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

> How does a cyclograph draw an arc without its centre?

- Category: Mechanisms › Linkages
- Kind: Hardware
- Basis: Xem Studio design following Brown No. 403 (1868)
- Page: https://xemstudio.com/objects/cyclograph-arc-mechanism/ · JSON: https://xemstudio.com/api/geometry?object=cyclograph-arc-mechanism · Plain text: https://xemstudio.com/objects/cyclograph-arc-mechanism/sheet.txt
- License: dimension sheet CC BY 4.0 (Xem Studio); files Xem Studio Design License (free to use and sell from, not to redistribute as files: https://xemstudio.com/license/)
- Use in every chat: add https://xemstudio.com/mcp as a connector (https://xemstudio.com/use-with-ai/)

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

## 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.

## Sizes

| parameter | value |
|---|---|
| chord c | 100 mm |
| angle φ | 120° |
| radius | 57.74 mm |
| rise (versed sine) | 28.87 mm |
| rules | 2 × 150 × 15 × 4 |

## Where the files are

- Henry T. Brown (1868, public domain): [Five Hundred and Seven Mechanical Movements — scan on archive.org](https://archive.org/details/fivehundredseven00browiala) — web — Publisher's page; dimensions are facts
- 507movements.com: [Brown's movement No. 403: figure, his text and an animation](https://507movements.com/mm_403.html) — web — Publisher's page; dimensions are facts

## Free STEP/STL (1 parts, Xem Studio Design License, Xem Studio)

Index: https://xemstudio.com/files/hardware/cyclograph-arc-mechanism/index.json · license: https://xemstudio.com/files/hardware/cyclograph-arc-mechanism/LICENSE.md

- cyclograph-arc-mechanism-assembly: [STEP](https://xemstudio.com/files/hardware/cyclograph-arc-mechanism/cyclograph-arc-mechanism-assembly.step) · [STL](https://xemstudio.com/files/hardware/cyclograph-arc-mechanism/cyclograph-arc-mechanism-assembly.stl) — Assembled mechanism, all parts placed at one pose (STEP keeps each part separate)

## 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.

## Fits with

- [Cyclograph rule 150 × 15 × 4 with pivot and brace holes](https://xemstudio.com/objects/cyclograph-rule-150x15x4/)
- [Cyclograph brace link, 165.71 mm centres](https://xemstudio.com/objects/cyclograph-brace-165-71/)
- [Cyclograph base plate 180 × 100 × 4 with two Ø3 pin holes 100 mm apart](https://xemstudio.com/objects/cyclograph-base-plate-180x100-c100/)
- [Dowel pins, 2–6 mm (ISO 2338 / DIN 6325)](https://xemstudio.com/objects/dowel-pins-metric/)