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Cyclograph: fixed-angle rules sliding on two pins draw a circular arc

How does a cyclograph draw an arc without its centre? · Hardware · basis: Xem Studio design following Brown No. 403 (1868)

Also called: cyclograph, arc drawing instrument, inscribed angle arc tool, arch setting out, ciclógrafo, Zyklograph

Exact dimensions. True size, with no printing tolerance added.

sizes
5 in table

Notes

  • Two rules fastened at a fixed angle φ slide with their edges against two pins at the ends of a chord: the apex always sees the chord under the same angle, so (inscribed-angle theorem) it moves on a circular arc through the two pins — no centre needed, which is how flat arches of large radius are set out.
  • Chord 100 mm and φ = 120° give R = c / (2 sin φ) = 57.74 mm and a rise of 28.87 mm. The rules' edges are tangent to the Ø3 pins, so the edge apex runs 1.73 mm inside the pin-centre arc. Drawn at the top of the arc, pins touching both edges.
  • Brown's No. 403 (cyclograph: two rules from the chord's ends to the apex of the versed sine, braced, pencil at the angle).
  • Parts (each has its own page): cyclograph-rule-150x15x4, cyclograph-brace-165-71, cyclograph-base-plate-180x100-c100, dowel-pins-metric.
  • Brown's movement No. 403 (1868) shows it; this model is a Xem Studio design following his figure, every part at true size.

Cyclograph: fixed-angle rules sliding on two pins draw a circular arc size chart

parametervalue
chord c100 mm
angle φ120°
radius57.74 mm
rise (versed sine)28.87 mm
rules2 × 150 × 15 × 4

Where the files are

Other publishers' files are linked, not hosted. "verify" = check the license at the link before commercial use.

Henry T. Brown (1868, public domain)Five Hundred and Seven Mechanical Movements — scan on archive.orgwebPublisher's page; dimensions are facts
507movements.comBrown's movement No. 403: figure, his text and an animationwebPublisher's page; dimensions are facts

Free files

★ Free files 1 parts. Xem Studio design: free to use, print and sell things from; not to be republished as files. Xem Studio Design License. index.json

cyclograph-arc-mechanism-assemblyAssembled mechanism, all parts placed at one pose (STEP keeps each part separate)STEP · STL
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Plain text: sheet.txt · JSON: /api/geometry?object=cyclograph-arc-mechanism

Cyclograph: fixed-angle rules sliding on two pins draw a circular arc, isometric render of a generated part
Render of cyclograph-arc-mechanism-assembly from the free STEP; not to scale.

Fits with

Cyclograph rule 150 × 15 × 4 with pivot and brace holes, isometric render of a generated part
Cyclograph rule 150 × 15 × 4 with pivot and brace holes150.0×15.0×4.0 mm★ 1 file
Cyclograph brace link, 165.71 mm centres, isometric render of a generated part
Cyclograph brace link, 165.71 mm centres175.71×10.0×4.0 mm★ 1 file
Cyclograph base plate 180 × 100 × 4 with two Ø3 pin holes 100 mm apart, isometric render of a generated part
Cyclograph base plate 180 × 100 × 4 with two Ø3 pin holes 100 mm apart180.0×100.0×4.0 mm★ 1 file
Dowel pins, 2–6 mm (ISO 2338 / DIN 6325), isometric render of a generated part
Dowel pins, 2–6 mm (ISO 2338 / DIN 6325)5 sizes★ 23 files

Questions

How does a cyclograph draw an arc without its centre?

Two rules fastened at a fixed angle φ slide with their edges against two pins at the ends of a chord: the apex always sees the chord under the same angle, so (inscribed-angle theorem) it moves on a circular arc through the two pins — no centre needed, which is how flat arches of large radius are set out.