Peaucellier–Lipkin linkage

Description

The Peaucellier–Lipkin linkage is a planar linkage that turns motion on a circle into motion on an exact straight line. It is built from these bars, joined by revolute joints, with two fixed pivots $O$ and $C$ at distance $r$.

1. Two bars of length $\ell$ join $O$ to two joints $A$ and $B$.
2. Four bars of length $s < \ell$ form a rhombus $A P B Q$.
3. One bar of length $r$ joins $C$ to $P$.

Check: by symmetry $O$, $P$ and $Q$ lie on the perpendicular bisector of $AB$; with $M$ the midpoint of $AB$, $$\abs{OP} \cdot \abs{OQ} = \abs{OM}^2 - \abs{MP}^2 = (\ell^2 - \abs{AM}^2) - (s^2 - \abs{AM}^2) = \ell^2 - s^2.$$ So $Q$ is the image of $P$ under the inversion in the circle of centre $O$ and radius $\sqrt{\ell^2 - s^2}$. By bar 3, $P$ moves on the circle of centre $C$ and radius $r$, which passes through $O$. Inversion maps a circle through the centre of inversion to a straight line, so $Q$ moves on a straight line, perpendicular to $OC$.