Trammel for Drawing Ellipses: How the Archimedes Mechanism Works, Parts and Uses

How the Trammel for Drawing Ellipses Works

The trammel works because of a parametric identity hiding in plain sight. Lock one slider to the X-slot and one to the Y-slot, connect them with a rigid bar, and pick any point on that bar at distance a from the X-slider and distance b from the Y-slider. As the bar rotates and the sliders glide, that point traces x = a·cos θ and y = b·sin θ — the textbook parametric form of an ellipse with semi-major axis a and semi-minor axis b. No approximation, no four-centre cheat. Move the marking tip closer to the X-slider and the ellipse stretches along Y. Move it past both sliders and you get a larger ellipse with the slider positions inside the curve.

The geometry only stays true if the two slots cross at exactly 90°. Slot a cross-piece at 89° instead of 90° and the traced curve is no longer a true ellipse — it becomes a skewed conic, and on a 200 mm major axis you will see the error as a 1.5 mm bump where the curve closes. The slider-to-slot fit matters just as much. Loose sliders chatter and the marking tip wanders by 0.3-0.5 mm at the ends of the major axis, where slider velocity peaks and any clearance shows up as backlash.

Failures come from three places. The arm flexes if it is too long and too thin, which throws off the a and b distances under marking pressure. The slot ends crash into the slider housings if you size the slots shorter than 2a and 2b — the slider must travel the full ±a or ±b, not less. And a sloppy pivot between the marking tip and the arm lets the tip cock over and scribe a fat, fuzzy line instead of a sharp one. A 6.0 mm tip pin in a 6.1 mm hole is fine; a 6.0 mm pin in a 6.3 mm hole gives you a 0.3 mm wobble that prints straight onto the work.

Key Components

• Cross-slot Base: Two perpendicular slots machined into a single plate, intersecting at exactly 90°. The slot length on each axis must be at least twice the longest semi-axis you plan to draw — for a 300 mm × 200 mm ellipse that means slots of 300 mm and 200 mm minimum. Squareness tolerance should be held to ±0.05° or the traced curve drifts off true.
• Sliders (Shoes): Two pivoting blocks that run in the slots. Each carries a small post that connects to the arm. Slider clearance in the slot needs to sit at 0.05-0.10 mm — tight enough to kill backlash, loose enough that the slider does not bind at the slot ends where dust collects.
• Trammel Arm (Beam): The rigid bar joining the two sliders and carrying the marking tip. Stiffness matters more than length — a 12 mm × 25 mm hardwood or aluminium bar resists flex under typical 2-3 N marking pressure. The two slider posts and the marking tip lie on a single straight line; collinearity within 0.1 mm is the hard rule.
• Marking Tip Holder: A pin or sleeve at the chosen radius along the arm, carrying a pencil, scriber or knife. The holder must fit the pin with no more than 0.1 mm play, and the tip itself must register against the work face — not float above it — or the line thickness varies along the curve.
• Adjustable Tip Position: The mechanism for shifting the marking tip along the arm to set a and b. On precision trammels this is a calibrated slot with a clamp screw; on shop-built versions it is a series of drilled holes at known increments. Position accuracy directly sets the ellipse accuracy — a 0.5 mm error in a maps to a 0.5 mm error in the major axis.

Who Uses the Trammel for Drawing Ellipses

Anywhere a true ellipse needs to land on real material at full scale, the trammel earns its place. CAD plotters and CNC routers handle the digital case, but on plywood, sheet steel, stone and leather, you still want one pass with a physical tool. The same instrument shows up in classrooms as a kinematics demonstrator — the famous "do-nothing machine" novelty toy is a trammel with the marking tip removed.

• Boatbuilding: Lofting elliptical transom outlines and oval porthole cutouts at full scale on plywood loft floors, used routinely in traditional yards like Brooklin Boat Yard in Maine for cold-moulded hulls.
• Architectural Stonework: Laying out elliptical arch templates for Georgian and Regency restoration work — Historic Royal Palaces conservation teams use plywood trammels when scribing replacement keystones for elliptical openings at Hampton Court.
• Sheet Metal Fabrication: Marking elliptical duct transitions and oval access hatches on flat stock before shearing. HVAC fabricators cutting flat-to-flat oval transitions use a steel-armed trammel with a scriber tip.
• Sign and Display Making: Layout of elliptical signboards, frame routing templates and oval cabinet mirrors. Signpainters working in the British pub-sign tradition still use brass trammels for one-off layouts.
• Furniture Pattern Making: Drawing elliptical tabletop and inlay outlines on MDF templates for shaper and router work — Krenov-style cabinet shops use compact aluminium trammels for ellipses up to 1200 mm.
• Engineering Education: Demonstrating parametric curve generation in mechanism design courses. The MIT Hobby Shop and many UK university workshops keep a wooden trammel on the bench as a kinematics teaching aid.
• Landscape and Garden Design: Setting out elliptical flower beds, ponds and lawn shapes at site scale using a rope-and-stake trammel — Capability Brown's surveyors used the same principle in the 18th century.

The Formula Behind the Trammel for Drawing Ellipses

The governing equation is the parametric form of an ellipse, with the slider angle θ as the parameter. What matters in practice is what changes as you adjust the marking tip position along the arm. Slide the tip close to the inner slider and a shrinks toward zero — you draw a near-flat sliver. Slide it out past both sliders and a and b both grow, but so does the arm overhang and the flex error. The sweet spot for most shop work sits with the tip beyond both sliders by 30-50% of the longer slot length, where the arm geometry stays stiff and the slider posts carry useful load instead of fighting each other.

Variables

Worked Example: Trammel for Drawing Ellipses in a heritage railway carriage shop

A heritage railway carriage shop at the Bluebell Railway in Sussex is laying out a full-scale plywood template for an elliptical droplight window in a restored Maunsell-design coach side. The window opening calls for a 600 mm major axis and a 360 mm minor axis, so the trammel must trace an ellipse with a = 300 mm and b = 180 mm. The shop's trammel has a 1000 mm aluminium arm with adjustable tip positions in 5 mm increments, and the cross-slot base has 700 mm and 500 mm slots machined at 90.0° ± 0.03°.

Given

• a = 300 mm
• b = 180 mm
• Slot length X = 700 mm
• Slot length Y = 500 mm

Solution

Step 1 — set the marking tip position. The tip must sit at distance a = 300 mm from the Y-slider and distance b = 180 mm from the X-slider, with both sliders and the tip collinear on the arm. The two sliders therefore sit 300 − 180 = 120 mm apart on the arm:

Step 2 — verify slot travel. Each slider must traverse ±a or ±b through its slot. The Y-slider needs 2a = 600 mm of travel; the X-slider needs 2b = 360 mm. The shop's 700 mm and 500 mm slots both clear the requirement with 50-70 mm of overrun at each end, which is exactly what you want — enough margin that the slider never crashes into the slot end at the extreme positions where the marking pressure peaks.

Step 3 — predict the traced curve at the nominal setting. With θ swept through 0 to 2π, the tip plots:

This gives a 600 × 360 mm ellipse — exactly the droplight outline. At the low end of the shop's typical operating range, suppose the same trammel is used for a small 200 × 120 mm porthole template with a = 100 mm, b = 60 mm. The arm geometry stays well inside the slot envelope and the traced curve closes to within 0.2 mm at the seams, because slider velocity stays low and clearance does not show up. At the high end, push the trammel to its limit with a 700 × 500 mm ellipse (a = 350 mm, b = 250 mm). The Y-slider now needs 700 mm of travel — exactly the slot length — and the slider crashes into the slot end at θ = 0 and θ = π, leaving a flat spot 1-2 mm long at each end of the major axis. The sweet spot for this trammel lives in the 200-600 mm major-axis range.

Result

The trammel traces a true 600 × 360 mm ellipse in a single pass, with the marking tip set at 300 mm from one slider and 180 mm from the other on a collinear arm. The curve will close cleanly with the seam invisible to the eye — typically under 0.3 mm error on a well-built trammel. Compare that to the 200 × 120 mm porthole case where error stays under 0.2 mm, and the 700 × 500 mm extreme case where the slider runs out of slot and the major-axis ends flatten by 1-2 mm. If the finished line shows a noticeable bulge at one end of the major axis, the slot squareness has drifted off 90° — measure it with a machinist's square. If the line is fuzzy or doubled, the slider clearance in its slot has opened past 0.15 mm from grit or wear, and you need to clean and re-shim. If the ellipse closes short by 1-2 mm overall, the arm has flexed under marking pressure — drop your scriber pressure or stiffen the arm with a backing rib.

Trammel for Drawing Ellipses vs Alternatives

Drawing an ellipse on real stock has three practical methods, and each one wins in a different size and accuracy band. The trammel is mechanical and analogue; the string-and-pins method is older and simpler; CNC plotting is digital and exact but needs the part on a machine bed.

Frequently Asked Questions About Trammel for Drawing Ellipses