An Ellipsograph Turntable is a draughting instrument that traces a true mathematical ellipse by sliding two pins of a rigid arm along two perpendicular guide slots cut into a rotating base plate. Pattern-makers, jewellers and architectural draughtsmen rely on it whenever a freehand or template ellipse will not hold its true shape. As the arm rotates, the pins constrain the geometry so a pen at the arm's end traces an ellipse whose semi-axes equal the pin-to-pen distances. The result is a drawn curve accurate to the slot tolerance — typically within 0.1 mm on a quality Stanley or Haff instrument.
Ellipsograph Turntable Interactive Calculator
Vary slider fit and pen force to estimate how slot clearance and arm deflection affect the traced ellipse accuracy.
Equation Used
This calculator estimates ellipsograph tracing error from the article example: slot clearance is the slot width minus the slider diameter, and the pen wander is taken as roughly the same amount. Arm deflection is scaled from the 200 g reference load and added as a conservative total error.
- Slot walls are parallel and the slider is square in the slot.
- Positive slot clearance produces pen wander roughly equal to the clearance.
- Arm deflection scales linearly with pen force near the 200 g reference load.
- Slot perpendicularity and bent-arm errors are not included.
How the Ellipsograph Turntable Works
The mechanism is a trammel of Archimedes mounted on a rotating turntable. Two perpendicular slots are machined into the base, and a rigid bar — the trammel arm — carries two sliders that ride in those slots. A pen is fixed at one end of the arm, beyond both sliders. As you rotate the arm by hand, slider A is forced to travel only along the X-slot and slider B only along the Y-slot. That double constraint forces the pen tip into an ellipse, mathematically. No fudging, no approximation. The semi-major axis equals the distance from the pen to the far slider, and the semi-minor axis equals the distance from the pen to the near slider.
The quality of the drawn ellipse depends entirely on the slot fit. If the sliders rock in the slots — say a 6.0 mm slider in a 6.15 mm slot — the pen wanders off the true curve by roughly the same amount, and you'll see a faint wobble or a closing-error gap where the pen comes back to its start point. The slots must be parallel-walled, the sliders ground square, and the trammel arm must be stiff enough that the pen pressure does not flex it. On a typical 300 mm draughting instrument the arm is brass or steel section about 4 × 12 mm, sized so that 200 g of pen-tip pressure produces less than 0.05 mm of deflection.
Failure modes are predictable. Worn slots produce an out-of-true closing point. A bent trammel arm produces a curve that is elliptical but tilted off the intended axes. Slider play in the vertical direction lets the pen lift unevenly, producing a broken ink line. And if the two slots are not exactly 90° to one another — a defect you'll see on cheap imports where the slots are routed rather than precision-milled — the traced figure is not a true ellipse at all but a distorted oval whose foci do not lie where the geometry says they should.
Key Components
- Turntable Base Plate: The flat reference plate carrying the two perpendicular guide slots. Typically 200-400 mm square, machined from cast iron, brass or stabilised hardwood. Slot perpendicularity must hold to within 0.05° or the traced figure deviates from a true ellipse.
- Cross Slots (X and Y guides): Two parallel-walled channels cut at exactly 90° to one another, intersecting at the centre of the plate. Slot width is typically 6.0 mm with a sliding fit of H7/g6 — about 0.012 mm clearance — to keep slider rock below the line-width of a 0.3 mm drafting pen.
- Sliders (Trammel Pins): Two ground-steel or brass blocks that ride in the slots and pivot on the trammel arm. Each slider must move freely along its slot but not lift or yaw. Typical slider length is 12-15 mm so the contact patch resists tipping under pen pressure.
- Trammel Arm: The rigid bar carrying the two sliders and the pen holder. The pen-to-near-slider distance sets the semi-minor axis; pen-to-far-slider sets the semi-major axis. Both distances are usually adjustable on a graduated scale reading to 0.5 mm.
- Pen or Pencil Holder: Holds a drafting pen, lead, or scribe at the end of the arm. Spring-loaded vertical compliance of 1-2 mm absorbs paper thickness variation without lifting the pen off the sheet during the arc.
- Adjustable Stops or Locks: Thumb-screw clamps that fix the slider positions on the arm once the desired axes are dialled in. They must hold against repeated drawing strokes — a slip of 0.1 mm during the trace produces a visible step in the ellipse.
Industries That Rely on the Ellipsograph Turntable
Anywhere a true ellipse is needed on a drawing or workpiece, an Ellipsograph Turntable beats a string-and-pins method or a plastic ellipse template. Templates only come in fixed sizes and ratios, and the string method depends on knot accuracy and paper friction. The turntable gives you any axis ratio, any size within the arm's range, and a curve that closes on itself within the slot tolerance. You see them on real benches in a handful of trades.
- Architectural draughting: Drawing elliptical arch springers and oval window heads on heritage restoration drawings, where a Haff 590 ellipsograph holds the true curve for arches up to 600 mm major axis.
- Jewellery and silversmithing: Laying out elliptical bezels and oval lockets on a brass plate before piercing, using a small bench ellipsograph with a scribe in place of the pen.
- Pattern making and joinery: Marking out elliptical table tops, mirror frames and inlaid oval panels on MDF templates at the Robert Thompson 'Mouseman' workshop in Kilburn, North Yorkshire.
- Sheet-metal fabrication: Scribing elliptical flange cutouts where a round pipe meets a cylindrical vessel at an angle — the intersection is a true ellipse, and a turntable scribes it directly onto the parent sheet.
- Stage and theatrical scenery: Drawing full-scale elliptical proscenium reveals on the scenic painter's deck at the Royal Opera House paint frame, where the curve must read true from the auditorium.
- Machine drawing and tooling: Producing isometric ellipses to represent circular features on inclined surfaces in third-angle projection, replacing the standard 35°16' ellipse template when a non-standard view angle is needed.
The Formula Behind the Ellipsograph Turntable
The governing equation tells you exactly where the pen sits as the arm rotates. The two pin-to-pen distances set the semi-major and semi-minor axes directly, and the rotation angle θ sweeps the pen through the full ellipse. At the low end of the typical range — say a 20 mm minor axis — slider play and pen line-width dominate the error budget, and you'll struggle to hold better than 1% accuracy. Around the nominal sweet spot of 80-200 mm axes, slot tolerance becomes negligible compared to the ink line itself. At the high end, beyond about 400 mm major axis, arm flex and pen-pressure variation start to show as a visible thinning of the line on the long sweeps.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| x, y | Pen-tip coordinates relative to the slot-intersection centre | mm | in |
| a | Semi-major axis — distance from pen to far slider on the trammel arm | mm | in |
| b | Semi-minor axis — distance from pen to near slider on the trammel arm | mm | in |
| θ | Rotation angle of the trammel arm measured from the X-slot | rad or ° | rad or ° |
Worked Example: Ellipsograph Turntable in a luthier laying out a classical guitar rosette
A luthier in Granada is laying out the elliptical inlay ring around the soundhole of a classical guitar. The design calls for a true ellipse with a 95 mm major axis and a 70 mm minor axis, scribed into a 2 mm spruce top before the rosette channel is routed. The luthier sets the trammel arm of a bench ellipsograph so the pen-to-far-slider distance is 47.5 mm (a) and the pen-to-near-slider distance is 35.0 mm (b), then rotates the arm a full 360° to scribe the curve.
Given
- a = 47.5 mm
- b = 35.0 mm
- Slot width = 6.00 mm
- Slider width = 5.98 mm
Solution
Step 1 — at the nominal rotation θ = 45°, the pen sits at:
Step 2 — at θ = 0° (low end of the sweep, pen on the major-axis end), the pen sits at the extreme of the major axis:
This is where slot play matters most. With a 5.98 mm slider in a 6.00 mm slot, the slider can rock 0.02 mm laterally, and at full extension that translates to roughly 0.04 mm of pen wander — invisible against a 0.3 mm ink line, perfectly acceptable for a rosette.
Step 3 — at θ = 90° (high end, pen on the minor-axis end), the pen sits at:
The closing-error check happens back at θ = 360°. If the pen lands more than 0.1 mm off its starting point, the slots are not truly perpendicular or the arm has flexed during the sweep. On a quality instrument with milled slots, closing error sits below 0.05 mm. On a budget plastic ellipsograph the same trace can close 0.3-0.5 mm off, which shows as a visible step in the inked line.
Result
The traced ellipse has a 95 mm major axis and 70 mm minor axis, with the pen passing through (33. 59, 24.75) mm at the 45° point. In practice the luthier sees a clean continuous scribe line that closes on itself within the width of the scribe point — about 0.08 mm on a sharpened steel scriber. At small axes (20 mm range) slot play dominates and you'll see closing errors approaching 0.2 mm; in the 80-200 mm sweet spot the curve closes within line-width; above 400 mm the arm starts to flex and the line thins on the long sweeps. If your measured curve is off, the three usual culprits are: (1) slot non-perpendicularity — check with a precision square across the slot walls, (2) a bent trammel arm causing axis tilt rather than scale error, or (3) a loose slider clamp letting one axis dimension drift mid-sweep.
Ellipsograph Turntable vs Alternatives
An Ellipsograph Turntable is one of three practical ways to draw a true ellipse. Each has a clear envelope where it wins and a clear envelope where it loses. Pick on accuracy required, axis size, and how often you need to repeat the same ellipse.
| Property | Ellipsograph Turntable | String-and-Two-Pins Method | Plastic Ellipse Template |
|---|---|---|---|
| Accuracy on closing point | ±0.05-0.1 mm on quality instrument | ±0.5-2 mm depending on string stretch | ±0.2 mm but only at the fixed template ratio |
| Range of axis sizes | 10-600 mm continuous, any ratio | 50 mm to several metres, any ratio | Fixed sizes only, typically 6-150 mm |
| Setup time per ellipse | 1-2 minutes to dial in axes | 3-5 minutes to set focal pins and string | Seconds — pick the template |
| Cost | £150-800 for a precision instrument | Pennies — string and two pins | £10-40 for a full template set |
| Repeatability across multiple traces | Excellent once locked, <0.1 mm drift | Poor — string stretches each pull | Perfect for the fixed sizes available |
| Best application fit | Custom ellipses, repeated production work | One-off large architectural curves | Standard isometric and small detail work |
Frequently Asked Questions About Ellipsograph Turntable
The closing-error step is almost always slot non-perpendicularity rather than slider play. If the X and Y slots are 89.5° instead of a true 90°, the figure traced is not a closed ellipse but an open curve that misses its start by roughly the angle error times the major axis. On a 100 mm ellipse, half a degree of slot error gives you about 0.9 mm of closing miss.
Check by laying a precision square across both slots. If the slots are correct, look next at slider clamp slip — a thumb-screw that loosens 0.1 mm during the sweep will produce the same symptom because one axis is no longer a constant.
No — the 35°16' isometric template is a fixed 1 : cos(35°16') ≈ 1 : 0.816 ratio, which is roughly 1.225:1. A 1.5:1 needs a different template ratio entirely, and scaling a template doesn't change its ratio anyway, only its size.
Either buy a multi-ratio template set that includes 1.5:1, or use the Ellipsograph Turntable — set a = 150 mm and b = 100 mm and it traces exactly the curve you need. For 300 mm major axis you'll want an instrument with at least a 350 mm arm, like the Haff 591 or equivalent.
The geometry is mathematically exact regardless of size. What breaks down is the physical realisation. Below about 15 mm minor axis, the two sliders are so close together that any rock in either one dominates the curve — you'll trace a wobbly oval rather than a clean ellipse.
Above about 400 mm major axis, the trammel arm itself becomes the limit. A 4 × 12 mm brass arm at 500 mm length deflects roughly 0.3 mm under typical pen pressure, which shows as a slightly flattened curve at the major-axis extremes. For large architectural ellipses, switch to a string method or a beam-trammel with a much stiffer section.
That's a bent trammel arm or, more commonly, a misaligned pen holder on the arm. The geometry assumes the pen, the near slider pivot, and the far slider pivot all lie on a single straight line. If the pen is offset perpendicular to that line by even 1 mm, the major axis of the traced ellipse rotates off the slot axis by an angle equal to roughly atan(offset / a).
Sight along the arm with the pen down to check linearity, and re-seat the pen holder square. The slots themselves are not the cause here — slot misalignment distorts the curve, it doesn't rotate it cleanly.
For that size and a repeat-production job, the turntable wins. The slots-and-sliders configuration locks the geometry once you set it, so every panel comes out identical to within slot tolerance. A beam-trammel relies on you holding two pins against the paper and is more vulnerable to operator drift across a production run.
The exception is if you need to scribe directly onto the glass rather than transfer from a paper pattern. Then a beam-trammel with a glass-cutter wheel at the working end has the reach and rigidity to score the glass in one pass; the turntable is paper-and-board oriented.
Near 45° both sliders are moving simultaneously, so any backlash between slider and arm pivot adds to position error in both axes at once. A loose pivot pin that gives 0.1 mm in each direction shows up as roughly 0.14 mm of combined error at 45° — about 1% on a 20 mm axis but invisible on a 200 mm one.
Check by drawing the same ellipse twice in opposite rotation directions. If the two curves diverge near 45° but coincide at 0° and 90°, you've found pivot backlash. Tighten the slider-to-arm pivots or replace worn bushings.
You can, and small geared motors at 5-15 RPM work well, but be aware the pen-tip linear velocity is not constant — it varies between b·ω at the major-axis ends and a·ω at the minor-axis ends. For a 95×70 mm ellipse at 10 RPM that's roughly 73 mm/s versus 50 mm/s, a 45% swing.
Inks and scribes that depend on contact time will lay down a thicker line at the slow points and a thinner line at the fast points. Either accept the variation, use a felt-tip that is insensitive to speed, or program the motor to vary speed inversely with the instantaneous tangent length to keep ink-laydown constant.
References & Further Reading
- Wikipedia contributors. Trammel of Archimedes. Wikipedia
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