Discrete Comput Geom 22:297–315 (1999) Discrete & Computational Geometry © 1999 Springer-Verlag New York Inc. Configuration Spaces of Mechanical Linkages D. Jordan 1 and M. Steiner 2 1 Mathematisches Institut, Universit¨ at Bern, CH-3012 Bern, Switzerland jordan@math-stat.unibe.ch 2 D´ epartement de Math´ ematiques, EPF Lausanne, CH-1015 Lausanne, Switzerland marcel.steiner@epfl.ch Abstract. We present an easy to survey constructive method using only basic mathematics which allows us to define a homeomorphism between any compact real algebraic variety and some components of the configuration space of a mechanical linkage. The aim is to imitate addition and multiplication in the framework of weighted graphs in the euclidean plane that permit a “mechanical description” of polynomial functions, and thus of varieties. 1. Introduction A mechanical linkage G is a mechanism in the euclidean plane R 2 that is built up exclusively from rigid bars joined along flexible links. Some links of the linkage may be pinned down with respect to a fixed frame of reference. The configuration space [G] of a mechanical linkage is the totality of all its admissible positions in the euclidean plane. Configuration spaces of such linkages have been studied for centuries as one of the basic topics of kinematics, and it is a known fact that their configuration spaces are compact real algebraic varieties naturally embedded in (R 2 ) n , where n is the number of vertices in the graph. Therefore it is natural to ask whether, conversely, every compact real algebraic variety arises as the configuration space of some mechanical linkage in the euclidean plane. In [9] Lebesgue gives an account of several results, including Kempe’s universality theorem, not for the configuration space of the mechanism itself, but for the orbit of one of its vertices: “Toute courbe alg´ ebrique peut ˆ etre trac´ ee ` a l’aide d’un syst` eme articul´ e. ” The existence of the following universality theorem for some components of the configuration space has been part of folklore for at least two decades.