Geodesic
Curves whose arcs with a fixed origin are similar
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2023), pp. 26-40.

Voir la notice de l'article provenant de la source Math-Net.Ru

The author previously put forward the hypothesis that in the $n$-dimensional Euclidean space $E^n$, curves, any two oriented arcs of which are similar, are rectilinear. The same statement was proven for dimensions $n=2$ and $n=3$. In a space of arbitrary dimension, the hypothesis found its confirmation in the class of rectifiable curves. The work provides a complete solution to the problem, and in a stronger version: a) a curve in $E^n$, any two oriented arcs of which with a common origin (not fixed) are similar, is rectilinear; b) if a curve in $E^n$ has a half-tangent at its boundary point and any two of its oriented arcs with a beginning at this point are similar, then the curve is rectilinear; c) if a curve in $E^n$ has a tangent at an interior point and all its oriented arcs starting at this point are similar, then the curve is rectilinear. Examples of curves in $E^2$ and $E^3$ are given, in which all arcs with a common origin are similar, but they are not rectilinear, and a complete description of such curves in $E^2$ is also given. Research methods are topological, set-theoretic, using the apparatus of functional equations.
Keywords: straight line, curve with similar arcs, criterion for the straightness of a curve, logarithmic spiral, concho-spiral, functional equation, functional exponential Cauchy equation.
Mots-clés : tangent cone 
@article{IVM_2023_11_a2,
     author = {I. V. Polikanova},
     title = {Curves whose arcs with a fixed origin are similar},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {26--40},
     publisher = {mathdoc},
     number = {11},
     year = {2023},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2023_11_a2/}
}
TY  - JOUR
AU  - I. V. Polikanova
TI  - Curves whose arcs with a fixed origin are similar
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2023
SP  - 26
EP  - 40
IS  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2023_11_a2/
LA  - ru
ID  - IVM_2023_11_a2
ER  - 
%0 Journal Article
%A I. V. Polikanova
%T Curves whose arcs with a fixed origin are similar
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2023
%P 26-40
%N 11
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2023_11_a2/
%G ru
%F IVM_2023_11_a2
I. V. Polikanova. Curves whose arcs with a fixed origin are similar. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2023), pp. 26-40. http://geodesic.mathdoc.fr/item/IVM_2023_11_a2/

[1] Mandelbrot B., Fraktalnaya geometriya prirody, In-t komp. issledov., M., 2002

[2] Polikanova I.V., “Kriterii pryamolineinosti krivoi”, Materialy Mezhdunarodnoi konferentsii «Klassicheskaya i sovremennaya geometriya», posvyaschennoi 100-letiyu so dnya rozhdeniya professora Levona Sergeevicha Atanasyana (15 iyulya 1921 g. - 5 iyulya 1998 g.) (Moskva, 1-4 noyabrya 2021 g.), v. 1, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 220, VINITI RAN, M., 2023, 86–98 | DOI

[3] Norden A.P., Differentsialnaya geometriya, Uchebn. posobie dlya ped. in-tov, Uchpedgiz, M., 1948

[4] Savelov A.A., Ploskie krivye: sistematika, svoistva, primeneniya $($spravochnoe rukovodstvo$)$, FIZMATLIT, M., 1960

[5] Aleksandrov P.S., Vvedenie v teoriyu mnozhestv i obschuyu topologiyu, Nauka, M., 1977

[6] Atsel Ya., “Nekotorye obschie metody v teorii funktsionalnykh uravnenii odnoi peremennoi. Novye primeneniya funktsionalnykh uravnenii”, UMN, 11:3 (1956), 3–68 | MR

[7] Atsel Ya., Dombr Zh., Funktsionalnye uravneniya s neskolkimi peremennymi, FIZMATLIT, M., 2003

[8] Polikanova I.V., “O liniyakh s affinno-ekvivalentnymi dugami v n-mernom affinnom prostranstve”, Sib. matem. zhurn., 63:1 (2022), 180–196 | MR | Zbl

[9] Polikanova I.V., “O ploskikh liniyakh s affinno-ekvivalentnymi dugami”, Sib. elektron. matem. izv., 15 (2018), 882–889 | MR | Zbl

[10] Tetenov A.V., “Samopodobnye zhordanovy dugi i graf-orientirovannye sistemy podobii”, Sib. matem. zhurn., 47:5 (2006), 1147–1159 | MR | Zbl

[11] Bandt C., Kravchenko A.S., “Differentiability of fractal curves”, Nonlinearity, 24 (2011), 2717–2728 | DOI | MR | Zbl

[12] Tetenov A.V., “Teoremy o stroenii samoaffinnykh zhordanovykh dug”, Mezhdunarodnaya konferentsiya po geometricheskomu analizu, posvyaschennaya pamyati akademika Yu.G. Reshetnyaka, Tez. dokl. (23–29 oktyabrya 2022 g.), ed. Basalaev S.G., IPTs NGU, Novosibirsk, 2022, 115–116