A Form 6 Parallel Ruler is a drafting and navigation instrument made of two straightedges connected by two equal-length pivoting link arms in a parallelogram four-bar linkage. The link arms are the critical component — they constrain the upper rule to translate parallel to the lower rule no matter how far you walk it across the paper. The geometry transfers a bearing or reference line from one location on a chart to another without rotation, which is exactly what you need for plotting courses on a paper Admiralty chart or stepping out parallel construction lines in a hand-drafted plan.
Parallel Ruler Form 6 Interactive Calculator
Vary link-arm mismatch, nominal arm length, step count, and bearing angle to see per-step and cumulative angular drift.
Equation Used
The calculator estimates how unequal link arms rotate the transferred line. Per-step drift is dL divided by nominal link length, converted from radians to degrees; total drift multiplies that by the number of walking steps. The 60 nmi error shows the approximate cross-track error from the accumulated bearing error over a 60 nautical mile run.
- Small-angle approximation for link length mismatch.
- dL is the absolute difference between the two link-arm pivot distances.
- Cumulative drift is per-step drift multiplied by the number of walks.
- Pivot looseness, rule bowing, and paper slip are not included.
How the Parallel Ruler (form 6) Works
The Form 6 is a parallelogram four-bar linkage. Two rules — typically 250 to 400 mm long in boxwood, rosewood, or acrylic — sit one above the other, joined by two brass link arms of identical centre-to-centre length, usually 60 to 100 mm. Pin the arms at matching positions on each rule and you have a four-bar where opposite sides stay parallel through the full motion envelope. Walk the upper rule away from the lower one, then anchor the upper and walk the lower, and the rule steps across the paper while preserving its original angle to within a fraction of a degree.
The geometry is unforgiving on one spec: the two link arms must be exactly equal in pivot-to-pivot distance, and the pivot pairs on each rule must be exactly equal in spacing. Mismatch the arms by even 0.2 mm on a 80 mm link and the upper rule rotates roughly 0.14° per step — small per step, but cumulative. Step across an 800 mm chart in 10 walks and you have drifted more than 1°, which on a coastal passage translates to a position error of nearly 1 nautical mile per 60 sailed. That is why heritage rule makers grind the link arms as a matched pair and ream the pivot holes with a single jig.
Failure modes are mechanical, not kinematic. Loose pivots from a worn rivet let the link arm wobble in plane, which shows up as a rule that no longer holds its bearing when you lift your hand. Bowed rules — common in cheap plastic stock left in a sunny chartroom — mean the straightedge is no longer straight, so you transfer a curved line. And dirt under the pivot heads adds friction that makes the rule jump rather than glide, which spoils precise step-overs.
Key Components
- Lower Straightedge: The reference rule held against the chart or drawing during the step. Typical length 250-400 mm, edge straightness held to better than 0.1 mm over the full length on a quality boxwood or rosewood instrument. The bevelled drawing edge sits flush to the paper so the pencil or divider point reads a true line.
- Upper Straightedge: The mirror of the lower rule, identical in length and edge profile. Carries the matching pivot positions for the link arms. On a Form 6 both rules are the same dimension — unlike the Captain Field pattern which adds a protractor scale to one rule.
- Link Arms (×2): Two flat brass or nickel-silver bars, typically 3 mm thick and 60-100 mm pivot-to-pivot. They MUST be a matched pair — equal length to within 0.05 mm — or the parallelogram becomes a trapezium and the upper rule rotates as it translates. The arms cross over each other in the closed position and fan out as the rule opens.
- Pivot Pins: Brass or steel rivets, usually 2-3 mm shank, peened or screwed through the rule and arm. Need a slip fit (~0.05 mm clearance) — tight enough that the joint has no detectable wobble, loose enough that the arm swings under finger pressure without binding. Worn pivots are the single most common reason an old rule loses accuracy.
- Bevelled Drawing Edge: The working edge of each rule, ground to a shallow chamfer (around 30°) so the pencil rides against the bevel without ink wicking under the rule. On marine pattern rules the edge is graduated in degrees of latitude or compass bearing for direct chart work.
Who Uses the Parallel Ruler (form 6)
The Form 6 has the simplest geometry of the parallel-rule family, which is why it survived in serious use long after rolling-ball and Captain Field patterns appeared. Anywhere a line must be transferred parallel across a flat surface, the Form 6 does the job with no rolling drift, no protractor calibration, and nothing to maintain beyond the four pivots.
- Marine Navigation: Admiralty paper-chart plotting on UK Hydrographic Office charts — RYA Yachtmaster candidates still learn passage planning with a 380 mm boxwood Form 6 to step a bearing from the compass rose to the vessel's DR position.
- Architectural Drafting: Heritage practices in Edinburgh and Kyoto producing listed-building survey drawings by hand — the Form 6 ruler steps parallel construction lines for elevation work on B1 boards where CAD output is not accepted by the conservation officer.
- Shipwright Modelling: Scratch-built plank-on-frame model ship workshops in Bristol and Hamburg use 200-250 mm brass Form 6 rules to transfer waterlines and frame stations on 1:48 plans.
- Cartography Restoration: Map and chart conservators at institutions like the British Library use Form 6 rules with felt-faced edges to retrace and verify parallel rhumb lines on 18th-century portolan reproductions without scratching the original.
- Tattoo and Calligraphy: Bespoke tattoo studios and calligraphers use small 150 mm acrylic Form 6 rulers to lay out parallel guide lines for lettering layouts directly on skin or paper without rolling motion.
- Quilt and Pattern Making: Couture pattern shops use long-arm 500 mm Form 6 rules to step parallel seam lines across full-size pattern paper for tailored garments.
The Formula Behind the Parallel Ruler (form 6)
What matters in a Form 6 is the angular drift of the upper rule per step, given a manufacturing mismatch between the two link arms. At the low end of the typical mismatch range (well-made instrument, 0.05 mm difference between arms) the drift is essentially invisible across a chart. At the high end (a worn or cheap rule with 0.5 mm mismatch) the drift is large enough to ruin a coastal navigation plot. The sweet spot for serious work sits below 0.1 mm mismatch on a 60-100 mm link arm.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Δθ | Angular drift of the upper rule per single step (open or close cycle) | degrees | degrees |
| ΔL | Difference in pivot-to-pivot length between the two link arms | mm | in |
| Larm | Nominal pivot-to-pivot length of the link arm | mm | in |
| Nsteps | Number of walking steps across the chart (used to compute cumulative drift Δθ × N<sub>steps</sub>) | — | — |
Worked Example: Parallel Ruler (form 6) in a sailmaker's loft Form 6 rule
A traditional sailmaker's loft in Bremen is producing a small batch of 320 mm pearwood Form 6 parallel rules for chart-table use aboard restored 1920s pilot cutters. The link arms are nominally 75 mm pivot-to-pivot. The shop wants to know what manufacturing tolerance on the link-arm length pair is acceptable for a navigator stepping a bearing 8 times across an 800 mm chart without losing more than 0.5° total bearing accuracy.
Given
- Larm = 75 mm
- Nsteps = 8 steps
- Δθbudget = 0.5 degrees total
Solution
Step 1 — work out the per-step drift budget by dividing the total budget by the number of steps:
Step 2 — at the nominal target, solve for the allowable arm-length mismatch ΔL:
Step 3 — at the low end of typical workshop tolerance, ΔL = 0.05 mm (a careful jig-reamed pair on a single setup):
Across 8 steps that gives a cumulative 0.31° — well inside the 0.5° budget, and on an 800 mm chart that is roughly 0.4 nautical miles of position error over a 60 nm coastal leg. The navigator will not see it.
At the high end of typical tolerance, ΔL = 0.30 mm (sloppy hand-fit arms or a worn antique with elongated pivot holes):
Across the same 8 steps that is 1.83° cumulative drift — almost four times the budget, and now the navigator is putting the boat 2 nm off track on the same coastal leg. That is the difference between clearing a headland and going aground.
Result
The Bremen shop must hold matched-pair link-arm length tolerance to ΔL ≤ 0. 082 mm to meet the 0.5° / 8-step budget on a 75 mm arm. At a tight 0.05 mm mismatch the cumulative drift is a comfortable 0.31° — invisible on the chart. At 0.30 mm mismatch the drift balloons to 1.83°, enough to put the vessel a full mile off its plotted track. If a finished rule measures worse than the predicted drift on a test sweep, the most common causes are: (1) pivot holes drilled separately on each arm rather than stack-drilled, leaving 0.1-0.3 mm pin-to-pin error; (2) brass arm bowed during peening so effective pivot spacing changes through the swing; or (3) the two rules themselves not parallel-ground to the same edge length, which mimics arm mismatch even if the arms are perfect.
When to Use a Parallel Ruler (form 6) and When Not To
The Form 6 is the baseline parallel-rule pattern. Captain Field adds a protractor scale, the rolling parallel rule swaps linkages for two coupled cylinders, and a draftsman's parallel-glide straightedge uses cables and pulleys to span an entire drawing board. Each option trades simplicity for a different capability.
| Property | Form 6 Parallel Ruler | Rolling Parallel Rule | Drafting-board Parallel Glide |
|---|---|---|---|
| Per-step angular accuracy (typical, well-made) | ±0.04° per step | ±0.1° per metre rolled (slip-dependent) | ±0.02° across full board |
| Maximum useful span | ~2 × link arm length, ~150 mm | Limited only by paper size, ~1 m+ | Full board width, 600-1500 mm |
| Mechanical complexity | 4 pivots, 2 arms, 2 rules | 2 ground rollers + axle + frame | Cable, 4 pulleys, counterweight, slide rails |
| Cost (quality instrument, 2024) | £40-£250 | £60-£300 | £400-£1500 board-mounted |
| Failure mode | Pivot wear, arm length mismatch | Roller slip on dusty paper, axle bend | Cable stretch, pulley bearing wear |
| Best application fit | Marine chart work, hand drafting transfers | Long-distance line transfer on smooth paper | Full-board architectural drafting |
| Setup / calibration time | Zero — pick up and use | Zero, but check roll on test paper | 30+ min board installation, periodic tension |
Frequently Asked Questions About Parallel Ruler (form 6)
Arm length is only one of three matched dimensions. The pivot-pair spacing on each rule must also match the other rule's pivot-pair spacing — if one rule has pivots at 65.00 mm centres and the other at 65.15 mm, the parallelogram becomes a trapezoid even with perfect arms. The other usual culprit is pivot-hole concentricity. A 2 mm pin in a 2.2 mm reamed hole gives 0.2 mm radial slop per joint, and four joints in series stack to roughly 0.4 mm of effective length variation as the rule walks. Use a pin gauge on each hole and replace any that take a 2.1 mm gauge freely.
If the navigator works exclusively from the printed compass rose on the chart, a Form 6 is faster — you set the rule on the rose, walk it to the DR position, read the bearing. If the navigator wants to read a bearing directly off the rule without referencing the rose (useful in a seaway where you cannot reach the rose), the Captain Field's integrated protractor scale earns its keep. The Form 6 is also lighter and less prone to damage, since there is no engraved scale to wear off. For RYA training boats most instructors still issue Form 6 because the workflow forces students to learn the compass rose properly.
Mixed materials are fine and common — brass arms on acrylic or boxwood rules is the heritage construction. What matters is that the two link arms are the same material and made from the same stock, because differential thermal expansion between two arms changes the matched-pair length. A 75 mm brass arm and a 75 mm steel arm will diverge by about 0.04 mm across a 30°C chartroom temperature swing, which is half your tolerance budget gone before you start. Keep the arms identical; the rules can be whatever your customer wants.
That is a friction problem at the rule-paper interface, not the linkage. The Form 6 relies on you holding one rule firm while the other walks. If the bevelled drawing edge has been polished smooth or waxed by handling, it slides on the paper instead of staying planted. Heritage makers often leave the underside of the rule slightly matte, or fit small cork or felt patches at each end of the underside. If the rule slips, lightly abrade the underside with 600-grit paper or fit two 5 mm cork dots near the ends — do not add anti-slip across the full underside, because that breaks the smooth lift you need between steps.
Tape a sheet of A1 paper to a flat board. Draw a single reference line along the lower rule with a 0.3 mm pencil. Walk the rule across the full sheet in 10 steps and draw a final line. Measure the angle between the two lines with a precision protractor or by trigonometry off the pencil-line endpoints. A quality Form 6 should show under 0.4° total drift across 10 steps on a 75-80 mm arm. If you measure more, check arm-length matching first, then pivot slop, then rule-edge straightness with a steel reference straightedge — bowed rules masquerade as linkage error.
The geometric limit is when the link arms approach 180° from the rule, but you lose stiffness long before that. As the arms open past about 60° from the closed position, the lever arm holding the upper rule against your guiding hand shortens, and any side load on the rule deflects the linkage. Most experienced chart-table users keep individual steps to roughly 1.2 × the link-arm length and take more steps if needed. On a 75 mm arm that means 90 mm steps, so a 360 mm transfer takes 4 walks rather than 2 long ones. Smaller steps also reduce cumulative angular drift proportionally.
References & Further Reading
- Wikipedia contributors. Parallel rulers. Wikipedia
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