Recognition of plane paths and plane curves under linear pseudo-similarity transformations
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Journal of Geometry
Recognition of plane paths and plane curves under linear pseudo-similarity transformations ˙ ¨ Idris Oren and Djavvat Khadjiev Abstract. E12 be the 2-dimensional pseudo-Euclidean space of index 1, G = SimL (E12 ) be the group of all linear pseudo-similarities of E12 and 2 G = Sim+ L (E1 ) be the group of all orientation-preserving linear pseudo2 2 similarities of E12 . In this paper, groups Sim+ GL (E1 ) and SimGL (E1 ) + 2 2 2 are defined. For the groups G = SimGL (E1 ), SimGL (E1 ), SimL (E1 ), 2 2 Sim+ L (E1 ), G-invariants of paths in E1 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 Ginvariants, evident forms of all linear pseudo-similarity transformations, carrying the paths and the curves, are obtained. Mathematics Subject Classification. 53A35, 53A55, 51M10. Keywords. Hyperbolic number, Pseudo-similarity, Curve, Invariant.
1. Introduction Let R be the field of real numbers, n and p are integers such that 0 ≤ p < n. Let Epn be the n-dimensional pseudo-Euclidean space of index p and O(n, p) be the group of all pseudo-orthogonal transformations of Epn . Put O+ (n, p) = {g ∈ O(n, p) | detg = 1}. Clearly, E1n and E0n are known as the n-dimensional pseudo-Euclidean space of index 1 and the n-dimensional Euclidean space, respectively (see [22,29]). Let Sim(Epn ) be the group of all pseudo-similarities of Epn , Sim+ (Epn ) be the group of all orientation-preserving pseudo-similarities of Epn , SimL (Epn ) be the n group of all linear pseudo-similarities of Epn and Sim+ L (Ep ) be the group of all n orientation-preserving linear pseudo-similarities of Ep . The groups Sim(E0n ) 0123456789().: V,-vol
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˙ Oren, ¨ I. D. Khadjiev
J. Geom.
and Sim(E1n ) are known as the fundamental groups of the Euclidean similarity geometry and Lorentzian similarity geometry, resp.(For these definitions, see [18,21,27]). The theory of local invariants of curves in E02 for the group Sim(E02 ) is given without proofs by [11]. In [18], geometric invariants of curves in similarity geometry considered. Existence and rigidity theorems for a curve obtained only for the group Sim+ (E0n ). As a generalization of the paper [18], some characterizations of a curve α using the arc length parameters of its Vi indicatrix and an arc length parameter of the curve α are given in [5]. Thus invariant theory of curves in the similarity geometry in E0n was developed only for the group Sim+ (E0n ). As it is well known, global differential G-invariants are an important tool for many areas in sciences. Therefore, using complex numbers, the differential geometry of plane paths and plane curves under linear similarity transformations have been improved by [24]. Similarity transformations play an important role in mathematics and it has many applications in physics, mechanics and computer sciences (see some references [1,2,7–9,14,15,19,20,25,26,
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