How to draw an approximate straight line!

Introduction

A Newcomen Engine

How to draw a straight line? This seems like a very strange question, but to the Victorian engineers this was a crucial issue. They needed to constrain the piston in a steam engine to move in a straight line. For example, the Newcomen engine shown in the picture uses a chain. This can only pull in one direction. When engines were developed to push and pull this approach no longer worked.

So, how can you draw a straight line without a reference edge? This title is taken from the book by A. B. Kempe of the same name, and describes plane linkages which were designed to constrain mechanical linkages to move in a straight line.

James Watt's linkage

James Watt's Linkage

James Watt (1736-1819) is remembered for his pioneering work on steam engines. He invented the first straight line linkage, and the idea of its genesis is contained in a letter he wrote in June 1784.

I have got a glimpse of a method of causing a piston rod to move up and down perpendicularly by only fixing it to a piece of iron upon the beam, without chains or perpendicular guides [...] and one of the most ingenious simple pieces of mechanics I have invented.

In a letter to Boulton on 11th September 1784 he describes the linkage as follows.

The convexities of the arches, lying in contrary directions, there is a certain point within the connecting-lever, [the movement of] which has very little sensible variation from a straight line.

Notice that he does not claim that his linkage generates an exact straight line.

Click here to download a GeoGebra worksheet.

You can see an application of this linkage to a steam engine designed by Phinehas Crowther (1763-1818).

Chebyshev's linkage

Chebyshev's Linkage

Following Watt's discovery a whole range of mechanisms which approximate a straight line were developed. The first we consider was invented by the Russian mathematician Pafnuty Chebyshev (1821-1894) and of his many designs this is the best known. For many years he was of the firm belief that linkages could never be designed to produce exact straight lines.

Chebyshev's linkage - alternative form

While the picture here looks quite different from Chebyshev's linkage above, these two configurations actually generate the same curve. It is curious that a particular curve may be generated by more than one linkage. For a clue why this might work, have a play with the GeoGebra applet below, in which the two linkages are superimposed.

Roberts' linkage

A further development was made by Richard Roberts (1789-1864). It is another example of a Watt-type linkage, and here the restrictions on link lengths can be relaxed.

All these examples of linkages involve two fixed points and three bars. Notice the progression from a point P on the link between the arms, then on an extension arm in line with the ends of the arms, and finally in Roberts' mechanism to an arbitrary point fixed relative to the ends of the arms. This is the most general situation possible without the addition of extra linkages.

Which of these linkages draws the closest approximation to a straight line?

It is possible to draw an exact straight line?

How to draw an approximate straight line!

Introduction

Conclusion