Equiform bundle motions in $E_3$ with spherical trajectories. I.
Beitr. Algebra Geom., 39 (2):
307-316 (1998)Abstract
In 1906 and 1908 Bricard and Borel published two long papers in which
they determined Euclidean motions with spherical trajectories ıt
R. Bricard, J. l'Ecole Polytechnique, II. Ser. 11, 1-93 (1906; JFM
37.0705.04); E. Borel, Mem. Sav. Etrangers, Paris 33, 1-128
(1908; JFM 39.0749.02). Based on these papers the authors determine
equiform bundle motions which have spherical trajectories. They introduce
a 10-dimensional parameter space $P(W)$ in which a 6-dimensional
surface $B_6$ represents the condition that a point remains on a
sphere during an equiform motion. Furthermore, an embedding of the
motion group into a 10-dimensional projective space $P(V)$ is used
to give an interpretation of the sphere condition as an orthogonality
relation between the points of $P(V)$ and $P(W)$. It is shown that
the images of sphere conditions given by any motion are on a linear
intersection of $B_6$. Because of this fact, linear subspaces of
$P(W)$ and their intersection with $B_6$ are studied. The case that
a line is contained in $B_6$ results in either one point having a
circle as trajectory or the points of a circle or a line of the moving
system are constrained to move on spheres or in planes. The case
of the intersection of a 2-dimensional linear space with $B_6$ yields
a one parameter motion which moves the points of a bundle plane on
spherical trajectories. For Part II, see the following review
Zbl 907.53009.
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