Аннотация
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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