COMPUTER AIDED GEOMETRIC MODELING OF SOLUTIONS TO THE TASKS OF APPLIED CYCLOGRAPHY
Abstract and keywords
Abstract (English):
In the present paper the solutions based on cyclographic method are considered on the example of two applied tasks: generation of road surface forms and pocket machining process engineering. Geometric structures based on cyclographic mapping of space E3 on plane E2 and the corresponding mathematical models in the form of systems of parametric equations are provided. On the basis of the developed models, analytical solutions to the problems of shaping the surface and linear forms of the studied objects in the areas of road design and surface treatment of mechanical engineering products were obtained. The models develop the authors’ previous research and are aimed at comprehensive solution to the problems of surface form generation in application to the two mentioned tasks.

Keywords:
cyclographic method, mapping, mathematical modeling, roads, offset curves, pocket surfaces
References

1. Automotive roads: Construction Norms and Rules 2.05.02-85 Introduced 1987-01-01. Gosstroy of USSR. Moscow,TSITP Gosstroya USSR, 1986.

2. Boykov V. N., Fedotov G. A., Purkin V. I. AutomatedAutomotive Road Design (on the example ofIndorCAD/Road) Moscow, MADI, 2005

3. Choi H.I., Han C.Y., Moon H.P ., Roh K.H., Wee N.S.:Medial axis transform and offset curves by MinkowskiPythagorean hodograph curves, Computer-Aided Design31 (1999), 5 p.p. 9–72.

4. Choi H. I., Choi S. W., and Moon H. P. Mathematicaltheory of medial axis transform. Pacific J. Math.,181(1):56–88, 1997.

5. Cho H.Ch., Choi H.I., Kwon S.-H., Lee D.S. and Wee N.-S. Clifford algebra, Lorentzian geometry and rationalparametrization of canal surfaces. Computer AidedGeometric Design, 21:327–339, 2004.

6. Dr. Emil Muller. Vorlesungenüber DarstellendeGeometrie. II. Band: Die Zyklographie. Edited from themanuscript by Dr. Josef Leopold Krames. Leipzig andVienna, Franz Deuticke,1929. - 476 pp.

7. Held M 1991 On the computational geometry of pocketmachining Lect. Notes in Comp. Sci. p 184.

8. Myasoedova T. M., Panchuk K. L. Geometric model ofgeneration of family of contour-parallel trajectories(equidistant family) of a machine tool. // IOP Conf. Series:Journal of Physics: Conf. Series. –2019. – vol. 1210(1). –p. 012104. doi: 10.1088/1742-6596/1210/1/012104

9. Panchuk K. L., Lyubchinov E.V., Myasoedova T.M.Cyclography. Aspects of Theory and PracticalApplications. GraphiCon 2018: 28th The 28thInternational Conference on Computer Graphics andVision. Conf. Proceedings. Tomsk, Tomsk PolitechnicalUniv., 2018. pp. 336 - 340.

10. Panchuk K. L., Kaygorodtseva N. V. CyclographicDesctiptive Geometry. Omsk, OmGTU, 2017 232 p.

11. Panchuk K.L., A. S. Niteyskiy, E. V. Lyubchinov.Cyclographic Modeling of Surface Forms of Highways.IOP Conf. Series: Materials Science and Engineering 262(2017) 012108. doi:10.1088/1757-899X/262/1/012108

12. Panchuk K. L., Lyubchinov E. V., Krysova I. V.Surfacetriads with optical properties. IOP Conf. Series: Journal ofPhysics: Conf. Series 944 (2018) 012086. doi:10.1088/1742-6596/944/1/012086

13. Peternell M. Rational two-parameter families of spheresand rational offset surfaces. J. Symbolic Computation 45(2010), 1-18.

14. Peternell M., Pottmann H. Computing rationalparametrizations of Canal Surfaces. J. SymbolicComputation 23 (1997), 255–266.

15. Pottmann H., Peternell M. Applications of LaguerreGeometry in CAGD, Comp. Aided Geometric Design 15(1998), 165–186.

16. Pottmann H., Wallner J. Computational Line Geometry.Berlin. Heidelberg: Springer Verlag, 2001. 565 p.

17. Purkin V. I. Basic Automated Design of AutomotiveRoads. Moscow, MADI, 2000.

18. Salkov N.A. Modeling of Geometric Forms of MotorRoads. Moscow, INFRA-M, 2012.