A cyclograph is a draughting instrument that traces cycloidal curves, ellipses, and other conic sections directly onto a drawing without the user plotting points by hand. Mechanical engineers and toolmakers depend on it when laying out cam profiles, isometric circles, and shaft cross-sections in technical drawings. The device uses a guided arm or pin-and-slot linkage so the pencil follows a defined locus as you move the handle. The outcome is a smooth, mathematically correct curve drawn in seconds — work that would take 20 minutes by ordinates.
Cyclograph Interactive Calculator
Vary the trammel arm length, pencil offset, and rotation angle to see the traced ellipse point and semi-axis dimensions.
Equation Used
The trammel of Archimedes constrains two sliders in perpendicular slots. With arm length L and pencil offset d from the horizontal slider, the pencil traces an ellipse with semi-axes a = L - d and b = d; theta sets the current point on the ellipse.
- Slots are perfectly perpendicular.
- Trammel arm is rigid.
- Slider clearance and pencil play are ignored.
- Theta is entered in degrees.
The Cyclograph in Action
A cyclograph constrains a pencil point to a specific mathematical path. The most common workshop variant — sometimes called an ellipsograph or trammel of Archimedes — uses two sliders running in perpendicular slots, with a rigid arm pinned to both sliders and carrying the pencil at a third point. As you swing the arm, the two sliders trade displacement and the pencil tip traces a true ellipse. Change the position of the pencil along the arm and you change the ratio of major to minor axis. That is the whole trick.
The geometry only works if the slot intersection is square to within about 0.1° and the slider clearance is tight — typically 0.02 to 0.05 mm of play. If the slots wear or the sliders rattle, the traced ellipse picks up a visible kink near the axis crossings. You see it as a flat spot on what should be a smooth curve. Worn cyclographs from the 1950s draughting room at places like Vickers-Armstrongs would often get rebuilt because of this exact symptom — sliders polished oversize, slots galled.
The cycloidal version uses a different linkage: a small rolling disc inside or outside a larger circular guide, with the pencil offset from the disc centre. That offset is what makes the path a cycloid, an epicycloid, or a hypocycloid depending on which side of the guide the disc rolls on. If the rolling disc slips even 0.5 mm during a full traverse, the curve closes off-pattern and you get a visible cusp displacement. Tooth-engaged versions exist — basically a small planetary gear set on a drafting board — and those eliminate slip entirely.
Key Components
- Cross slot frame: Two perpendicular slots machined into a flat brass or aluminium plate. Slot squareness must hold to 0.1° or better, and slot width is typically 6.0 mm with sliders ground to 5.95 mm for a 0.05 mm running fit.
- Sliders (shoes): Two short brass blocks, each carrying a pivot pin. They constrain the trammel arm endpoints to move only along their respective slot axes. Worn sliders are the number one source of inaccurate ellipses on second-hand instruments.
- Trammel arm: The rigid bar that links both sliders and carries the pencil. Length sets the major axis; the pencil position along the arm sets the minor axis. Adjustment is usually by a sliding collar with a thumbscrew clamping to 1 N·m.
- Pencil holder or stylus: Holds a 0.5 mm lead or a fine ink nib at the locus point. Holder must clamp the lead vertical to within 1° or the line weight will vary as the arm rotates.
- Rolling disc (cycloidal variant): A toothed or friction wheel rolling inside or outside a fixed guide ring. The pencil sits on a radial arm offset from the disc centre. Tooth engagement eliminates slip; friction-rolling versions need a clean, dust-free guide surface.
- Base plate clamps: Hold the cyclograph fixed to the drawing board. Any movement of the base during a traverse shows up directly as a translated curve — usually a 0.3 to 0.5 mm step where you stopped to reposition your hand.
Where the Cyclograph Is Used
Cyclographs earn their keep wherever a draughtsman needs a true ellipse, cycloid, or epicycloid drawn at production-drawing scale. Before CAD, every shaft section in an isometric assembly view came off one of these. Today they survive in restoration shops, classical drafting courses, and a handful of trades where hand-drawn templates still beat plotted output.
- Mechanical engineering drafting: Drawing isometric circles on shaft cross-sections in assembly drawings — a Staedtler Mars ellipsograph set was standard issue at Rolls-Royce Derby through the 1970s.
- Cam and gear design: Laying out cycloidal cam profiles for textile machinery, where the rise-dwell-fall geometry is mathematically a cycloid. Used historically at Northrop Loom Company drafting offices.
- Architectural drafting: Tracing elliptical arches and oval window profiles on heritage building restoration drawings — the Society for the Protection of Ancient Buildings still teaches the technique.
- Marine naval architecture: Drawing the elliptical waterline sections on yacht lines plans where a spline batten can't reach a tight enough radius. Camper & Nicholsons drafting room kept a Haff trammel for exactly this.
- Watch and clockmaking: Laying out epicycloidal gear-tooth profiles on master drawings before the work is photographically reduced for cutter manufacture — Patek Philippe's archive drawings show the technique through the 1960s.
- Education and technical training: Teaching conic section geometry in mechanical drafting courses — the Kern Aarau Swiss instrument sets are still in use at trade schools across central Europe.
The Formula Behind the Cyclograph
The defining equation for a trammel-type cyclograph relates the pencil's traced position to the arm length and the pencil offset along that arm. At the low end of the typical operating range — short arm lengths around 50 mm — you get tight ellipses suitable for small shaft sections, but slot wear shows up as visible flats near the axis crossings. At the nominal range of 100 to 200 mm arm length, the geometry is most forgiving and the curves come out clean. Push the arm beyond 300 mm and the unsupported end deflects under pencil pressure — typically 0.3 to 0.5 mm of sag — distorting the major-axis end of the ellipse. The sweet spot is 120 to 180 mm.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| x | Pencil position along the major axis from centre | mm | in |
| y | Pencil position along the minor axis from centre | mm | in |
| L | Total trammel arm length between slider pivots | mm | in |
| d | Distance from outer slider pivot to pencil point | mm | in |
| θ | Angular position of the trammel arm | rad or ° | rad or ° |
Worked Example: Cyclograph in a vintage motorcycle restoration drawing
A specialist restoring a 1952 Vincent Black Shadow needs to redraw the missing crankcase assembly drawing in third-angle projection. The main shaft sits in an isometric view, so the 60 mm-diameter shaft must appear as an ellipse with a major axis of 60 mm and a minor axis of 34.6 mm (60 × cos 54.74° for the standard isometric foreshortening). The shop has a Haff trammel ellipsograph set up on a drafting board. The draughtsman needs to set the arm length and pencil offset to produce the correct ellipse.
Given
- Major axis (2 × (L − d)) = 60 mm
- Minor axis (2 × d) = 34.6 mm
- Slot squareness tolerance = 0.1 °
Solution
Step 1 — solve for the pencil offset d from the minor axis requirement:
Step 2 — solve for the trammel arm length L from the major axis requirement at the nominal isometric setting:
This is at the low end of the typical Haff trammel range — workable, but slot wear becomes visible as small flats near the four axis crossings if the instrument is older than about 20 years. The drawn ellipse will look correct from a metre away but reveal subtle kinks under a 5× loupe.
Step 3 — check the result at the nominal sweet-spot range. If the same isometric circle were drawn at 2:1 scale (a common detail-view convention), L scales to 94.6 mm and d to 34.6 mm:
This sits squarely in the 80 to 180 mm forgiveness band — slot wear barely shows, arm deflection is negligible, and the ellipse traces clean in a single sweep. At 4:1 scale the arm runs to 189 mm, which is still acceptable but the unsupported pencil end starts to deflect 0.2 mm under typical 2 N drawing pressure. Beyond that — say a wall-chart enlargement at 8:1 — the arm reaches 378 mm and you should switch to a beam compass with an elliptical attachment instead.
Result
The draughtsman sets L = 47. 3 mm and d = 17.3 mm to produce the required 60 × 34.6 mm isometric ellipse at 1:1 scale. At this setting the curve traces in roughly 4 seconds of arm sweep — fast, but in the wear-sensitive zone. Compare across the operating range: at 1:1 (47.3 mm arm) you risk visible kinks from slot slop, at 2:1 (94.6 mm) the ellipse comes out cleanest, and beyond 4:1 (189 mm) arm deflection becomes the dominant error. If the drawn ellipse looks lopsided when measured, the most likely causes are: (1) a loose thumbscrew on the pencil collar letting d drift mid-traverse — check the clamping torque, should be a firm 1 N·m, (2) the base plate not clamped flat to the drafting board, producing a 0.3 mm step where the arm rotated past your hand, or (3) lead deflection from holding the pencil non-vertical, which produces a curve that's correct in shape but offset by half a lead-width on one side.
When to Use a Cyclograph and When Not To
A cyclograph is one of three practical ways to draw a true ellipse on a board. The alternatives are the string-and-pins method (two focal pins and a closed loop) and a printed ellipse template set. Each has a different sweet spot, and the right choice depends on how many ellipses you draw, what size, and how precisely.
| Property | Cyclograph (trammel) | String-and-pins method | Ellipse template set |
|---|---|---|---|
| Accuracy on a 100 mm major axis | ±0.1 mm | ±0.5 mm | ±0.2 mm (nearest stock size) |
| Setup time per ellipse | 30 seconds | 2 minutes | 5 seconds |
| Range of sizes from one tool | 10 mm to 300 mm continuously | 50 mm to 1 m continuously | Fixed: only stocked sizes |
| Cost (typical 2024 prices) | $150 to $600 for a Haff or Kern set | Effectively zero | $40 to $120 for a 30-template set |
| Wear-related accuracy loss | Slot/slider wear after ~10 years heavy use | None | Edge nicks reduce smoothness over time |
| Best application fit | Production drafting, repeat ellipses at varying sizes | One-off large ellipses, layout work | Quick small ellipses at standard sizes |
| Skill required | Moderate — must set arm length correctly | Low — geometry self-corrects | Minimal — pick template, trace |
Frequently Asked Questions About Cyclograph
This is almost always slider clearance. As the arm rotates through 0°, 90°, 180° and 270°, one slider reverses direction. If there is more than about 0.05 mm of play in the slot, the slider rocks at reversal and the pencil dwells for a fraction of a millimetre before the new direction takes over. That dwell shows up as a tiny flat.
Check by gently rocking each slider in its slot — you should feel almost no perceptible movement. The fix on a worn instrument is to fit oversized sliders ground to a 0.02 mm running fit. Cleaning old grease out of the slots also helps; gummed lubricant adds drag that mimics the same symptom.
Rolling slip. On a friction-driven cycloidal cyclograph, even a 0.3% slip over one full revolution leaves the start and end points displaced by roughly 1% of the circumference. On a 100 mm guide ring that's 3 mm of error — a visible mismatch.
Solutions: clean the guide ring with isopropyl alcohol to remove pencil graphite and skin oil, then check the disc for flat spots from storage. If the instrument is a toothed-gear version, look for a chipped tooth — one missing tooth produces a single sharp kink in the otherwise smooth curve at exactly the same angular position every revolution.
Templates win for P&IDs. Pipe sections on process drawings are almost always at standardised nominal sizes — 50 mm, 80 mm, 100 mm, 150 mm — which match the stock template ellipses exactly. You'll draw 200 of them on a single drawing, and the 5-second-per-ellipse template speed crushes the 30-second cyclograph setup time.
Reach for the cyclograph when the diameters are non-standard or the projection angle isn't the standard 30° isometric. A custom oblique view at 35°, for instance, produces ellipses whose minor-axis ratio doesn't match any stock template.
Not reliably. Below 10 mm, the pencil offset d drops below 5 mm — close to the slider width itself. The geometry breaks down because the slider can no longer pass cleanly through the central crossing without the pencil collar fouling the slot frame. You'll see the curve bunch up near the centre on both sides of the minor axis.
For ellipses below 10 mm, switch to a small ellipse template or simply draw the construction by ordinates — eight points around the ellipse and a French curve to connect them gives better fidelity at small scale than any trammel can.
Two causes, usually combined. First, the base plate is shifting fractionally each time you reposition your hand — even 0.1 mm of base drift moves the entire traced curve by that amount, and successive sweeps land on slightly different paths. Re-check that all three or four base clamps are tight against the board.
Second, the pencil lead is rotating in its holder as the arm sweeps. Round leads develop a chisel point on the first traverse, and that point's effective contact moves laterally by half a lead-width on later passes. Use a 0.5 mm clutch pencil with a fresh lead per drawing, or rotate a wooden pencil 90° between sweeps to keep the chisel symmetric.
Draw a test circle. Set L = 2d exactly — pencil offset half the arm length — and the trammel should produce a perfect circle, not an ellipse. Measure the diameter at 0°, 45°, 90° and 135°. All four should match within 0.1 mm on a 100 mm circle.
If the 0°-90° diameter pair matches but the 45°-135° pair is consistently smaller by 0.3 mm or more, your slot squareness has drifted off 0.5° or worse. That's a workshop fix — usually involving a precision square against the slot edges and re-bedding the slot frame. Most users at this point retire the instrument or buy a replacement.
References & Further Reading
- Wikipedia contributors. Trammel of Archimedes. Wikipedia
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