Parallel manipulators and Borel–Bricard problem

In this paper we shall discuss connections between some old problems of elementary geometry from the beginning of the 20-th century and geometry of parallel manipulators.Beside of this we would like to demonstrate how dual quaternions give a natural tool for solution of problems connected with spatial motions. In order to give this demonstration a short excursion to the theory of kinematic spaces is necessary. As an application we show, how one problem of such kind can be solved by using computer algebra.At the end we explain how methods of approximate parameterization yield a natural tool for the visualization of trajectories of self-motions of parallel manipulators.