The cyclograph shown here uses three straight rules clamped into a rigid frame to trace a circular arc when its center cannot be reached. The frame slides against two fixed pins at the chord ends, and a pencil at the crossing of the two guide directions follows the circle.
Cyclograph Interactive Calculator
Set the two guide pins and arc rise, then choose a station for pencil coordinates. The rigid braced rules trace a circle without reaching its center.
Equation Used
- Three rules clamped into one rigid frame.
- Fixed point guide pins at the chord ends; no radius or pencil offset correction.
- Positive rise selects the displayed circular arc.
- Illustrative rule lengths provide reach; physical clearance is not calculated.
Ideal point pins and straight rule directions. Station 0% is the arc midpoint; ±100% would be the guide pins, excluded from the animated sweep.
A fixed angle determines the pencil path
Hiscox movement1521 identifies this three-rule instrument as a cyclograph. The neighboring movement1522 is a different device: the perpendicular-slider trammel for ellipses. This page now shows the circular-arc tool itself.
At the central setup position, the pencil is above the chord by the required rise. The third rule braces the two sloping rules, fixing their included angle. Keeping those directions against the guide pins constrains the pencil to a circle through the pins.
Draw an arc whose center is off the sheet
Shallow circular arcs may have centers far beyond the available drawing surface. This construction sets the arc from two chord points and a rise, then guides the pencil locally.
The selected-station input gives coordinates relative to the chord midpoint. It does not alter the circle or resize the rigid frame. The outlined gold point marks that selected location while the solid pencil continues through the illustrative sweep.
Arc and coordinate geometry
For chord c and positive rise h, R=(c²/4+h²)/(2h). Let φ=2 atan(2h/c). The arc has central angle 2φ and length 2Rφ.
For station u as a percentage of the half-arc angle, θ=φu/100. Pencil coordinates are x=R sin θ and y=h−R+R cos θ. The coordinate origin is the chord midpoint, with positive y toward the selected rise.
The fixed angle between guide directions is π−φ. If rise exceeds half the chord, this describes a major arc; the calculation retains that selection.
400 mm chord, 80 mm rise
The circle radius is 290 mm and the full arc length is approximately 441.39 mm. Station0% places the pencil at x=0 mm, y=80 mm.
Negative and positive stations lie symmetrically on either side of the midpoint. Their y coordinates match and their x coordinates have opposite signs. The full arc dimensions remain unchanged.
Practical guide limitations
This model treats each guide pin as a point and each rule direction as a straight line. Pin diameter, edge offsets, pencil clearance, clamp rigidity and interference need checking in a real instrument.
The animation stops short of the chord endpoints, where the pencil would coincide with a pin and the ideal guidance direction becomes indeterminate. The former ellipse formula and unsupported manufacturing tolerances have been replaced with this instrument’s circular-arc geometry.
Cyclograph questions
Does this tool draw an ellipse?
No. This is the three-rule circular-arc cyclograph. A perpendicular-slider trammel is a separate ellipse-drawing mechanism.
What remains stationary?
The two blue guide pins. All three braced rules move together.
Why does the pencil move while the coordinate tiles remain fixed?
The tiles report the selected station; the drawing also shows the current moving pencil coordinates.
What does station80% mean?
Eighty percent of the central angle from the arc midpoint toward the positive endpoint, not eighty percent of the chord length.
Reference
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances, printed page365, movement1521. The adjacent movement1522 distinguishes the ellipse trammel. Circle and coordinate relationships are derived from the illustrated point-guide construction.
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