Research Article
BibTex RIS Cite

Minkowski Düzleminde Eğrilerin Anti-Pedalları ve İlkelleri

Year 2022, , 92 - 99, 28.02.2022
https://doi.org/10.35414/akufemubid.1026512

Abstract

Verilen bir eğrinin teğet doğruları üzerindeki sabit bir noktanın dik izdüşümü, o eğrinin bir pedal eğrisini oluşturur. Bu çalışmanın amacı, düzgün eğriler için bile tekil noktaları olan pedal eğriler gibi bazı özel eğrileri Minkowski düzleminde incelemektir. Bunun için, pedal eğrisi ile yakından ilişkili olan, eğrinin anti-pedalını ve ilkelini araştırdık. Bir eğrinin ilkeli, pedal yapmak için ters yapı tarafından sağlanan bir eğridir. Bir fonksiyon ailesinin örtüsünü kullanarak, Minkowski düzlemindeki eğriler için ilkel kavramını elde ettik. Daha sonra, orijinal eğrinin bir anti-pedalının, pedal eğrisinin inversiyon görüntüsüne eşit olduğunu gösterdik. Dahası, inversiyonu kullanarak eğrinin ilkeli ve anti-pedalı arasındaki ilişkileri analiz ettik. Ayrıca, sonuçlarımızı sağlayan örnekler sunduk.

References

  • Arnold, V.I., 1989, Dynamical systems VIII, Encyclopedia of mathematical sciences, 39, Springer, 88-92.
  • Arnold, V.I., 1990, Singularities of caustics and wave fronts, 62, Springer, 1-56.
  • Aydın Şekerci, G. and Izumiya, S., 2021. Evolutoids and pedaloids of Minkowski plane curves. Bulletin of the Malaysian Mathematical Sciences Society, 44, 2813-2834.
  • Bakurova, V., 2013. On singularities of pedal curve in the Minkowski plane. Proceedings of symposium on computer geometry SCG, 22, 5-10.
  • Giblin, P.J. and Warder J.P., 2014. Evolving evolutoids. The American Mathematical Monthly, 121, 871-889.
  • Izumiya, S., Romero Fuster, M.C. and Takahashi, M., 2018. Evolutes of curves in the Lorentz-Minkowski plane. Advanced Studies in Pure Mathematics, 78, 313-330.
  • Izumiya, S. and Takeuchi, N., 2019a. Evolutoids and pedaloids of plane curves. Note di Matematica, 39, 13-23.
  • Izumiya, S. and Takeuchi, N., 2019b. Pedals and inversions of quadratic curves. Arxiv: 1912.03114v1, 1-18.
  • Izumiya, S. and Takeuchi, N., 2020. Primitivoids and inversions of plane curves. Beitrage zur Algebra und Geometrie, 61, 317-334.
  • Li, Y. and Sun Q.Y., 2019. Evolutes of fronts in the Minkowski plane. Mathematical Methods in the Applied Sciences, 42, 5416-5426.
  • Nishimura, T., 2008. Normal forms for singularities of pedal curves produced by non-singular dual curve germs in Sn. Geometriae Dedicata, 133, 59-66.
  • O’Neill, B., 1983, Semi-Riemannian geometry with applications to relativity, 1, Academic Press Inc., 46-58.
  • Thom, R., 1956. Les singularities des applications differentiables. Ann. Inst. Fourier, 6, 43-87.

Anti-pedals and Primitives of Curves in Minkowski Plane

Year 2022, , 92 - 99, 28.02.2022
https://doi.org/10.35414/akufemubid.1026512

Abstract

The orthogonal projection of a fixed point on the tangent lines of a given curve yields a pedal curve of that curve. The aim of this study is to examine some special curves, such as pedal curves, which have singular points even for regular curves, in the Minkowski plane. For this, we investigate an anti-pedal and a primitive of curve, which is closely related to the pedal curve. The primitive of a curve is a curve that is provided by the inverse construction to make pedal. Using the envelope of a family of functions, we obtain the notion of primitive for the curves in the Minkowski plane. Then, we show that an anti-pedal of the original curve is equal to the inversion image of the pedal curve. Moreover, we analyze the relationships between primitive and anti-pedal of the curve using the inversion. We also present examples that provide our results.

References

  • Arnold, V.I., 1989, Dynamical systems VIII, Encyclopedia of mathematical sciences, 39, Springer, 88-92.
  • Arnold, V.I., 1990, Singularities of caustics and wave fronts, 62, Springer, 1-56.
  • Aydın Şekerci, G. and Izumiya, S., 2021. Evolutoids and pedaloids of Minkowski plane curves. Bulletin of the Malaysian Mathematical Sciences Society, 44, 2813-2834.
  • Bakurova, V., 2013. On singularities of pedal curve in the Minkowski plane. Proceedings of symposium on computer geometry SCG, 22, 5-10.
  • Giblin, P.J. and Warder J.P., 2014. Evolving evolutoids. The American Mathematical Monthly, 121, 871-889.
  • Izumiya, S., Romero Fuster, M.C. and Takahashi, M., 2018. Evolutes of curves in the Lorentz-Minkowski plane. Advanced Studies in Pure Mathematics, 78, 313-330.
  • Izumiya, S. and Takeuchi, N., 2019a. Evolutoids and pedaloids of plane curves. Note di Matematica, 39, 13-23.
  • Izumiya, S. and Takeuchi, N., 2019b. Pedals and inversions of quadratic curves. Arxiv: 1912.03114v1, 1-18.
  • Izumiya, S. and Takeuchi, N., 2020. Primitivoids and inversions of plane curves. Beitrage zur Algebra und Geometrie, 61, 317-334.
  • Li, Y. and Sun Q.Y., 2019. Evolutes of fronts in the Minkowski plane. Mathematical Methods in the Applied Sciences, 42, 5416-5426.
  • Nishimura, T., 2008. Normal forms for singularities of pedal curves produced by non-singular dual curve germs in Sn. Geometriae Dedicata, 133, 59-66.
  • O’Neill, B., 1983, Semi-Riemannian geometry with applications to relativity, 1, Academic Press Inc., 46-58.
  • Thom, R., 1956. Les singularities des applications differentiables. Ann. Inst. Fourier, 6, 43-87.
There are 13 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Gülşah Aydın Şekerci 0000-0001-9130-2781

Publication Date February 28, 2022
Submission Date November 20, 2021
Published in Issue Year 2022

Cite

APA Aydın Şekerci, G. (2022). Anti-pedals and Primitives of Curves in Minkowski Plane. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, 22(1), 92-99. https://doi.org/10.35414/akufemubid.1026512
AMA Aydın Şekerci G. Anti-pedals and Primitives of Curves in Minkowski Plane. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. February 2022;22(1):92-99. doi:10.35414/akufemubid.1026512
Chicago Aydın Şekerci, Gülşah. “Anti-Pedals and Primitives of Curves in Minkowski Plane”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 22, no. 1 (February 2022): 92-99. https://doi.org/10.35414/akufemubid.1026512.
EndNote Aydın Şekerci G (February 1, 2022) Anti-pedals and Primitives of Curves in Minkowski Plane. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 22 1 92–99.
IEEE G. Aydın Şekerci, “Anti-pedals and Primitives of Curves in Minkowski Plane”, Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 22, no. 1, pp. 92–99, 2022, doi: 10.35414/akufemubid.1026512.
ISNAD Aydın Şekerci, Gülşah. “Anti-Pedals and Primitives of Curves in Minkowski Plane”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi 22/1 (February 2022), 92-99. https://doi.org/10.35414/akufemubid.1026512.
JAMA Aydın Şekerci G. Anti-pedals and Primitives of Curves in Minkowski Plane. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2022;22:92–99.
MLA Aydın Şekerci, Gülşah. “Anti-Pedals and Primitives of Curves in Minkowski Plane”. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi, vol. 22, no. 1, 2022, pp. 92-99, doi:10.35414/akufemubid.1026512.
Vancouver Aydın Şekerci G. Anti-pedals and Primitives of Curves in Minkowski Plane. Afyon Kocatepe Üniversitesi Fen Ve Mühendislik Bilimleri Dergisi. 2022;22(1):92-9.


Bu eser Creative Commons Atıf-GayriTicari 4.0 Uluslararası Lisansı ile lisanslanmıştır.