Complexes in three-dimensional quasi-hyperbolic space // BSU bulletin. Mathematics, Informatics. - 2016. №1. . - С. 9-15.

Complexes in three-dimensional quasi-hyperbolic space

In the article the canonical frame of a complex is constructed. This frame is

geometrically characterized by the fact that in normal correlation the points A0

and A1 (centers of complex-ray) correspond to the planes (A0 A1 A2) and (A0 A1 A3), which are polar conjugated with respect to the absolute and cross absolute line to the points

A2 and A3 . The theorem of existence is proved. We have given the geometric characteristics of the complex invariants using three simple ruled surfaces (central surface and two central torses) belonging to the complex.

Two main quadratic forms of the complex have been obtained. The ruled surfaces conjugated with respect to the first quadratic form are characterized by the harmonic conjugation of their adherent points. The surfaces conjugated with respect to the second quadratic form are characterized by the harmonic conjugation of the adherent points of one of them with the symmetry points of the other.

We have obtained the equation of inflectional centers of the complex generatrices, the conditions characterizing the linear complex, and found some spe- cial classes of the complexes.

geometrically characterized by the fact that in normal correlation the points A0

and A1 (centers of complex-ray) correspond to the planes (A0 A1 A2) and (A0 A1 A3), which are polar conjugated with respect to the absolute and cross absolute line to the points

A2 and A3 . The theorem of existence is proved. We have given the geometric characteristics of the complex invariants using three simple ruled surfaces (central surface and two central torses) belonging to the complex.

Two main quadratic forms of the complex have been obtained. The ruled surfaces conjugated with respect to the first quadratic form are characterized by the harmonic conjugation of their adherent points. The surfaces conjugated with respect to the second quadratic form are characterized by the harmonic conjugation of the adherent points of one of them with the symmetry points of the other.

We have obtained the equation of inflectional centers of the complex generatrices, the conditions characterizing the linear complex, and found some spe- cial classes of the complexes.

non-Euclidean space, quasi-hyperbolic space, absolute, com- plex, frame, normal correlation, invariants.

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2. Tsyrenova V. B. Kompleksy v trekhmernom kvaziellipticheskom pro- stranstve [Complexes in Three-Dimensional Quasi-Elliptic Space]. Geometri- cheskii sbornik – Geometric Collection. 1985. No. 25. Pp. 91-100.

3. Tsyrenova V. B., Proskuryakova I. V. Kompleksy v trekhmernom kvazi- giperbolicheskom prostranstve [Complexes in Three-Dimensional Quasi- Hyperbolic Space]. Vestnik Buryatskogo gosudarstvennogo universiteta. Ma- tematika, informatika – Bulletin of Buryat State University. Mathematics, In- formatics. 2011. No. 1. Pp. 92–94.

2. Tsyrenova V. B. Kompleksy v trekhmernom kvaziellipticheskom pro- stranstve [Complexes in Three-Dimensional Quasi-Elliptic Space]. Geometri- cheskii sbornik – Geometric Collection. 1985. No. 25. Pp. 91-100.

3. Tsyrenova V. B., Proskuryakova I. V. Kompleksy v trekhmernom kvazi- giperbolicheskom prostranstve [Complexes in Three-Dimensional Quasi- Hyperbolic Space]. Vestnik Buryatskogo gosudarstvennogo universiteta. Ma- tematika, informatika – Bulletin of Buryat State University. Mathematics, In- formatics. 2011. No. 1. Pp. 92–94.