Abstract and keywords
Abstract (English):
Investigation of singularity related to a mapping by the orthogonal projection in a four-dimensional space of a hypersurface given by parametric equations is presented in this paper. On this investigation’s basis have been proposed in a united way three approaches to the hypersurface’s discriminant determination. Thus, have been defined conditions which the investigated hypersurface’s discriminant set and criminant are satisfied. Have been obtained dependencies settling the relationship between parameters of the hypersurface at its discriminant points. They are used to determine the singularity of the hypersurface mapping by analytical methods in general form. The complexity of this approach (the first one) is that an equation connecting the hypersurface’s parameters contains its differential characteristics, and often is the transcendental one in applications, that causes certain difficulties when solving it. Have been obtained dependences in carrying out of which the hypersurface’s discriminant has an edge of regression. A study of hypersurface sections by hyperplanes which are parallel to coordinate hyperplanes has been performed. The last ones contain a coordinate axis along which the hypersurface mapping is performed. It has been established that curves obtained in these sections have extreme points belonging to the hypersurface’s discriminant. Such property is used to calculate the points of the hypersurface’s discriminant by numerical methods without using the hypersurface’s differential characteristics, and it is a basis for the second approach to solving the problem posed. It has been also demonstrated the use of 3D modeling for study of hypersurface’s different sections, as well as its discriminant that represents the third approach to the study. All three approaches having a common basis can be used both independently and complement each other in determining the envelope for one-parameter family of surfaces. As an example has been considered a hypersurface formed by a family of spheres. Based on stated results have been obtained equations determining the hypersurface’s discriminant and this family’s corresponding envelope, as well as various sections. These equations have been used for creating of polygonal 3D models of the hypersurface’s discriminant, and some of the hypersurface’s sections.

mapping singularity, hypersurface, discriminant, envelope, geometric modeling.

1. Введение
Понятие огибающей семейства линий или поверхностей широко используется в различных приложениях. В теории зацеплений огибающие используется для определения сопряженных поверхностей в зубчатых зацеплениях.


1. Arnol'd V.I. Osobennosti gladkikh otobrazheniy [Singularities of smooth maps]. Uspekhi matematicheskikh nauk [Progress in mathematical sciences]. 1968, V. XXIII, I. 1, pp. 4–44.

2. Brus Dzh., Dzhiblin P. Krivye i osobennosti [Curves and features]. Moscow, Mir Publ., 1988. 262 p.

3. Bykov V.I., Naykhanov V.V. Opredelenie konturnoy linii na poverkhnosti, zadannoy uravneniem v neyavnoy forme [Determination of a contour line on a surface defined by an equation in implicit form]. Tezisy Vsesoyuznogo nauchno-metodicheskogo simpoziuma “Primenenie sistem avtomatizirovannogo proektirovaniya konstruktsiy v mashinostroenii” [Theses of the All-Union Scientific and Methodological Symposium "Application of Automated Design of Structures in Mechanical Engineering"]. Rostov-on-Don, 1983, pp. 40–41.

4. Golovanov N.N. Geometricheskoe modelirovanie [Geometric modeling]. Moscow, Fiziko-matematicheskaya literature Publ., 2002. 472 p.

5. Zalgaller V.A. Teoriya ogibayushchikh [Theory of Envelopes]. Moscow, Nauka Publ., 1975. 104 p.

6. Karacharovskiy V.Yu. Geometricheskoe modelirovanie formoobrazovaniya prostranstvennykh poverkhnostey pri vintovom otnositel'nom dvizhenii [Geometric modeling of the formation of spatial surfaces under screw relative motion]. Problemy geometricheskogo modelirovaniya v avtomatizirovannom proektirovanii i proizvodstve: 1-ya Mezhdunarodnaya nauchnaya konferentsiya [Problems of geometric modeling in automated design and production: 1st International Scientific Conference]. Moscow, MGIU Publ., 2008, pp. 143–146.

7. Kozlov Yu.V. Modelirovanie protsessa frezerovaniya zubchatykh koles i otsenka ikh kinematicheskikh pogreshnostey [Modeling the process of milling gears and evaluating their kinematic errors]. Vestnik Belorussko-Rossiyskogo universiteta [Bulletin of the Belarusian-Russian University]. 2008, I. 3, pp. 82–89.

8. Korotkiy V.A. Geometricheskoe modelirovanie poverkhnosti posredstvom ee otobrazheniya na chetyrekhmernoe prostranstvo [Geometric modeling of a surface by means of its mapping to four-dimensional space]. Omskiy nauchnyy vestnik [Omsk Scientific Herald]. 2015, I. 137, pp. 8–12.

9. Korotkiy V.A., Khmarova L.I., Usmanova E.A. Komp'yuternoe modelirovanie kinematicheskikh poverkhnostey [Computer Simulation of Kinematic Surfaces]. Geometriya i grafika [Geometry and graphics]. 2015, V. 3, I. 4, pp. 19–26. (in Russian). DOI: 10.12737/17347.

10. Korotkiy V.A. Komp'yuternoe modelirovanie figur chetyrekhmernogo prostranstva [Computer modeling of four-dimensional space figures]. Vestnik komp'yuternykh informatsionnykh tekhnologiy [Herald of computer information technologies]. 2014, I. 7, pp. 14–20. DOI: 10.14489/vkit.2014.07.pp.014-020.

11. Lashnev S.I. Raschet i konstruirovanie metallorezhushchikh instrumentov s primeneniem EVM [Calculation and design of metal-cutting tools with the use of computers]. Moscow, Mashinostroenie Publ., 1975. 392 p.

12. Litvin F.L. Teoriya zubchatykh zatsepleniy [Theory of gearing]. Moscow, Nauka Publ., 1968. 584 p.

13. Lyashkov A.A. Modelirovanie formoobrazovaniya vintovykh poverkhnostey detaley reykoy i chervyachnoy frezoy [Modeling of the Forming of Screw Surfaces by Rack and Worm Cutter]. Metalloobrabotka [Metalloobrabotka]. St. Petersburg, Politekhnika Publ., 2011, I. 1, pp. 2–7.

14. Lyashkov A.A. Komp'yuternoe modelirovanie protsessa formoobrazovaniya diskovoy frezoy detaley s vintovoy poverkhnost'yu [Computer modeling of the process of shaping by disk milling of parts with a screw surface]. STIN [STIN]. 2012, I. 1, pp. 26–29.

15. Nesmelov I.P. Nedifferentsial'nyy podkhod k resheniyu zadachi ogibaniya [A non-differential approach to the solution of the envelope problem]. "Mekhanika mashin" ["Mechanics of machines"]. 1983, I. 61, pp. 3–10.

16. Platonova O.A. Osobennosti proektsiy gladkikh poverkhnostey [Peculiarities of projections of smooth surfaces]. Uspekhi matematicheskikh nauk [Progress in mathematical sciences]. 1984, V. 39, I. 1, pp. 149–150.

17. Platonova O.A. Proektsii gladkikh poverkhnostey [Projections of smooth surfaces]. Trudy Seminara im. I.G. Petrovskogo [Proceedings of the Seminar. I.G. Petrovsky]. 1984, V. 10, pp. 135–149.

18. Platonova O.A. Osobennosti proektirovaniy gladkikh poverkhnostey [Features of projections of smooth surfaces]. Uspekhi matematicheskikh nauk [Progress in mathematical sciences]. 1979, V. 34, I. 2, pp. 3–38.

19. Pshenichnyy B.N. Neobkhodimoe uslovie ekstremuma [Necessary condition for an extremum]. Moscow, 1969. 151 p.

20. Rashevskiy P.K. Kurs differentsial'noy geometrii [Course of differential geometry]. Moscow, Gos. tekhn.-teor. liter. Publ., 1956. 420 p.

21. Sal'kov N.A. Geometricheskoe modelirovanie i nachertatel'naya geometriya [Geometric modeling and descriptive geometry]. Geometriya i grafika [Geometry and graphics]. 2016, V. 4, I. 4, pp. 31–40. DOI: 10.12737/22841

22. Tolstov G.P. K otyskaniyu ogibayushchey semeystva ploskikh krivykh [To find the envelope of a family of plane curves]. UMN Publ., V. 7, I. 4, 1952, pp. 173–179.

23. Uayld D.Dzh. Metody poiska ekstremuma [Methods for finding an extremum]. Moscow, Nauka Publ., 1967. 267 p.

24. Sheveleva G.I. Teoriya formoobrazovaniya i kontakta dvizhushchikhsya tel [The theory of the formation and contact of moving bodies]. Moscow, Mosstankin Publ., 1999. 494 p.

25. Litvin F.L. Alfonso Fuentes Geometry and Applied Theory [Tekst] / F.L. Litvin – Cembridge University Press, 2004. – 816 pp.

26. Schulz T. Envelope Computation by Approximate Implicitization / T. Schulz, B. Juttler // Industrial Geometry, 2010. 20 p.

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