Here we shall examine a number of linkages which are arranged so that the geometry guarantees an exact straight line movement.
The first planar linkage was invented by Charles Nicolas Peaucellier (1832-1913) in 1864. It uses seven links.
There is also an alternative form of the linkage. This is more a compact arrangement and the movement is very satisfying indeed to watch.
The key to drawing a straight line lies in the mathematics of the inverse in a circle. The following GeoGebra worksheet explains this.
Can you work out how to combine Peaucellier's Cell with the inverse in a circle to draw a straight line?
Hart's Linkage reduces the number of links to only five. The GeoGebra applet below shows the essence of the cell in Hart's Linkage, which forms the inversor.
Can you work out how to combine Hart's Cell with the inverse in a circle to draw a straight line?
Hart's A-frame also uses only five links. This is compact and has the most delightful movement. Can you work out why this draws a straight line?
Historically, the first straight line linkage was described by Sarrus in 1853. It differs in that its parts move in three dimensions. It is applied widely in jacks, elevating platforms and similar devices.
All these linkages are used to generate an exact straight line. There are many other uses for general linkages.