LS(2)-Equivalence conditions of control points and application to planar Bezier curves

Abstract

Having an important role in CAD and CAM systems the Bezier and B- spline curves and surfaces and NURBS modelling are based on control points belongs to these curves and surfaces. So the invariants of these curves and surfaces are the invariants of the control points of these curves and surfaces. In this study we studied the equivalence conditions of compared two different control point systems under the linear similarity transformations LS(2) in R² according to the invariant system of these control points. Finally the equivalence conditions of two planar Bezier curves is examined. 

Keywords

• Linear Similarity,equivalence conditions,control points

References

1. Dj. Khadjiev, Some Questions in the Theory of Vector Invariants, Math. USSR- Sbornic, 1(3)(1967), 383-396.
2. Grosshans F., Obsevable Groups and Hilbert’s Problem, American Journal of Math., 95(1973), 229-253.
3. H. Weyl, The Classical Groups, Their Invariants and Representations, 2nd ed., with suppl., Princeton, Princeton University Press, 1946.
4. Dj. Khadjiev , An Application of the Invariant Theory to the Differential Geometry of Curves, Fan, Tashkent, 1988. (in Russian)
5. F. Klein, A comperative review of recent researches in geometry (translated by Dr. M.W. Haskell), Bulletin of the New York Mathematical Society, 2(1893),215-249.
6. G. Birkhoff, Hydrodynamics, second Ed. Princeton, New Jersey, Princeton Univ. Pres, 1960.
7. J. B. J. Fourier, Theoric Analytique de la Chaleur, 1822 (English Transl. By A. Freeman, The Analytical Theory of Heat, Cambridge University Press,1878).
8. P. W. Bridgman, Dimensional Analysis, 2 nd Ed. Yale University Press, New Heaven, 1931.

9. L. I. Sedov, Similarity and Dimensional Method in Mechanics, English Tr. By V.Kisin, Mir Publishers, USSR, 1982.
10. H. L. Langhaar , Dimensional Analysis and Theory of Models,Wiley, 1951.
11. M. Incesu, The Complete System of Point Invariants in the Similarity Geometry, Phd. Thesis, Karadeniz Technical University, Trabzon, 2008.
12. Marsh D., Applied Geometry for Computer Graphics and CAD, Springer-Verlag London Berlin Heidelberg, London, 1999.
13. Farin G.,Curves and Surfaces for Computer Aided Geometric Design A Practical Guide, 2nd edition, Academic Press Inc., San Diago, 1990.
14. Farouki R. and Rajan V. T., On the numerical condition of polynomials in Bernstein form. Computer Aided Geometric Design, 4(3)(1987),191-216 .
15. Sağiroğlu Y., "The Equivalence Problem For Parametric Curves In One-Dimensional Affine Space", International Mathematical Forum, 6(2011), 177-184.
16. Sağiroğlu Y., "Equi-affine differential invariants of a pair of curves", TWMS Journal of Pure and Applied Mathematics, 6(2015),238-245.
17. Sağiroğlu Y., Pekşen Ö., "The Equivalence Of Centro-Equiaffine Curves", Turkish Journal of Mathematics, 34(2010),95-104.
18. Ören İ., "Complete System of Invariants of Subspaces of Lorentzian Space", Iranian Journal of Science and Technology Transaction A-Science, 40(3) (2016),1-8.
19. Khadjiev (Hacioğlu) D., Ören İ., Pekşen Ö., "Generating systems of differential invariants and the theorem on existence for curves in the pseudo-Euclidean geometry", Turkish Journal of Mathematics, 37(2013),80-94.
20. O. Deveci and E. Karaduman, The cyclic groups via the Pascal matrices and the generalized Pascal matrices, Linear Algebra and its Appl., 437(2012),2538-2545.