A Conchoid Delineator is a mechanical draughting device that traces the conchoid of Nicomedes — a classical curve where every point lies a fixed distance from a straight directrix, measured along a line through a fixed pole. The Greek geometer Nicomedes designed the original around 200 BC to solve angle trisection and cube duplication problems classical compass-and-straightedge tools could not handle. The instrument uses a sliding bar pivoting on a pin at the pole and a stylus offset a constant length along that bar. Draughtsmen used it for cam profiles, optical reflector layouts, and architectural mouldings well into the 20th century.
Conchoid Delineator Interactive Calculator
Vary the pole-to-directrix distance, stylus offset, branch, and arm angle to see the traced conchoid point and linkage geometry.
Equation Used
The conchoid radius is found by first locating where the rotating arm crosses the directrix, a/cos(theta), then adding or subtracting the fixed stylus offset b along the same arm. The resulting polar point is converted to x-y coordinates for the plotted stylus position.
- Ideal linkage with no pole play, rail bow, or stylus collar drift.
- Directrix is the vertical line x = a, with the pole at the origin.
- theta is measured from the perpendicular line from pole to directrix.
- s = +1 places the stylus outward from the directrix crossing; s = -1 places it inward.
Inside the Conchoid Delineator
The mechanism is a guided-slider linkage. A fixed pin marks the pole, a straight slot or rail forms the directrix, and a rigid arm passes through both — sliding freely along the directrix while pivoting around the pole. A stylus mounts on the arm at a fixed offset distance from where the arm crosses the directrix. As the arm sweeps through its angular range, the slider on the directrix moves and the stylus traces the conchoid. Two branches exist — one above the directrix, one below — depending on which side of the directrix crossing you place the stylus.
The geometry is unforgiving. If the pole pin has 0.2 mm of radial slop, your traced curve develops a visible wobble near the asymptote where small angular errors amplify into large positional errors. The directrix slot must be straight to within 0.05 mm over its working length — a bowed slot bends the entire family of generated points. The stylus offset must hold constant; a loose stylus collar that drifts even 0.3 mm changes the curve identity entirely, since the offset b is one of only two parameters defining the conchoid. The classical failure mode on antique instruments is wear at the pole pivot — the pin loosens, the arm develops play, and the curve near the cusp (when b > a) loses its sharp loop and rounds off into mush.
Why build it this way rather than plotting points and joining them? Because for cam profiles and reflector contours, the curve must be a continuous physical track a follower or light ray will obey. Point-by-point plotting introduces interpolation error between points; the delineator produces a single unbroken locus that respects the underlying geometry exactly, limited only by mechanical tolerances of the linkage itself.
Key Components
- Pole pin: Fixed pivot pin clamped to the drawing board, typically 3-5 mm hardened steel. The arm rotates around it. Radial clearance must be under 0.05 mm — anything looser introduces angular play that grows into millimetres of error at the stylus end of a 200 mm arm.
- Directrix rail: The straight reference line, usually a precision-ground steel slot or T-rail mounted parallel to a board edge. Straightness tolerance of 0.05 mm over 300 mm working length. The arm carries a slider block that runs along this rail.
- Slider block: Bronze or brass block riding on the directrix rail. Carries a transverse hole through which the arm passes. Fit between block and arm is critical — too tight and the arm binds, too loose and the curve develops chatter marks.
- Pivoting arm: Rigid bar 200-400 mm long passing through both pole and slider. Must be straight and torsionally stiff. The arm encodes the line through the pole; any bow translates directly into curve error.
- Stylus carriage: Holds the marking point at a fixed offset distance b from the slider centre, along the arm. Lockable collar, usually with a vernier scale reading to 0.1 mm. Setting b accurately is the single most important adjustment on the instrument.
- Pole-to-directrix distance setting: The perpendicular distance a from pole to directrix. Adjustable on better instruments via a slide on the base. Determines whether the conchoid has a node, a cusp, or a smooth loop — corresponding to b<a, b=a, b>a respectively.
Where the Conchoid Delineator Is Used
The delineator looks like a museum piece, but the conchoid curve it draws still appears in working drawings today. Anywhere you need a curve where every point is a fixed offset from a line, measured along radials from a single point, you are drawing a conchoid — whether you know the name or not.
- Optical engineering: Layout of off-axis parabolic reflector mounting flanges at firms like Edmund Optics, where the flange profile is a conchoid of the optical axis.
- Cam design: Profile generation for radial-follower cams in vintage textile machinery — Schiffli embroidery looms used conchoid-derived dwell profiles for needle-bar timing.
- Architecture: Drawing entasis curves and moulding profiles for classical column restoration work, including projects at the American Academy in Rome's archive of measured drawings.
- Mathematics education: University geometry courses still demonstrate the instrument as the historical solution to angle trisection — Brown University's mathematical model collection holds an original 19th-century example.
- Antenna engineering: Layout of conchoidal feed-horn shrouds in early radar work at MIT Rad Lab, where the offset-from-axis property suppressed sidelobe scatter.
- Machine tool pattern making: Pattern shops at firms like Pratt & Whitney used conchoid templates for cutter-relief profiles on form tools before CNC made the curve trivial to compute.
The Formula Behind the Conchoid Delineator
The conchoid in polar form gives the radial distance from the pole as a function of angle. The two parameters that control everything are a, the perpendicular distance from pole to directrix, and b, the offset of the stylus from the directrix crossing. At small offset (b much less than a) the curve is a gentle bulge — useful for reflector flange layouts where you want a near-circular form. At b = a you get a cusp, sharp enough to use as a cam dwell point. At b > a the curve develops a loop, and that loop is what classical geometers exploited to trisect angles. The sweet spot for most practical draughting work is b between 0.5a and 1.5a, where the curve is well-behaved and the linkage doesn't approach its angular limits.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| r | Radial distance from pole to traced point | mm | in |
| θ | Angle of arm from perpendicular to directrix | rad or ° | rad or ° |
| a | Perpendicular distance from pole to directrix | mm | in |
| b | Stylus offset along arm from directrix crossing | mm | in |
| ± | Sign selects the outer (+) or inner (−) branch of the curve | — | — |
Worked Example: Conchoid Delineator in a heritage organ-pipe mouth profile
A pipe organ restorer at a Dutch workshop is drawing a new mouth-arch template for a 1735 Christian Müller flue pipe. The arch profile is a conchoid with pole-to-directrix distance a = 40 mm and a nominal stylus offset b = 30 mm. The restorer needs to know the radial reach r at three angles — 0°, 30°, and 60° from the perpendicular — to size the template blank.
Given
- a = 40 mm
- b = 30 mm
- θ range = 0 to 60 °
Solution
Step 1 — at the nominal centre angle θ = 30°, compute the arm length from pole to directrix crossing:
Step 2 — add the stylus offset to get the outer-branch radius at the nominal angle:
Step 3 — at the low-end of the working range, θ = 0° (arm perpendicular to directrix):
This is the closest approach of the curve to the pole — the tightest part of the arch where the pipe's mouth meets the foot. A 6 mm change from low to nominal sounds small, but on a 200 mm-tall pipe mouth it is the difference between a clean voicing edge and one that looks visibly cocked. Step 4 — at the high end of the practical range, θ = 60°:
Beyond about 70° the cos term collapses toward zero and r runs away to infinity — the curve heads for its asymptote. In practice you stop drawing well before then because the slider on the directrix is travelling fast for very little useful curve, and any pivot slop in the arm shows up as visible chatter on the template edge.
Result
The template blank needs to accommodate a radial sweep from 70 mm at θ = 0° to 110 mm at θ = 60°, with the nominal middle of the arch at 76. 19 mm. The 40 mm spread across the working range is what gives the organ-pipe mouth its characteristic flared profile — too narrow a range and the arch looks tubby, too wide and the voicing geometry skews the harmonics. If your traced curve disagrees with these numbers by more than 0.3 mm at any point, suspect three things in this order: (1) the stylus collar has shifted on the arm — re-check b with calipers against a hard reference, (2) the directrix rail is not square to the board datum, throwing the angle reference off by enough to matter at the 60° end, or (3) the arm itself is bowed from being stored under load, which adds a sinusoidal error that peaks mid-range.
Choosing the Conchoid Delineator: Pros and Cons
The conchoid delineator is one of several historic curve-tracing instruments. The choice between them comes down to which curve family you actually need and how much accuracy you can extract from a mechanical linkage versus a numerical method.
| Property | Conchoid Delineator | Ellipsograph (Trammel of Archimedes) | CAD spline plot |
|---|---|---|---|
| Curve family generated | Conchoid of Nicomedes only | Ellipses only | Any analytic or freeform curve |
| Typical accuracy on a 300 mm drawing | ±0.2 mm with well-maintained linkage | ±0.1 mm | ±0.01 mm limited by plotter resolution |
| Setup time per curve | 2-5 minutes | 1-3 minutes | 30 seconds once parameters known |
| Cost (current market for usable instrument) | $300-$1500 antique market | $80-$400 new | Software licence only, $0 to $5000/yr |
| Reliability over decades | High if pivot wear managed | Very high, simple geometry | Software dependent, file-format risk |
| Best application fit | Cam profiles, reflector flanges, classical mouldings | Architectural ovals, optical apertures | Any modern engineering drawing |
| Complexity to learn | Moderate — angle/offset interplay | Low — two-slider intuition | High — full CAD package |
Frequently Asked Questions About Conchoid Delineator
The two branches of the conchoid have different mechanical behaviour at the linkage. On the outer branch (stylus offset added beyond the directrix crossing) the slider moves slowly and the stylus tracks cleanly. On the inner branch — particularly when b approaches a — the slider can reverse direction near the cusp, and any backlash in the slider-to-rail fit shows up as a visible discontinuity right at the sharpest part of the curve.
The fix is to take up the slider clearance with a light spring preload, or to redraw the cusp region in two passes from opposite directions and average the result by eye.
The choice changes the topology of the curve entirely. With b < a you get a smooth, monotone profile — good for follower cams with gentle dwell. At b = a the curve has a cusp at the pole, which gives an instantaneous direction change useful for indexing cams but brutal on follower wear. With b > a the curve forms a loop near the pole, which is what makes the conchoid useful for trisection demonstrations but rarely useful in machine design because the loop self-intersects.
For 90% of cam work you want b between 0.4a and 0.8a. Anything tighter and you are asking the follower to track a near-cusp at speed.
You can, but it's almost never the right call. A well-maintained delineator gives you ±0.2 mm on a 300 mm drawing, which is fine for a paper template but coarse for CNC. The better workflow is to use the delineator to verify the curve visually, then digitise the formula directly in CAM — the conchoid has a closed-form polar equation that any CAM package will accept as a parametric curve. The instrument earns its place when you are restoring a piece that was originally laid out with one and you want the new template to carry the same minor mechanical signature as the original.
Asymmetry on a curve that should be symmetric about the perpendicular through the pole almost always means the directrix rail is not square to the line from the pole. Drop a precision square between the pole pin and the rail — if you are off by even 0.5°, the curve on one side reaches further than the other because the cosine term is no longer referenced to a true perpendicular.
Less commonly, the arm itself can be bent in the plane of the drawing. Sight along it against a steel rule. A 0.3 mm bow over 300 mm is enough to skew the trace visibly.
The formula r = a/cos(θ) + b has a singularity at θ = 90° — cos goes to zero and r runs to infinity. Mechanically, this means the slider on the directrix is racing along the rail for very little arm rotation, and the stylus is moving very fast for a small angular input. Any friction stick-slip in the slider, any pivot slop at the pole, and any flex in the arm all amplify into the trace.
Practically, stop drawing somewhere between 65° and 75°. If you need the curve further out, finish that section by computing points numerically and joining them with a French curve.
It depends on the application's sensitivity to b. For a decorative moulding, ±1 mm on a 30 mm offset is invisible. For an optical reflector flange, a 0.5 mm error in b shifts the focal alignment by enough to matter at the focal plane. For a cam profile, the error translates directly into follower position error at every angle.
Rule of thumb — if the part the curve generates has a positional tolerance T, hold b to roughly T/3 or better. On instruments without fine adjustment, shim the stylus collar with feeler gauge stock to dial b in to 0.05 mm.
References & Further Reading
- Wikipedia contributors. Conchoid (mathematics). Wikipedia
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
