How Do You Copy a Drawing at Twice the Size? Four Rods and a Pivot

Four rods are pinned together to form a parallelogram that can flex while keeping opposite sides parallel. One corner of the arrangement is fixed to the table, one point is used as the tracer and follows the original, and a third point carries the pencil. Because the parallelogram keeps its shape as it moves, the fixed point, the tracer and the pencil always lie on one straight line, and the distances from the fixed point to each of them keep a constant ratio. The pencil therefore moves along a path identical in shape to the tracer's path and larger by exactly that ratio, in every direction at once.

The same geometry does several quite different jobs:

• •Copying drawings and maps at a chosen enlargement or reduction
• •Engraving, where a large master pattern guides a small cutting tool
• •Cutting keys, where the tracer follows the original and a cutter copies it
• •Sculpture copying, in a three-dimensional version with extra joints
• •Milling machines guided by a template rather than by numbers
• •Signature machines that reproduce a handwritten signature repeatedly

A version that engraves at a large reduction was central to banknote and stamp production for over a century, and the reasoning is worth following. An engraver cuts a master at a comfortable large size, where fine control is possible, and the machine reproduces it at perhaps a tenth of that size onto the printing plate, which produces lines far finer and more regular than anybody could cut directly. A forger attempting to copy that by hand cannot match the precision, and attempting it photographically loses the depth of the engraved line. The same machines produced the geometric background patterns of interlocking curves, generated mechanically so that they are exactly repeatable and extremely hard to imitate.

Changing the scale means moving where the joints sit rather than changing the rods, which is why the instruments carry rows of holes along each arm. The rule is simply the ratio of the distance from the fixed point to the pencil against the distance from the fixed point to the tracer, so moving the pencil out to twice the tracer's distance doubles everything. Swapping the tracer and the pencil turns an enlarger into a reducer without any other change. Accuracy depends entirely on the joints, since any slack in a pin becomes error in the copy, and precision instruments used hardened pivots and jewelled bearings for exactly that reason.

The hinged frame on the roof of an electric train that presses a contact strip against the overhead wire carries the same name, and the reason is historical rather than functional. Early versions used exactly this jointed parallelogram, which has the useful property of raising and lowering the contact strip while keeping it level and in the same horizontal position, so it stays under the wire as it rises to meet it. Modern versions usually use a single-arm design that achieves the same thing with fewer joints, but the name stuck. The requirement is to keep constant pressure on a wire that varies in height while the train moves at speed, which is a harder problem than it appears.

Four rods pinned into a flexible parallelogram keep the fixed point, the tracer and the pencil on one line at a constant ratio of distances, so the pencil draws the tracer's path scaled exactly. Reducing versions engraved banknote plates at a tenth size, giving lines finer than any hand could cut and patterns a forger cannot match. The hinged current collector on an electric train took its name from the same linkage.