Integral formula for secantoptics and its application

Witold Mozgawa; Magdalena Skrzypiec

Annales UMCS, Mathematica (2012)

  • Volume: 66, Issue: 1, page 49-62
  • ISSN: 2083-7402

Abstract

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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 Cieślak, 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.

How to cite

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Witold Mozgawa, and Magdalena Skrzypiec. "Integral formula for secantoptics and its application." Annales UMCS, Mathematica 66.1 (2012): 49-62. <http://eudml.org/doc/268064>.

@article{WitoldMozgawa2012,
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 Cieślak, 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.},
author = {Witold Mozgawa, Magdalena Skrzypiec},
journal = {Annales UMCS, Mathematica},
keywords = {Secantoptic; isoptic; secant; secantoptic; oval; integral formula},
language = {eng},
number = {1},
pages = {49-62},
title = {Integral formula for secantoptics and its application},
url = {http://eudml.org/doc/268064},
volume = {66},
year = {2012},
}

TY - JOUR
AU - Witold Mozgawa
AU - Magdalena Skrzypiec
TI - Integral formula for secantoptics and its application
JO - Annales UMCS, Mathematica
PY - 2012
VL - 66
IS - 1
SP - 49
EP - 62
AB - 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 Cieślak, 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.
LA - eng
KW - Secantoptic; isoptic; secant; secantoptic; oval; integral formula
UR - http://eudml.org/doc/268064
ER -

References

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  1. Benko, K., Cieślak, W., Góźdź, S. and Mozgawa, W., On isoptic curves, An. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. 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. Zbl0739.53001
  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. Zbl0881.53003
  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. Zbl0039.18201
  6. Góźdź, S., On Jordan plane curves which are isoptics of an oval, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 42 (1996), no. 1, 127-130. Zbl0912.53003
  7. Hilton, H., Colomb, R. E., On orthoptic and isoptic loci, Amer. J. Math. 39 (1917), no. 1, 86-94. Zbl46.0954.02
  8. Langevin, R., Levitt, G. and Rosenberg, H., Hérissons et multihérissons (envellopes paramétrées par leur application de Gauss), Singularities (Warsaw, 1985), Banach Center Publ. 20, PWN, Warsaw, 1988, 245-253. Zbl0658.53004
  9. Martinez-Maure, Y., Geometric inequalities for plane hedgehogs, Demonstratio Math. 32 (1999), no. 1, 177-183. Zbl0931.53008
  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. Zbl1121.52011
  11. Miernowski, A., Mozgawa, W., Isoptics of pairs of nested closed strictly convex curves and Crofton-type formulas, Beiträge Algebra Geom. 42 (2001), no. 1, 281-288. Zbl1021.53050
  12. Mozgawa, W., Skrzypiec, M., Crofton formulas and convexity condition for secantoptics, Bull. Belg. Math. Soc. Simon Stevin 16 (2009), no. 3, 435-445. Zbl1178.53001
  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, Beiträge Algebra Geom. 49, (2008), no. 1, 205-215. Zbl1155.53003
  15. Szałkowski, D., Isoptics of open rosettes, Ann. Univ. Mariae Curie-Skłodowska, Sect. A 59 (2005), 119-128. Zbl1135.53004
  16. Szałkowski, D., Isoptics of open rosettes. II, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 53 (2007), no. 1, 167-176. Zbl1150.53002

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