×

Some new characteristic properties of the a-pedal hypersurfaces in \(E^{n+1}\). (English) Zbl 1258.53010

Summary: The primary purpose of this paper is to present the definition of the a-pedal hypersurface with respect to a point in the interior of a closed, convex and smooth hypersurface \(M\). The secondary purpose of this paper is to give some new characteristic properties of the a-pedal hypersurfaces related to the support function, Gauss curvature, mean curvature, the first and second fundamental forms and their coefficients of \(M\) (Section 3). Using the classical methods of the hypersurfaces in differential geometry we prove that the support function \(h_{a}\) of the a-pedal hypersurface \(M_{a}\) is equal to \(\frac{h^{a+1}}{P_{a}}\) where \(P_{a}^{2}=h^{2}+a^{2}\overset{III}{\nabla}(h,h)\).

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53A15 Affine differential geometry
PDFBibTeX XMLCite
Full Text: Euclid

References:

[1] C. Georgiou, T. Hasanis, and D. Koutroufiotis, The Pedal of a Hypersurface Revisited , Technical Report No. 96, 1983. · Zbl 0538.53048
[2] C. Georgiou, T. Hasanis, and D. Koutroufiotis, On the caustic of a convex mirror , Geometria Dedicata, 28 (1988), 153-158. · Zbl 0659.53004 · doi:10.1007/BF00147448
[3] W. H. Guggenheimer, Differential Geometry , McGraw-Hill, New York, 1963. · Zbl 0116.13402
[4] H. H. Hacısalihoğlu, Diferensiyel Geometri , Mat. No. 2, \.Inönü Üniversitesi Fen-Edebiyat Fakültesi Yayınları, Malatya, 1983.
[5] A. S. Hassan, Higher order Gaussian curvatures of parallel hypersurfaces , Commun. Fac. Sci. Univ. Ank., Series A1, 46 (1997), 67-76. · Zbl 0935.53005
[6] N. J. Hicks, Notes on Manifolds , Van Nostrand Reinhold Company, London, 1974.
[7] C.-C. Hsiung, Some global theorems on hyper-surfaces , Canad J. Math., 9 (1957), 5-14. · Zbl 0085.15901 · doi:10.4153/CJM-1957-002-1
[8] N. Kuruoğlu, Some new characteristic properties of the pedal surfaces in Euclidean space , Pure and Applied Mathematika Sciences, 23 (1986), no. 1-2, 7-11. · Zbl 0605.53003
[9] N. Kuruoğlu and A. Sarıoğlugil, On the characteristic properties of the hyperpedal surfaces in the \((n+1)\)-dimensional Euclidean space , Pure and Applied Mathematika Sciences, 55 (2002), no. 1-2, 15-21.
[10] N. Kuruoğlu and A. Sarıoğlugil, On the characteristic properties of the a-pedal surfaces in the Euclidean space , Communications Faculty of Sciences University of Ankara, Series A1, 42 (1993), 19-25. · Zbl 0865.53003
[11] B. O’Neil, Elementary Differential Geometry , Academic Press, New York, 1966.
[12] G. Salmon, Analytic Geometry of Three Dimensions , Vol. II, Chelsea Publishing Company, New York, 1965.
[13] A. Sarıoğlugil and N. Kuruoğlu, On the characteristic properties of the reciprocal surfaces in the Euclidean Space , International Journal of Applied Mathematics, 11.1 (2002), 37-48. · Zbl 1047.53003
[14] A. Sunma, A. Sarıoğlugil, and N. Kuruoğlu, Some new characteristic properties of the reciprocal hypersurfaces in the Euclidean space , Hadronic Journal, 32 (2009), 549-564. · Zbl 1207.53006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.