Download PDF - Integral formula for secantoptics and its applicationCC BY: Creative Commons Attribution 4.0, Opens in new tab

ArticleOriginal scientific text

Title

Integral formula for secantoptics and its application

Authors ,

Abstract

Some properties of secantoptics of ovals defined by Skrzypiec in 2008 were proved by Mozgawa and Skrzypiec in 2009. In this paper we generalize to this case results obtained by Cieslak, Miernowski and Mozgawa in 1996 and derive an integral formula for an annulus bounded by a given oval and its secantoptic. We describe the change of the area bounded by a secantoptic and find the differential equation for this function. We finish with some examples illustrating the above results.

Keywords

ENG
Secantoptic, isoptic, secant

Bibliography

  1. Benko, K., Cieślak, W., Góźdź, S. and Mozgawa, W., On isoptic curves, An. Stiint. Univ. Al. I. Cuza Iasi Sect¸. I a Mat. 36 (1990), no. 1, 47-54.
  2. Cieślak, W., Miernowski, A. and Mozgawa, W., Isoptics of a closed strictly convex curve, Global differential geometry and global analysis (Berlin, 1990), Lecture Notes in Math., 1481, Springer, Berlin, 1991, 28-35.
  3. Cieślak, W., Miernowski, A. and Mozgawa, W., Isoptics of a closed strictly convex curve. II, Rend. Sem. Mat. Univ. Padova 96 (1996), 37-49.
  4. Gage, M., On an area-preserving evolution equation for plane curves, Nonlinear Problems in Geometry (Mobile, Ala., 1985), Contemp. Math., 51, Amer. Math. Soc., Providence, RI, 1986, 51-62.
  5. Green, J. W., Sets subtending a constant angle on a circle, Duke Math. J. 17 (1950), 263-267.
  6. Góźdź, S., On Jordan plane curves which are isoptics of an oval, An. Stiint¸. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 42 (1996), no. 1, 127-130.
  7. Hilton, H., Colomb, R. E., On orthoptic and isoptic loci, Amer. J. Math. 39 (1917), no. 1, 86-94.
  8. Langevin, R., Levitt, G. and Rosenberg, H., Herissons et multiherissons (envellopes parametrees par leur application de Gauss), Singularities (Warsaw, 1985), Banach Center Publ. 20, PWN, Warsaw, 1988, 245-253.
  9. Martinez-Maure, Y., Geometric inequalities for plane hedgehogs, Demonstratio Math. 32 (1999), no. 1, 177-183.
  10. Michalska, M., A sufficient condition for the convexity of the area of an isoptic curve of an oval, Rend. Sem. Mat. Univ. Padova 110 (2003), 161-169.
  11. Miernowski, A., Mozgawa, W., Isoptics of pairs of nested closed strictly convex curves and Crofton-type formulas, Beitr¨age Algebra Geom. 42 (2001), no. 1, 281-288.
  12. Mozgawa, W., Skrzypiec, M., Crofton formulas and convexity condition for secantoptics, Bull. Belg. Math. Soc. Simon Stevin 16 (2009), no. 3, 435-445.
  13. Santalo, L., Integral geometry and geometric probability, Encyclopedia of Mathematics and its Applications, vol. 1. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976.
  14. Skrzypiec, M., A note on secantoptics, Beitrage Algebra Geom. 49, (2008), no. 1, 205-215.
  15. Szałkowski, D., Isoptics of open rosettes, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 59 (2005), 119-128.
  16. Szałkowski, D., Isoptics of open rosettes. II, An. Stiint. Univ. Al. I. Cuza Iasi. Mat. (N.S.) 53 (2007), no. 1, 167-176.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
2012/66/1
Export to PBN, Opens in new tab
Main language of publication
English
Published
2012
Published online
2016-07-24

DOI: 10.17951/a.2012.66.1.49-62

Exact and natural sciences