The Chebyshev linkage is a mechanical linkage that converts rotational motion to approximate straight-line motion.

It was invented by the 19th century mathematician Pafnuty Chebyshev who studied theoretical problems in kinematic mechanisms. One of the problems was the construction of a linkage that converts a rotary motion into an approximate straight line motion. This was also studied by James Watt in his improvements to the steam engine.[1]

The straight line linkage confines the point P — the midpoint on the link L3 — on a straight line at the two extremes and at the center of travel. (L1, L2, L3, and L4 are as shown in the illustration.) Between those points, point P deviates slightly from a perfect straight line. The proportions between the links are

$L_1 : L_2 : L_3 = 2 : 2.5 : 1 = 4 : 5 : 2. \,$

Point P is in the middle of L3. This relationship assures that the link L3 lies vertically when it is at the extremes of its travel.[2]

The lengths are related mathematically as follows:

$L_4=L_3+\sqrt{L_2^2 - L_1^2}. \,$