Proportion compasses are twin-armed dividers with a sliding pivot that scales any measured length by a fixed ratio — typically 1:1 to 1:10 — letting a drafter transfer dimensions at accuracies of about 0.2 mm over a 200 mm leg. A hyperbola-drawing linkage uses a similar two-arm geometry, but the pivot tracks a ruler so the marker traces a true hyperbolic curve defined by the difference of distances to two foci. Both tools solve the same practical problem: producing exact mathematical curves and scaled dimensions without computation. You still see them in pattern-shops, optical-bench layout, and museum drafting reproductions.
Proportion Compasses / Hyperbola Drawing Interactive Calculator
Vary the long-point opening, scale mark, and pivot slop to see the reduced short-point opening and estimated drafting error.
Equation Used
The proportion compass uses similar triangles about the sliding pivot. On the Lines scale, a mark of r gives a short-point opening equal to the long-point opening divided by r. The error estimate scales the article's stated 0.3 to 0.5 mm full-opening drift to the selected opening and pivot slop.
- Uses the Lines scale for direct length scaling.
- Compass arms are straight, symmetric, and share a common sliding pivot.
- Error estimate uses the article midpoint of 0.3 to 0.5 mm drift at full opening with 0.05 mm pivot slop.
- Full opening reference is 200 mm.
How the Proportion Compasses / Hyperbola Drawing Works
A proportion compass is two flat steel arms crossed and pinned by a sliding pivot screw. Each arm carries a scribed scale — usually Lines, Plans, Solids, Polygons — and the pivot screw clamps into a numbered notch that sets the ratio of the long-end opening to the short-end opening. Set the pivot at the 5 mark on the Lines scale and any distance you set with the long points appears as one-fifth of that distance between the short points. Set it at 3 and you get one-third. The geometry is dead simple: similar triangles sharing a common apex at the pivot. The accuracy depends almost entirely on the pivot fit. If the pivot screw has more than about 0.05 mm of radial slop, the short-end reading drifts as you rotate the instrument and you'll see scaling errors of 0.3 to 0.5 mm at full opening — enough to ruin a guitar rosette or a clock-wheel layout.
The hyperbola-drawing variant takes the same two-arm idea and constrains one arm to slide along a fixed ruler pinned at one focus, while a string of fixed length runs from the second focus, around a pin at the moving end of the arm, and back to a marker held against the arm. As the arm rotates about the first focus, the marker traces a curve where the difference between its distances to the two foci stays constant — that is the definition of a hyperbola. The eccentricity is set by the ratio of the focal separation to the string slack. If the string stretches even 0.5 mm under tension, the asymptotes shift and the curve drifts off-true near the vertices, where the geometry is most sensitive. Common failure modes are a worn pivot bushing, a stretched cord (use braided Dacron, not cotton), and a ruler that isn't properly clamped at the focus pin so it walks during the sweep.
Why build the curve mechanically at all when CAD will plot one in seconds? Because plenty of restoration, pattern-making, and optical-layout work happens on physical stock — wood, brass, ground glass — where you need the line drawn directly on the workpiece, not printed. A linkage gives you a continuous true curve, not a polyline approximation, and it does it in the coordinate frame of the part itself.
Key Components
- Crossed Arms: Two hardened steel or brass arms, typically 150-300 mm long, slotted along their centreline so the pivot can clamp at any of the scribed ratio marks. The arms must be flat to within 0.05 mm over their length or the points won't close cleanly when folded.
- Sliding Pivot Screw: A knurled clamp screw passing through both arm slots. Sets the ratio by locking at numbered notches — Lines 2 through 10, Plans for area scaling, Solids for volume scaling. Radial play above 0.05 mm is the single biggest source of scaling error.
- Hardened Points: Tool-steel needles ground to a 30° included angle. The points must be matched in length to within 0.1 mm so that a closed instrument shows zero gap. Replaceable on better instruments — the Stanley and Riefler patterns use a small grub screw.
- Ruler Track (hyperbola variant): A straight steel rule clamped at one focus pin, providing the linear constraint that converts the rotating arm into a hyperbola tracer. Straightness must be better than 0.1 mm over 300 mm or the asymptotes won't converge correctly.
- Fixed-Length Cord: Braided Dacron or fine wire of length L = (focal separation) − 2a, where 2a is the transverse axis of the desired hyperbola. Stretch under marker drag is the dominant error term — keep cord tension light and consistent.
- Marker Carrier: A small sliding block on the rotating arm that holds a pencil or scribe and the cord-return pin. Must run smoothly along the arm — any stiction shows up as visible kinks in the traced curve near the vertices.
Where the Proportion Compasses / Hyperbola Drawing Is Used
These instruments still earn their keep wherever a true geometric curve or an exact ratio must appear directly on the workpiece. Pattern-makers, museum conservators, optical technicians, and a surprising number of architectural model shops keep them in the drawer.
- Stained Glass Restoration: A studio in Chartres reproducing a 14th-century rose window cartoon uses a Riefler-pattern proportion compass set at 1:4 to scale measurements from a fragment tracing onto fresh kraft paper before cutting lead came.
- Optical Bench Layout: A laboratory laying out a Cassegrain telescope baffle traces the hyperbolic secondary-mirror profile directly onto 3 mm aluminium stock using a focus-pinned hyperbola linkage with a 240 mm transverse axis.
- Luthiery: An archtop guitar maker scales rib heights from a master template at 0.92:1 using brass proportional dividers when fitting a slightly smaller body variant.
- Naval Architecture Loftwork: A traditional wooden-boat yard in Maine transferring offsets from a 1:10 lines plan to full-size mould stations on the loft floor uses 600 mm proportion compasses to read directly from the table without arithmetic.
- Watchmaking and Clock Restoration: A horologist laying out a new escape-wheel blank scales tooth pitch from a published 1:5 drawing of a Graham deadbeat using a Stanley-pattern sector compass at the 5 mark.
- Museum Drafting Reproduction: An archive in Florence reproducing original Galileo sector drawings for display uses period-correct brass proportional compasses to maintain authenticity in the demonstration plates.
The Formula Behind the Proportion Compasses / Hyperbola Drawing
The core relation tells you the short-end opening you'll get for any long-end opening at a chosen pivot setting. At the low end of the typical range — pivot set near 2 — the instrument behaves almost like ordinary dividers, halving distances with very little mechanical advantage and very low error sensitivity. At the high end — pivot at 10 — small pivot wobble blows up tenfold at the short end, which is why you see drafters tap the screw twice and re-check before they trust a 1:10 reading. The sweet spot for daily work is between 3 and 6, where ratio range is useful but pivot slop hasn't yet dominated the error budget.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Lshort | Distance between the short-end points | mm | in |
| Llong | Distance between the long-end points | mm | in |
| a | Length from pivot to short-end point | mm | in |
| b | Length from pivot to long-end point | mm | in |
Worked Example: Proportion Compasses / Hyperbola Drawing in a heritage cartography studio reducing a survey plan
A heritage cartography studio in Lisbon is transferring detail from a 1:2,500 cadastral survey sheet onto a 1:10,000 display reproduction. The drafter sets a 250 mm proportion compass to a 4:1 ratio (pivot at the Lines 4 notch) and needs to know what short-end reading corresponds to a measured 180 mm feature on the original sheet, plus what the same compass produces at the bottom and top of its useful ratio band.
Given
- Llong = 180 mm
- Pivot setting (nominal) = 4 (Lines) ratio 4:1
- a / b at pivot 4 = 0.25 —
- Compass arm length = 250 mm
Solution
Step 1 — at the nominal 4:1 setting, plug the ratio directly into the relation:
That 45 mm reading is what the drafter pricks onto the new sheet. At this ratio the instrument is comfortable — pivot wobble in the 0.05 mm class translates to about 0.01 mm at the short end, well below the line-width of a 0.3 mm pencil.
Step 2 — at the low end of the useful range, pivot set to 2 (a 2:1 reduction):
Here the compass is barely working — you could almost step it off with ordinary dividers. Error sensitivity is low because the pivot is near the centre of the arms and any rotational slop barely changes the short-end opening.
Step 3 — at the high end, pivot set to 10 (a 10:1 reduction):
This is where the geometry gets unforgiving. The short arm is now only about 22 mm from pivot to point, and a 0.05 mm radial slop in the pivot screw shows up as roughly 0.12 mm of jitter at the points — a quarter of a typical pencil-line width. Most drafters won't trust 1:10 work without re-checking the pivot torque and verifying closure on a known reference distance first.
Result
The nominal answer is Lshort = 45. 0 mm, which the drafter steps directly onto the reproduction sheet. Across the operating band, the same 180 mm input gives 90.0 mm at 2:1, 45.0 mm at 4:1, and 18.0 mm at 10:1 — and the sweet spot is unmistakably the 3-to-6 range where mechanical advantage is useful but pivot error is still buried below the line-weight. If your measured short-end reading drifts from the predicted value, three causes account for nearly all of it: (1) pivot screw torqued unevenly so the arms cant under load, showing as a reading that changes when you flip the instrument over; (2) the arms themselves bowed by 0.1 mm or more, which you can spot by laying them on a surface plate; (3) point-length mismatch where one needle has been resharpened more than the other, showing as a non-zero gap when the instrument is fully closed.
Choosing the Proportion Compasses / Hyperbola Drawing: Pros and Cons
Proportion compasses solve a narrow problem extremely well, but they aren't always the right tool. The realistic alternatives are a pantograph for continuous tracing and CAD plus a printer for arbitrary geometry. Each one wins on different axes.
| Property | Proportion Compasses | Pantograph | CAD + Plotter |
|---|---|---|---|
| Typical scaling accuracy over 200 mm | ±0.2 mm | ±0.5 mm | ±0.05 mm |
| Ratio range | 1:1 to 1:10 (discrete) | 1:1 to 1:8 (continuous) | Any ratio |
| Setup time | 10 seconds | 5-10 minutes | Minutes to hours per drawing |
| Output medium | Direct on workpiece | Direct on workpiece | Paper or film only |
| Initial cost | $80-400 (period or new) | $200-1500 | $2000+ for plotter and software |
| Best application fit | Discrete dimension transfer, conic curves | Continuous freehand reduction or enlargement | Production drawings, arbitrary curves |
| Skill required | Low — read scale, prick points | Moderate — tracer technique | High — CAD proficiency |
Frequently Asked Questions About Proportion Compasses / Hyperbola Drawing
That's pivot-screw asymmetry, not arm wear. When you flip the instrument, gravity and any uneven clamp pressure on the pivot pad load the arms in the opposite direction, and if the screw isn't perfectly perpendicular to the arm faces the apparent ratio shifts.
Diagnostic check: close the compass fully and look down the long axis — the points should overlap with no visible gap and the arms should sit parallel. If they aren't parallel, loosen the pivot, seat the arms flat on a surface plate, and re-torque the screw evenly. A drop of light oil on the pivot faces also helps.
You don't interpolate the scale — that path leads to compounding error. Either pick the nearest marked ratio and accept the rounding, or use the Plans scale if your need is area-based, since Plans marks are pre-computed for square-root ratios.
For a one-off odd ratio like 1:3.7, the cleaner approach is to scale in two stages — say 1:2 then 1:1.85 — using known marks each time. The accumulated error of two clean settings is usually less than the error of one estimated mid-scale setting.
Reach for the linkage when you need the curve drawn directly on the workpiece — ground glass, brass, leather pattern stock — and a continuous true curve rather than a connect-the-dots polyline. The classic case is laying out a telescope baffle or a hyperbolic gear-tooth profile on the part itself.
If the curve is going on paper for documentation or CNC import, plot it numerically. The linkage wins specifically when transferring CAD output to physical stock would introduce its own registration error.
The vertices are where the geometry is most sensitive to cord stretch. Near the vertex the difference of focal distances is small, and any elongation in the cord directly shifts where that minimum lands.
Switch from cotton or nylon to braided Dacron or fine stainless wire — Dacron stretches under 0.1% at typical drawing tensions, cotton can stretch 1-2%. Also check that the marker carrier slides freely along the arm; stiction at the carrier shows up specifically near the vertex because that's where direction-change is sharpest.
A well-preserved 19th-century instrument from Stanley, Riefler, or Kern is often more accurate than current production because the scales were hand-divided on a precision dividing engine and the pivot fits were lapped. The catch is condition — pivot wear and bent points kill accuracy faster than any other failure mode.
Before buying, check three things: closure (no gap at full close), arm flatness (lay on a surface plate, look for light), and pivot freedom (smooth resistance, no notchy spots indicating worn threads). A clean period instrument in the $150-300 range typically outperforms a $50 new one.
2-3% at 1:10 is large enough to rule out pivot slop alone — that would give you under 1%. The most likely cause is that the pivot isn't seating in the correct notch. The Lines 10 mark sits very close to the arm end and the screw can clamp just outside the true notch, effectively giving you a 1:9.7 ratio.
Check by setting the long end to exactly 100 mm against a steel scale, reading the short end, and comparing. If you get 10.3 mm instead of 10.0 mm, loosen and re-seat the pivot, watching that the screw drops fully into the detent. If the error persists, the notch itself may be worn — common on heavily-used shop instruments — and the only fix is re-cutting the detent or using the next mark over.
References & Further Reading
- Wikipedia contributors. Sector (instrument). Wikipedia
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