Peaucellier–Lipkin Linkage

The Peaucellier–Lipkin linkage (1864) was the first planar linkage that converts circular motion into exact straight-line motion. Adjust the lengths below and watch how the curve traced by the point Q (red) changes. You can also drag the crank point P.

How it works

The points O and C are fixed to the ground. The two long arms OA and OB have length L, and A, P, B, Q form a rhombus with side s. By symmetry, O, P and Q always lie on a line, and applying the Pythagorean theorem to the triangles formed with the diagonal AB gives

|OP| · |OQ| = L² − s².

So Q is the inversion of P in the circle with center O and radius √(L² − s²) (drawn dashed). The crank CP forces P onto a circle with center C and radius r. When d = r this circle passes through O, and inversion maps it to a straight line. When d ≠ r, the image is another circle, and Q traces an arc of it.

The linkage can only be assembled when L − s ≤ |OP| ≤ L + s, which limits how far the crank can turn.