Recognition of plane paths and plane curves under linear pseudo-similarity transformations
Tóm tắt
$$E^{2}_{1}$$
be the 2-dimensional pseudo-Euclidean space of index 1,
$$G=Sim_{L}(E^{2}_{1})$$
be the group of all linear pseudo-similarities of
$$E^{2}_{1}$$
and
$$G=Sim_{L}^{+}(E^{2}_{1})$$
be the group of all orientation-preserving linear pseudo-similarities of
$$E^{2}_{1}$$
. In this paper, groups
$$Sim_{GL}^{+}(E^{2}_{1})$$
and
$$Sim_{GL}(E^{2}_{1})$$
are defined. For the groups
$$G=Sim_{GL}^{+}(E^{2}_{1}),Sim_{GL}(E^{2}_{1})$$
,
$$Sim_{L}(E^{2}_{1}),$$
$$Sim_{L}^{+}(E^{2}_{1})$$
, G-invariants of paths in
$$E^{2}_{1}$$
are investigated. Using hyperbolic numbers, a method to detect G-similarities of paths and curves is presented. We give an evident form of a path and a curve in terms of their G-invariants. For two paths and two curves with the common G-invariants, evident forms of all linear pseudo-similarity transformations, carrying the paths and the curves, are obtained.
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