Curves

Guilloché and spirograph

The engraving of banknotes and watch dials: hundreds of fine lines woven into rosettes, trochoids and ornamental borders.

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Under a magnifier, the background of a banknote turns out to be a net of hundreds of fine lines weaving into rosettes. The same net covers watch dials, cigarette cases and the gold beneath the enamel of Fabergé eggs. This is guilloché: engraving cut by a machine rather than a hand.

How it works

A rose engine is a lathe whose spindle rocks as it follows a cam, the rosette. As the work turns, the cutter moves in and out and cuts a closed wavy line; then the work is turned by a tiny angle and the line is cut again. A hundred shifted copies cross and shimmer with moiré.

The “rosette” figure does exactly this. Its nested rings are the “Bands”; each holds “Lines”, copies of one wave shifted in phase. The wave runs between the band's edges, which ripple round the circle as many times as there are “Lobes”. “Loops” sets the waves per lobe, “Amplitude” how deeply the edges ripple.

r_j(θ) = a(θ) + [b(θ) − a(θ)] · (1 + sin(L·k·θ + 2πj/N)) / 2 L — lobes, k — loops, N — lines, a and b — the band's edges hypotrochoid: z(t) = (R − r)·e^(it) + d·e^(−i(R − r)t/r)

The “hypotrochoid” and “epitrochoid” figures are the spirograph: a wheel of radius r rolls inside or outside a ring of radius R, with the pen at distance d from its centre. If R : r is the reduced fraction P : Q (“Lobes” : “Loops”), the curve closes after Q trips round the ring and has P lobes. The “epicycles” figure adds up three rotations, as in Ptolemy's astronomy; all their frequencies leave remainder 1 when divided by the number of lobes, which guarantees the symmetry. Such curves were studied by the mathematician Frank Farris.

A bit of history

Ornamental lathes appeared in the 16th and 17th centuries, cutting ivory and wood, and moved on to gold and silver in the 18th. In 1840 the background and border of the first postage stamp, the Penny Black, were made on a rose engine to foil forgers; guilloché has guarded banknotes and passports ever since. From the 1880s Carl Fabergé laid translucent enamel over engine-turned metal, so the cutting shimmers beneath it like silk. The Spirograph toy was shown by the British engineer Denys Fisher in 1965.

What to tweak

  • For the “hypotrochoid”, make “Lobes” and “Loops” coprime, say 13 and 5, for one long unbroken curve.
  • Push “Amplitude” above 0.5 and the spirograph grows loops: the pen now sits beyond the wheel's rim.
  • Raise “Lines” towards a hundred and lower “Line width” for moiré like real engraving.
  • The “border” figure is the straight-line engine: the same waves along a straight line.
  • Put the “Relief” effect on top in “metal” mode and the rosette becomes an engine-turned watch dial.

Parameters

Figure
rosette · hypotrochoid · epitrochoid · epicycles · border
Lobes
Order of symmetry: how many times the pattern repeats around (waves across the width for a border)
Loops
Waves per lobe in a rosette; for a spirograph, turns of the wheel before the curve closes
Lines
Phase-shifted copies of the wave in each band (rotated copies of the curve for a spirograph)
Bands
Nested rings of a rosette or rows of a border
Inner radius
The empty centre, as a share of half the short side; for a border, the gap between rows
Outer radius
Edge of the rosette; for a border, how much of the height the rows take
Amplitude
Depth of the waves along the band edges; for a spirograph, the pen offset: above 0.5 it loops
Line width
Rotation
Colouring
one colour · gradient across lines · gradient across bands
Line colour
Speed
Turns of the phase per animation loop