Equiform Bundle Motions in $E_3$ with Spherical Trajectories I

Abstract: The 3-parameter group of Euclidean bundle transformations moves every point of the 3-space on a ball centered in the bundle vertex. In this paper we investigate motions of the 4-parameter group of equiform bundle transformations with the additional property, that some of the points have spherical orbits, i.e. orbits lying on a plane or a ball. There are classical results for the (6-parameter) group of Euclidean transformations by R. Bricard [Br], G. Darboux [Da] and E. Duporcq [Du]. Here we present a systematical approach to our question, which is based on considerations of E. Borel and R. Bricard. The "principle of transfer" used in this paper was developed in J. Lang [L1], [L2], [L3] and applied for the flag space. A very elegant description and a generalisation is due to H. Vogler. This principle of transfer leads to a number of one- and two-parametric motions not investigated until now. We can also give parametric representations of these motions.

{\parindent25pt \item{[Br]} Bricard, R.: Sur un deplacement remarquable. Comptes rendus des Seances, seance du 30 nov. 1896. \item{[Da]} Darboux, G.: Sur le deplacement d'une figure invariable. Comptes rendus Paris 92 (1881). \item{[Du]} Duporcq, E.: Sur le deplacement le plus generale d'une droite dont tous les points decrivent des trajectoirs spheriques. Journal de Math. $5^e$ serie, tome IV, fasc. II, Paris 1898. \item{[L1]} Lang, J.: Bewegungsvorgänge des Flaggenraumes mit sphärischen Bahnen - ein Übertragungsprinzip. J. Geom. 47 (1993), 94-106. \item{[L2]} Lang, J.: Bewegungsvorgänge des Flaggenraumes mit sphärischen Bahnen I: Die vierparametrigen Bewegungsvorgänge. Math. Pannon. 4 (1993), 3-22. \item{[L3]} Lang, J.: Bewegungsvorgänge des Flaggenraumes mit sphärischen Bahnen II: Die dreiparametrigen Bewegungsvorgänge. Math. Pannon. 5 (1993), 91-104.

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