spacing. Calibrate against a known steel ball or a certified concave test plate every shift if you are doing serious optical work.
Key Components
- Three outer legs: Equilateral tripod with leg-spacing typically 50 mm or 100 mm centre-to-centre. Tips are hardened steel ground to a fine point. The three points define the reference plane from which the sagitta is measured, and their coplanarity must hold to within 2 µm.
- Central micrometer screw: Fine-pitch screw, usually 0.5 mm pitch, threaded through the frame's centre boss. The screw raises or lowers a hardened tip until it contacts the surface under test. Backlash in the thread must stay below 2 µm or repeatability collapses.
- Graduated drum (head): Circumferential scale with 100 or 500 divisions giving a direct reading of 5 µm or 1 µm per division. A reference index on the frame fixes the zero. Many instruments add a vernier scale for an extra factor of 10 in resolution.
- Vertical linear scale: Engraved on the screw column, marked in 0.5 mm increments to count whole revolutions of the drum. Without this you cannot tell a 0.5 mm sagitta from a 1.0 mm sagitta — both look identical on the rotary drum.
- Frame body: Cast or machined block, often brass or stainless, that ties the three legs and the central screw into one rigid assembly. Frame stiffness matters because hand pressure on the drum bends the body and falsifies the reading.
Where the Spherometer Is Used
The spherometer earns its keep wherever a curved surface needs verifying without a profilometer or interferometer. Optical workshops, ophthalmic labs, and instrument restorers all reach for one. The reason it persists in the digital era is simple — for shop-floor radius checks on telescope mirrors, contact lens moulds, and watch crystals, no electronic instrument matches its speed-to-accuracy ratio at this price.
- Amateur telescope making: Verifying the radius of curvature of a Newtonian primary mirror during the rough-grinding stage. A 200 mm f/6 mirror has a target radius of 2400 mm and needs the sagitta held to within 0.02 mm to keep focal length on target before figuring.
- Ophthalmic lens manufacturing: Checking base-curve radius on semi-finished spectacle lens blanks at companies like Essilor and Carl Zeiss Vision. The base curve drives prescription power and must hold to ±0.05 dioptres, which on a 6.00 D blank means a radius tolerance of about ±0.4 mm.
- Watchmaking: Measuring the dome radius of mineral and sapphire watch crystals. A typical dive-watch crystal has a 30 mm radius dome, and replacement crystals must match the bezel seat radius within 0.1 mm to seal correctly.
- Contact lens tooling: Bausch & Lomb and similar contact lens producers use ring spherometers to verify the radius of brass and steel mould inserts before injection moulding. Mould radius drives base curve, which must hold to ±0.025 mm on a 8.6 mm radius soft lens insert.
- Antique scientific instrument restoration: Conservators at the Science Museum London use period spherometers to non-destructively measure the curvature of original 19th-century achromat objectives during cataloguing, where contacting interferometry is forbidden.
- Concrete and material science: Modified large-leg spherometers measure surface flatness and dish on polished concrete and granite surface plates, calling out deviations from the certified Grade A flatness specification.
The Formula Behind the Spherometer
The formula converts a sagitta reading and the known leg-spacing into a radius of curvature. The trade-off across the operating range is simple — at small sagitta values, on long-radius surfaces, your radius result is highly sensitive to sagitta error, so a 5 µm misread on the drum can shift the calculated radius by hundreds of millimetres. At large sagitta values, on short-radius highly curved surfaces, the reading is robust but the surface curls away from the legs and you risk the central tip slipping off the apex. The sweet spot is a sagitta between roughly 0.2 mm and 2.0 mm — large enough that drum resolution doesn't dominate, small enough that the geometry stays well-conditioned.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| R | Radius of curvature of the surface | mm | in |
| l | Mean distance between adjacent outer legs (side of the equilateral triangle) | mm | in |
| h | Sagitta — height between the surface and the plane of the three legs | mm | in |
Worked Example: Spherometer in an amateur telescope mirror grinding session
An amateur optician in Bend, Oregon is rough-grinding a 200 mm diameter Pyrex disc into a Newtonian primary mirror with a target focal length of 1200 mm, which means a target radius of curvature of 2400 mm. The spherometer in use has an equilateral leg-spacing of 60.00 mm and a 0.5 mm-pitch micrometer screw graduated to 5 µm per division. After zeroing on a known optical flat, the operator reads a sagitta of 0.188 mm at the disc centre and wants to know the current radius of curvature, plus what reading would correspond to the low and high ends of a useful grinding range.
Given
- l = 60.00 mm
- hnominal = 0.188 mm
- Rtarget = 2400 mm
Solution
Step 1 — at the nominal measured sagitta of 0.188 mm, plug into the formula:
The mirror is currently far too shallow — radius is 3192 mm against a target of 2400 mm. The grinder needs to deepen the curve.
Step 2 — at the low end of useful grinding sagitta, say 0.150 mm (a very shallow early stage):
At this reading the mirror is barely curved — focal length would be 2000 mm, useless for a Newtonian of this aperture. The 38 µm difference between 0.150 mm and 0.188 mm sagitta translates into 800 mm of radius difference. That is the geometry penalty for working at long radii — drum resolution dominates the answer.
Step 3 — at the high end, when the operator hits target sagitta:
So the operator needs to deepen the sagitta from 0.188 mm to 0.250 mm — a further 62 µm of glass removed at centre. At this short-radius end the reading is robust: a 5 µm drum misread shifts radius by only ~50 mm, against ~85 mm at the nominal point and ~130 mm at the shallow end.
Result
Current radius is 3192 mm against a target of 2400 mm — the operator has another 62 µm of centre depth to grind. Across the range, a sagitta of 0.150 mm reads as a 4000 mm radius (mirror nearly flat, useless), 0.188 mm reads as 3192 mm (mid-grind), and 0.250 mm hits the 2400 mm target — so the working sweet spot is the deeper end of that range, where drum resolution buys you the most stable answer. If your computed radius drifts shot-to-shot by more than 30-50 mm at this geometry, suspect three things in this order: a dust speck or grit particle trapped under one leg lifting the frame by 5-10 µm, leg-tip wear that has rounded the points and shifted the effective leg-spacing l (recheck l on a calibrated steel ball), or thermal drift in the brass frame from hand contact — let the instrument equilibrate on the bench for 10 minutes before a critical reading.
When to Use a Spherometer and When Not To
Spherometers compete with several other ways to measure radius of curvature. Each tool fits a different operating range and budget. The choice usually comes down to whether you need sub-micron interferometric accuracy on a finished optic or a fast, repeatable shop-floor reading during grinding.
| Property | Spherometer | Optical test plate (Newton's rings) | Phase-shifting interferometer |
|---|---|---|---|
| Radius accuracy on a 2400 mm mirror | ±20-50 mm (±1-2%) | ±2 mm (matched plate) | ±0.5 mm (±0.02%) |
| Capital cost | $80-$600 | $200-$2000 per matched plate | $30,000-$150,000 |
| Measurement time per part | 20-40 seconds | 1-2 minutes plus interpretation | 2-5 minutes including setup |
| Surface finish required | Any — works on rough-ground glass | Polished only | Polished only |
| Operator skill | Low — touch and arithmetic | Moderate — fringe interpretation | Moderate — software-driven |
| Best application fit | Grinding-stage radius checks, mould tooling | Production matching to a master | Final figure verification on finished optics |
Frequently Asked Questions About Spherometer
On long-radius surfaces the formula amplifies any sagitta error by a factor of l2/(6h2). At a 0.2 mm sagitta with 60 mm leg-spacing, that amplification factor is around 15,000 — meaning a single 5 µm drum misread becomes a 75 mm radius swing. The two most common non-obvious causes are screw backlash (always approach the surface from the same direction, never reverse during a reading) and frame-flex from leaning your hand on the drum. Lift the instrument cleanly, rotate the drum with thumb and forefinger only, and average three readings.
Use the larger leg-spacing whenever the part diameter allows it. The sagitta scales with l2, so a 100 mm leg-spacing gives four times the sagitta of a 50 mm spacing on the same surface. That moves your reading from the noisy bottom of the drum range up into the well-conditioned middle. The hard limit is that all three legs must sit comfortably inside the polished or ground zone of the part — never on the bevel. For a 150 mm mirror, a 100 mm leg-spacing is fine; for a 75 mm mirror you are stuck with the 50 mm instrument.
The h/2 correction term in the radius formula flips sign depending on whether you measure on the concave or convex side. Many beginners drop this term entirely because it looks small. On a long-radius optic that is fine, but on a 50 mm radius watch crystal the term is 0.25 mm — a real fraction of the answer. Re-derive with the correct sign: R = l2/(6h) + h/2 for concave (legs on the rim, screw drops to the surface), and R = l2/(6h) − h/2 for convex (legs on the rim, screw rises off the apex). That alone usually closes the discrepancy.
The leg-spacing l is almost certainly not exactly what the instrument's nameplate says. Manufacturing tolerance on a hand-built brass spherometer is often ±0.1 mm on l, and l2 means a 0.2% error on l becomes a 0.4% error on R — close to the discrepancy you describe. Calibrate l directly: place the spherometer on a precision steel ball of certified diameter (a bearing ball works well), measure the sagitta, and back-calculate l from the formula using the ball's known R. Engrave the corrected l on the frame and use it forever.
Only crudely. The instrument assumes the surface under the three legs and the central tip is a section of a perfect sphere. On a parabola the local radius at the centre differs from the local radius at the 70% zone by a few percent, so a single spherometer reading averages those zones into a meaningless mean. For figuring you switch to a Foucault knife-edge test or a Ronchi grating, which resolve zonal radius differences directly. The spherometer is a rough-grinding and sphere-stage tool — retire it once you start parabolising.
Look at the tips under a 10× loupe. Fresh tips show a sharp conical point; worn tips show a small bright flat 0.1-0.3 mm across. A flat that size moves the effective contact point inward by roughly half its diameter, shrinking the effective l by 0.1-0.3 mm. Re-running the steel-ball calibration described above will quantify the shift. If the corrected l has drifted more than 0.2 mm from nameplate, send the instrument out for tip regrinding or replace it — re-pointing in a home shop almost never restores all three legs to the same height.
References & Further Reading
- Wikipedia contributors. Spherometer. Wikipedia
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