A geared ellipsograph is a draughting instrument that traces a true mathematical ellipse using a small pinion gear rolling inside a fixed internal ring gear at exactly a 2:1 ratio. The principle traces back to Gerolamo Cardano in the 16th century and was refined into precision drawing instruments by makers like Stanley of London in the 1800s. Any point on the pinion follows a perfect ellipse — the Cardan-circle theorem — so the pen carrier rides on a radial slot and draws the curve in one continuous stroke. Engineers used these for cam profiles, optics, and architectural ovals long before CAD existed.
Geared Ellipsograph Interactive Calculator
Vary the ring pitch diameter, pen offset, and backlash to see the ellipse axes, area, and cusp-risk response.
Equation Used
The Cardan-circle geometry uses a pinion of radius r rolling inside a fixed internal ring at a 2:1 diameter ratio. With ring pitch diameter D_ring, the pinion radius is D_ring/4. A pen mounted at offset e traces an ellipse with semi-major axis a = e + r and semi-minor axis b = |e - r|. Backlash is compared with the article's 0.05 mm visible-cusp threshold.
- Internal ring gear is exactly 2:1 relative to the pinion pitch diameter.
- Pinion radius r is one quarter of the fixed ring pitch diameter.
- Pen offset e is measured from the pinion centre along the carrier slot.
- Backlash risk is scaled to the article threshold of 0.05 mm.
The Geared Ellipsograph in Action
The mechanism rests on a tidy piece of geometry called the Cardan circles or hypocycloidal straight-line motion. Roll a small circle of radius r inside a fixed ring of radius 2r and any point on the small circle's circumference traces a straight line — a diameter of the big circle. Pick a point that is NOT on the circumference but offset somewhere on the pinion disc, and that point traces an ellipse. The 2:1 internal gear ratio is the whole game. Get the ratio wrong by even a tooth and the curve closes into a rosette or a limaçon instead of an ellipse, which is exactly what you see if a worn pinion skips engagement on an old Stanley instrument.
In a working geared ellipsograph the fixed internal ring gear is mounted to the baseplate and centred on the ellipse centre. A pinion with exactly half the pitch diameter meshes inside it. A pen carrier sits on the pinion at an adjustable eccentricity — slide it toward the pinion centre to shrink the semi-minor axis, slide it past the pitch circle to grow the semi-major axis. The semi-major axis a equals the distance from pinion centre to pen plus the pinion radius, and the semi-minor axis b equals that distance minus the pinion radius. That single sliding adjustment lets one instrument draw any ellipse from a near-circle down to a thin sliver.
Tolerances matter more than people expect. Backlash above about 0.05 mm at the gear mesh shows up as a visible kink at the four points where the pen reverses direction — you'll see a tiny cusp on the major and minor axes. A bent pen carrier arm or a worn central bearing produces an ellipse that is no longer symmetric about its axes; the curve looks slightly egg-shaped, called the limaçon error. The bore of the pinion shaft must be ground concentric with the pitch circle to within roughly 0.02 mm, otherwise the ellipse precesses each revolution and you draw a fat band instead of a single line.
Key Components
- Fixed Internal Ring Gear: The stationary outer gear with internal teeth, mounted concentric with the ellipse centre. Pitch diameter is exactly twice the pinion's. On a Stanley-pattern instrument the ring is typically 100-200 mm pitch diameter with 80-120 internal teeth and ground to AGMA Q10 or better.
- Pinion Gear: The rolling internal gear — half the ring's pitch diameter. It rotates about its own axis while its centre orbits the ring centre at half the input crank speed. Tooth profile must match the ring exactly; mismatched modules cause periodic chatter at every tooth pass.
- Pen Carrier Slot: A radial slot or T-track on the pinion face that holds the pen or pencil at an adjustable distance e from the pinion centre. The eccentricity e directly sets the difference between semi-major and semi-minor axis. Slot straightness must be within 0.02 mm over its length or the ellipse develops a wobble.
- Drive Crank: The handle that orbits the pinion around the ring. One full crank rotation produces one full ellipse. On precision instruments the crank runs on a ball bearing pressed into the baseplate to keep the orbit centred within ±0.01 mm.
- Baseplate and Centring Pin: Holds the ring gear rigidly and provides the registration point for the ellipse centre on the drawing. The centring pin drops into a pricked dot on the paper and locates the figure within roughly 0.1 mm of the intended position.
Who Uses the Geared Ellipsograph
Anywhere you needed a true ellipse before computer-driven plotters arrived, a geared ellipsograph earned its keep. The instrument shows up in surprising places — optical lens layouts, stage-set design, ship hull lofting, cam profile generation. The reason is simple: a string-and-pin construction gives you only two foci of one specific eccentricity, and a French curve is just a guess. A geared ellipsograph draws the exact curve in one continuous, repeatable stroke, which matters when the part it represents has to be machined to fit.
- Optical Instrument Design: Layout of elliptical mirrors and off-axis paraboloid templates at workshops like Carl Zeiss Jena, where draftsmen used geared ellipsographs to mark the projected outline of tilted circular optics on flat drawings.
- Architectural Drawing: Drawing elliptical arches, vaults, and oval rooms — the elliptical staircase plans for buildings like the Vatican Museums spiral were laid out with geared ellipsographs by mid-20th-century restoration draftsmen.
- Cam and Gear Design: Marking elliptical cam profiles for textile machinery and elliptical gears used in instruments and printing presses, including profile templates for Reishauer-style elliptical gear pairs.
- Shipbuilding Loft Floors: Lofting elliptical hull station outlines and porthole rings — yards like Harland & Wolff kept large geared ellipsographs in their loft floors for marking ovaloid bulkhead penetrations.
- Stage and Film Set Drafting: Drawing oval procenium openings and trompe-l'œil ellipses on set elevation drawings at studios like Pinewood, where the curve had to be exact so the carpentry shop could cut matching plywood templates.
- Technical Education: Teaching the Cardan-circle theorem in mechanical engineering programs — instruments by Stanley, Riefler, and Coradi survive in university teaching collections at places like the Deutsches Museum.
The Formula Behind the Geared Ellipsograph
What you need to predict before turning the crank is the size of the ellipse you'll get for a given pinion radius r and pen eccentricity e. The relationship is exact and clean. At low eccentricity (e close to zero) the curve collapses toward a straight line along a diameter — useful for the hypocycloidal straight-line trick but not for drawing ellipses. At high eccentricity (e close to r) the ellipse approaches a circle of radius 2r. The sweet spot for instrument work sits at e between roughly 0.3r and 0.8r, where the ellipse has a clear, useful aspect ratio between about 1.3:1 and 4:1.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| a | Semi-major axis of the traced ellipse | mm | in |
| b | Semi-minor axis of the traced ellipse | mm | in |
| r | Pinion pitch radius (half of ring gear pitch radius) | mm | in |
| e | Pen eccentricity — distance from pinion centre to pen tip | mm | in |
| ε | Eccentricity of the resulting ellipse (dimensionless, 0 to 1) | — | — |
Worked Example: Geared Ellipsograph in an antique telescope restoration draughting job
A telescope restoration workshop in Florence is redrawing the layout for the elliptical secondary-mirror mounting flange of an 1890s Amici reflector. The original drawing is missing and the mirror sits at a 45° tilt, so its projected outline on the tube wall is a true ellipse. The pinion in their Stanley geared ellipsograph has a pitch radius r = 50 mm. They need to set the pen eccentricity e to draw the projected outline of a 60 mm circular mirror tilted at 45°, which gives a target semi-major axis a = 30 mm / cos(45°) ≈ 42.4 mm and a target semi-minor axis b = 30 mm.
Given
- r = 50 mm
- a (target) = 42.4 mm
- b (target) = 30 mm
Solution
Step 1 — check whether this ellipse is even reachable on a 50 mm pinion. Required pen eccentricity e = a − r and also e = r − b. Both must give the same number, otherwise no setting works:
The two values disagree, which means a 50 mm pinion cannot draw this exact ellipse — the pinion is too large. The instrument can only draw ellipses centred on r, where a + b = 2r. Here a + b = 72.4 mm, so we need r = 36.2 mm. The restorer swaps to the 36 mm pinion that comes with the Stanley set.
Step 2 — recompute with r = 36 mm at the nominal target. Pen eccentricity:
The 0.4 mm shortfall on b tells the restorer the projected ellipse for a perfectly 45° tilt is slightly off — either the mirror sits at 44.6° or the pinion is 36.2 mm not 36 mm. Either way, set e = 6.4 mm and the ellipse will measure 42.4 × 29.6 mm.
Step 3 — bracket the operating range. At low eccentricity e = 2 mm the instrument draws a 38 × 34 mm ellipse — almost a circle, with ε = 0.45. At the high end e = 30 mm the instrument draws a 66 × 6 mm sliver, ε = 0.996, looking like a flattened needle that barely registers as an ellipse. Our nominal e = 6.4 mm puts the resulting eccentricity at:
That is a comfortable, clearly oval shape — exactly the aspect ratio you'd expect from a 45° projection of a circle, which is the geometric truth being drawn.
Result
Setting the pen carrier to e = 6. 4 mm on the 36 mm pinion produces a 42.4 × 29.6 mm ellipse with eccentricity ε = 0.716. That is the exact projected outline of a 60 mm circle tilted at 45°, ready for the workshop to cut a brass mounting flange to match. Across the operating range the same instrument draws anywhere from a near-circle at e = 2 mm (ε = 0.45) to a needle-thin sliver at e = 30 mm (ε = 0.996); the restorer's setting sits comfortably mid-range where the ellipse is well-defined and the mesh runs smoothly. If your traced ellipse measures 42.4 × 28.8 mm instead of 42.4 × 29.6 mm, the most likely causes are: (1) the pen tip flexing 0.4 mm under drawing pressure — switch to a stiffer ruling pen or reduce ink load; (2) gear mesh backlash exceeding 0.05 mm allowing the pinion to lag at the four reversal points, leaving visible cusps on the minor axis; or (3) the centring pin sitting in an oversized pricked dot, letting the whole ring gear shift mid-stroke and producing a precessed double-line.
Geared Ellipsograph vs Alternatives
Three instruments draw ellipses by hand: the geared ellipsograph, the trammel of Archimedes (the elliptic trammel), and the string-and-two-pins method. They are not interchangeable. Pick on accuracy, ellipse-size range, setup time, and how often the ellipse repeats.
| Property | Geared Ellipsograph | Trammel of Archimedes | String-and-Pins |
|---|---|---|---|
| Geometric accuracy | ±0.05 mm on a quality instrument | ±0.2 mm typical, slot wear adds error | ±0.5 mm, string stretches |
| Ellipse size range (typical) | 10-300 mm major axis (ring-gear bound) | 20-1500 mm major axis | 50-5000 mm major axis |
| Aspect ratio range | 1.0:1 down to ~30:1 | 1.0:1 down to ~10:1 before slot binding | 1.05:1 down to ~5:1 (string sag limits) |
| Setup time per ellipse | ~30 seconds (set e, drop pin) | ~1 minute (set both slot pins) | ~2 minutes (calculate foci, drive pins, knot string) |
| Repeatability over 10 traces | Excellent — same e gives same curve | Good — slot wear shifts curve over time | Poor — string elongation per trace |
| Cost (vintage instrument market) | £400-£2000 (Stanley, Riefler) | £80-£300 (common drafting tool) | Pennies — string and two pins |
| Common failure mode | Gear mesh backlash, pinion bore wear | Slot wear, slider play | String stretch, pin movement |
| Best application fit | Small precise ellipses repeated often (cams, optics) | Mid-size architectural/loft ellipses | Very large one-off ellipses (set design, gardens) |
Frequently Asked Questions About Geared Ellipsograph
That is the limaçon error and it almost always traces back to the pinion bore not being concentric with the pitch circle. If the bore is offset by even 0.02-0.03 mm, the pen orbits a slightly shifted centre while the gear engagement assumes a true centre, so the curve becomes asymmetric about its minor axis — fatter on one end than the other.
Check by running the instrument with a sharp pencil and measuring the distance from the centring pin to the curve at the two ends of the major axis. If they differ by more than 0.1 mm the pinion needs re-bushing. A worn central bearing on the drive crank produces the same symptom and is cheaper to fix — replace the bearing first.
No, and this catches people out. The geometry is hard-locked: a + b = 2r, where r is the pinion pitch radius. The maximum a you can ever draw equals 2r (a circle of pinion-diameter), and any ellipse must satisfy that sum constraint. If your target ellipse has a + b greater than the ring gear's pitch diameter, you cannot draw it on that instrument — period.
That is why precision sets like the Stanley geared ellipsograph ship with three or four pinions of different diameters. Pick the pinion where 2r equals or just exceeds your target a + b, then compute e = a − r.
Those are reversal cusps caused by gear backlash. At the four points where the pen direction reverses (top, bottom, left, right of the ellipse), the meshing teeth briefly unload and the pinion lags by the backlash distance before re-engaging. The pen pauses, then jumps, leaving a tiny visible cusp.
Backlash above 0.05 mm at the gear mesh is the threshold where these become visible to the eye. The fix on a vintage instrument is usually a fresh thin film of grease and tightening any adjustable mesh preload. On a worn instrument the ring gear teeth need re-cutting, which is rarely economic — accept the cusps and ink over them, or sell and buy a tighter instrument.
Trammel for that range. A geared ellipsograph with a 400 mm capacity needs a ring gear of 400 mm pitch diameter — these exist (Coradi made them) but they are rare, heavy, and cost north of £3000 on the second-hand market. A trammel of Archimedes scales to 1500 mm major axis on a desktop and costs a tenth as much.
Where the geared ellipsograph wins is small repeated work below about 200 mm major axis, especially when you need the same ellipse drawn ten or fifty times — cam template work, optical layouts. The trammel's slot wears with each cycle; the geared instrument's gear mesh does not.
Crank the instrument through one full turn with the pen down and watch whether the curve closes cleanly on itself. A true 2:1 ratio gives a closed ellipse after exactly one crank revolution. Any other ratio produces an open hypotrochoid — a rosette pattern that takes many revolutions to close, or never closes if the ratio is irrational.
Even a 2.05:1 mismatch (one extra tooth somewhere) makes the second trace land 0.5-1 mm offset from the first. If you crank twice and the second pass does not lie exactly on top of the first, the ratio is wrong. Count teeth on both gears — ring teeth must be exactly twice pinion teeth.
Periodic chatter usually means a tooth-form defect or a single damaged tooth. Each time that tooth comes into mesh it loads up differently than its neighbours, the pinion jolts, and the pen jumps. Count how many chatter points you see per revolution — if it equals the pinion tooth count, the bad tooth is on the pinion; if it equals the ring tooth count, it's on the ring.
The other common cause is a high spot on the baseplate under the paper. The pen height varies as the carrier orbits, and where the paper sits on the high spot the pen presses harder and either skips or floods ink. Check baseplate flatness with a straightedge — anything above 0.1 mm of bow across 200 mm causes visible variation.
References & Further Reading
- Wikipedia contributors. Ellipsograph. Wikipedia
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