The Trammel of Archimedes is a planar linkage that traces a perfect ellipse using two sliders constrained in perpendicular slots, connected by a rigid rod with a tracing point at one end. The classic novelty version is the wooden "do nothing machine," sold in countless gift shops since the 1940s. It exists to convert simple linear-slot motion into precise elliptical motion without gears, cams, or curved guides. That makes it the simplest known mechanical ellipsograph — accurate, cheap to build, and reliable for centuries.
Trammel of Archimedes Interactive Calculator
Vary the tracing-point distances to the two trammel shoes and see the resulting ellipse axes, area, eccentricity, and animated linkage motion.
Equation Used
The trammel constrains two points on a rigid rod to perpendicular slots. The tracing point follows an ellipse whose semi-major axis is the larger tracing-point-to-shoe distance and whose semi-minor axis is the smaller distance.
- Horizontal and vertical slots are exactly perpendicular.
- The rod is rigid and the slider shoes have negligible play.
- Distances are measured from the tracing point along the rod to each shoe.
Operating Principle of the Trammel of Archimedes
The Trammel of Archimedes, also called the Ellipsograph in drafting circles, works by exploiting a clean piece of geometry: if two points on a rigid rod are forced to slide along two perpendicular lines, every other point on that rod traces an ellipse. The rod is the trammel. The two sliders are sometimes called the trammel shoes or trammel blocks. Extend the rod past either slider and put a pen, a router bit, or a cutting tool there, and you get a perfect ellipse — no curve template, no CNC, no math beyond the geometry of the linkage itself.
The semi-major axis equals the distance from the tracing point to the farther slider. The semi-minor axis equals the distance from the tracing point to the nearer slider. Change those distances and you change the ellipse. That is the entire device. The same mechanism is sometimes called a Trammel Crank when one slider is driven rotationally to power the motion, or a Trammel Gear when the input comes through a gear pair coupled to one of the shoes — both are kinematically the same linkage with different prime movers.
Where builds go wrong is tolerance in the slots. If the two slot axes are not exactly 90° to each other, the curve is no longer an ellipse — it becomes a slightly skewed oval, and a 1° error in slot perpendicularity produces a visible asymmetry on a 200 mm ellipse. Slot width matters too: if the shoe has more than about 0.1 mm of side play in a typical 50 mm trammel, the tracing point wanders by roughly the same amount and you see chatter marks on a router cut. Worn or sticky shoes are the usual failure mode — the rod binds at one end of travel, the tool jumps, and the ellipse comes out lumpy near the apexes.
Key Components
- Cross-slot base: Two perpendicular slots, machined or routed at exactly 90° ± 0.1° to each other. Any squareness error in the slots transfers directly into the traced curve. On a precision Ellipsograph (trammel) used for drafting, the slots are typically ground steel; on a wooden do-nothing machine they are simply routed grooves.
- Trammel rod (beam): The rigid bar carrying both sliders and the tracing point. Length sets the maximum ellipse dimensions. Stiffness matters — a flexible rod under cutting load deflects mid-stroke and rounds off the ends of the ellipse. Aluminium 12 mm × 25 mm bar stock is a common choice for woodworking jigs.
- Sliders (trammel shoes): Two blocks that ride in the slots. Clearance must be tight: 0.05–0.10 mm side play is the practical limit before tracing accuracy degrades. PTFE or bronze shoes outlast plain wood-on-wood by an order of magnitude.
- Adjustable tracing point: Pen holder, router bit mount, or scribe. Distance from tracing point to each slider sets the two ellipse axes independently. A locking screw with a marked scale lets you dial in axis lengths to ±0.5 mm in a typical shop build.
- Drive crank (optional): When used as a Trammel Crank for powered motion — for example in a do-nothing-grinder demonstration — a crank handle rotates one slider's pivot point, which drives the whole linkage at constant input RPM. Output point speed is not constant; it varies sinusoidally around the ellipse.
Where the Trammel of Archimedes Is Used
The Trammel of Archimedes shows up wherever someone needs a true ellipse drawn, cut, or machined without resorting to a CNC path. Drafting offices used it for centuries. Woodworkers still use it for elliptical mirror frames and tabletops. Educational kits use it to demonstrate parametric motion. And the novelty industry has sold it as a desk toy — the so-called do-nothing machine — for nearly a hundred years.
- Drafting and technical drawing: Precision Ellipsograph instruments sold by Keuffel & Esser and Dietzgen from the late 1800s through the 1970s used the trammel principle to draw ellipses on engineering drawings before CAD took over.
- Woodworking: Elliptical mirror frames, oval picture frames, and racetrack-shaped tabletops are routinely cut on shop-built trammel jigs — Rockler and Lee Valley both sell modern versions of the basic Trammel Gear router fixture for ellipses up to about 600 × 400 mm.
- Stone and tile cutting: Stonemasons cut elliptical openings for fireplaces and arched windows using a trammel-guided router or grinder, exactly the same Trammel Crank linkage scaled up to 1.5–2 m beam length.
- Education and STEM kits: The classic wooden do-nothing machine — sometimes called a do-nothing grinder — is sold by Tedco, Channel Craft, and dozens of small woodworkers as a desk toy that demonstrates how circular input motion produces elliptical and linear output simultaneously.
- Mechanical engineering teaching: MIT, Cornell, and Cambridge all maintain Reuleaux-style mechanism collections that include a brass Trammel of Archimedes for teaching planar kinematics and the parametric equations of an ellipse.
- Watchmaking and instrument design: Miniature trammel linkages historically generated elliptical cam profiles for early astronomical clocks and orrery mechanisms, where smooth non-circular motion was needed without complex gear cutting.
The Formula Behind the Trammel of Archimedes
The position of the tracing point as a function of the driving angle is the key formula — it tells you both the shape of the curve and the instantaneous velocity at any point around the ellipse. At low driving speeds (say a hand-cranked drafting Ellipsograph at 10–20 RPM) the tracing point moves smoothly and the linkage's small clearances do not matter. At the typical mid-range of a router-driven woodworking jig (60–120 RPM crank input), the tool's tangential speed changes by a factor of a/b around the ellipse, and you have to slow your feed rate at the minor-axis ends or the bit grabs. Push past 200 RPM input on a wooden trammel and shoe friction heats up, the rod whips, and the ellipse loses its shape near the major-axis tips. The sweet spot for shop work sits around 80–100 RPM crank speed with a 300 mm beam.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| x, y | Coordinates of the tracing point in the plane of the slots | m | in |
| a | Semi-major axis — distance from tracing point to the farther slider | m | in |
| b | Semi-minor axis — distance from tracing point to the nearer slider | m | in |
| θ | Driving angle, measured from the x-axis slot to the rod | rad | rad |
Worked Example: Trammel of Archimedes in a router-driven oval tabletop jig
You are cutting an oval walnut tabletop on a shop-built trammel jig. Target ellipse is 800 mm major axis by 500 mm minor axis. The rod is 600 mm long with the router-bit mount at one end. You want to know the tool tip's tangential speed at the major-axis end versus the minor-axis end, given a 90 RPM hand-crank input.
Given
- a = 0.400 m
- b = 0.250 m
- N = 90 RPM
Solution
Step 1 — convert crank speed to angular velocity in rad/s. This is the rate at which θ advances:
Step 2 — at the major-axis end (θ = 0), x = a and y = 0. The tangential speed of the tracing point is dominated by the y-direction component, so v = b × ω. This is the slow end of the cut:
Step 3 — at the minor-axis end (θ = π/2), the tangential speed is set by a × ω. This is the fast end:
So at the nominal 90 RPM crank input, your router bit sweeps the ends of the major axis at 2.36 m/s and the ends of the minor axis at 3.77 m/s — a 60% speed swing around the ellipse. At a low-end 30 RPM hand crank, those numbers drop to 0.79 m/s and 1.26 m/s, slow enough that the bit barely cuts and burns the walnut on the slow ends. Push to a high-end 180 RPM crank speed and you hit 4.71 m/s and 7.54 m/s — the bit chatters at the major-axis tips because the trammel rod whips and the shoes start unloading the slot walls.
Result
At 90 RPM nominal, the bit speed varies from 2. 36 m/s at the major-axis ends to 3.77 m/s at the minor-axis ends. In practice that means you must slow your feed rate by roughly 40% as you approach the long-axis tips or the cut quality drops off — most amateur ellipse cuts fail right at those four points. Across the operating range, 30 RPM is too slow (burning), 90 RPM is the sweet spot for a 600 mm beam, and above 150 RPM rod whip starts to dominate. If your measured ellipse comes out 5–10 mm undersized on the major axis, the most likely causes are: (1) tracing-point lock screw slipping under cutting load — a common failure on aluminium beam jigs without a positive shoulder, (2) shoe side-play above 0.15 mm letting the rod cock in the slot, or (3) the two slots not being perpendicular within 0.5°, which produces a measurable foreshortening on the longer axis.
When to Use a Trammel of Archimedes and When Not To
The Trammel of Archimedes is one of three common ways to generate an ellipse mechanically. The other two are the string-and-two-pins method (the gardener's ellipse) and a CNC-controlled tool path. Each has different strengths depending on size, accuracy, and how many copies you need to make.
| Property | Trammel of Archimedes | String-and-pins method | CNC tool path |
|---|---|---|---|
| Accuracy on a 500 mm ellipse | ±0.5 mm with tight shoes | ±2–3 mm depending on string stretch | ±0.05 mm |
| Setup cost | $30–$150 shop-built jig | Two nails and string — under $1 | $2,000+ CNC plus software |
| Practical size range | 50 mm to 2 m | 100 mm to 10 m+ | Limited by machine bed, typically up to 1.5 m |
| Repeatability across multiple parts | Excellent — set axes once, cut many | Poor — string stretches, pins shift | Excellent — digital file |
| Speed/RPM constraint | Up to ~150 RPM crank before rod whip | Manual only | Limited only by spindle and feed |
| Skill required | Low — anyone can dial in axes | Low but inconsistent results | High — CAM software and machine setup |
| Failure mode | Worn shoes, slot non-perpendicularity | String slip or stretch | Tool deflection, programming error |
Frequently Asked Questions About Trammel of Archimedes
Almost always because the two slots are not perpendicular to each other. A 1° squareness error in the cross-slot base produces a visibly skewed curve on anything bigger than about 200 mm — the long axis is no longer perpendicular to the short axis, and one quadrant of the ellipse looks fatter than the opposite quadrant.
Check by laying a precision square in the slot intersection. If it rocks at all, re-cut the slots. The fix is unforgiving — the trammel only produces a true ellipse when the slot axes are exactly 90° apart. Geometry, not tolerance.
Yes, and this is a useful self-check. When the tracing-point distance to the near slider equals the distance to the far slider, the ellipse degenerates to a circle of radius a. If you set a = b = 200 mm and the result is not a clean circle, your linkage has a defect — usually shoe play or slot non-perpendicularity — and you should fix it before trying any non-circular shapes.
It is also the most sensitive test you can run, because any geometric error shows up immediately as an oval instead of a circle.
Use the trammel any time you need to actually cut, rout, or machine the ellipse — not just mark it. The string method is fine for chalking out a flowerbed or a one-off plywood mark, but a string cannot guide a router bit, and it stretches under any side load.
The trammel is also the right choice when you need to make multiple matching parts. Once you set the two axis distances and lock them, every ellipse you cut is identical to within the slot tolerance. With string and pins, every copy drifts.
At the major-axis ends, one slider is essentially stationary while the other is moving at maximum velocity. Any side-play in the stationary shoe lets the rod cock in its slot, and the tracing point shifts off the true ellipse path. You will see this as small flat spots or hooks at the four cardinal points of the ellipse.
Tighten the shoes to 0.05–0.10 mm side clearance, or replace wood-on-wood shoes with PTFE or bronze. The other common cause is the rod itself flexing under cutting load — a 12 mm thick aluminium beam handles a trim router fine, but a 6 mm beam will deflect noticeably at full extension.
Rule of thumb for a hand-cranked or router-driven setup: the beam length should not exceed about 50× its smallest cross-section dimension if you want to stay below 150 RPM input. A 12 mm × 25 mm aluminium bar tops out around 600–700 mm before mid-span deflection under cutting load gets noticeable.
If you need a bigger ellipse — say a 1.5 m architectural arch — go to 25 mm × 50 mm steel tube and slow the input to 60 RPM or below. Beyond about 2 m, you are better off scribing the ellipse with a trammel and cutting freehand to the line.
For drawing or scribing, no — the curve is purely geometric and does not care about speed. You can hand-crank at any rate and still get a true ellipse.
For cutting, yes, it matters. The tangential bit speed varies by a factor of a/b around the ellipse — in our 800 × 500 mm example, that is a 1.6× swing. If your crank speed is also varying because you are pushing harder at one end, the cut quality changes around the perimeter. A geared hand crank or a slow-rev DC gearmotor at 60–90 RPM gives much better surface finish than freehand cranking on anything past about 400 mm major axis.
References & Further Reading
- Wikipedia contributors. Trammel of Archimedes. Wikipedia
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
