A fourfold generalization of Peaucellier's inversion cell (Q676954)

The so-called inversion cell has the geometric property to invert planar curves into their inverse shape with respect to a fixed unit circle, and vice versa. Such an inversion converts a circle, for instance, into another one of different size, sometimes used in order to produce very large circles mechanically. This leads to applications in robot mechanisms and in calculating devices, as part of a machine or an instrument. With an inversion cell one can produce a true straight line. An inversion cell is quite different from a pantograph; the latter just transforms a curve into a similar one but never transforms a circle into a straight line. As the input curve may be arbitrary and planar, the cell must have two degrees of freedom. In this paper a combined transformation is presented in a vectorial manner, but also in a manner based on complex numbers instead of vectors. Inversion cells are, moreover, classified based on their number of links, their number of turning joints, and their crucial number of free design parameters. Finally, the most general inversion cell containing 7 links, 8 turning joints, and 6 free design parameters is introduced. This cell consists of an arbitrary four-bar linkage and an adjoined dyad which is dependent on the arbitrary choice of a single joint. The presentation is clear and easy to follow due to the large number of figures.