| Title:
|
Evolutes and isoperimetric deficit in two-dimensional spaces of constant curvature (English) |
| Author:
|
Cufí, Julià |
| Author:
|
Reventós, Agustí |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
50 |
| Issue:
|
4 |
| Year:
|
2014 |
| Pages:
|
219-236 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We relate the total curvature and the isoperimetric deficit of a curve $\gamma $ in a two-dimensional space of constant curvature with the area enclosed by the evolute of $\gamma $. We provide also a Gauss-Bonnet theorem for a special class of evolutes. (English) |
| Keyword:
|
curvature |
| Keyword:
|
evolutes |
| Keyword:
|
isoperimetric deficit |
| Keyword:
|
Gauss-Bonnet |
| MSC:
|
52A10 |
| MSC:
|
52A55 |
| MSC:
|
53A04 |
| idZBL:
|
Zbl 06487008 |
| idMR:
|
MR3291851 |
| DOI:
|
10.5817/AM2014-4-219 |
| . |
| Date available:
|
2014-12-09T14:42:04Z |
| Last updated:
|
2016-04-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144020 |
| . |
| Reference:
|
[1] Bruna, J., Cufí, J.: Complex Analysis.European Mathematical Society, 2013. MR 3076702 |
| Reference:
|
[2] Chern, S. S.: Curves and surfaces in Euclidean space.Studies in Global Geometry and Analysis 4 (1967), 16–56. MR 0212744 |
| Reference:
|
[3] Escudero, C. A., Reventós, A.: An interesting property of the evolute.Amer. Math. Monthly 114 (7) (2007), 623–628. Zbl 1144.53007, MR 2341325 |
| Reference:
|
[4] Escudero, C. A., Reventós, A., Solanes, G.: Focal sets in two-dimensional space forms.Pacific J. Math. 233 (2007), 309–320. Zbl 1152.53063, MR 2366378, 10.2140/pjm.2007.233.309 |
| Reference:
|
[5] Fenchel, W.: On the differential geometry of closed space curves.Bull. Amer. Math. Soc. (N.S.) 57 (1951), 44–54. Zbl 0042.40006, MR 0040040, 10.1090/S0002-9904-1951-09440-9 |
| Reference:
|
[6] Hurwitz, A.: Sur quelques applications géométriques des séries de Fourier.Annales scientifiques de l' É.N.S. 19 (1902), 357–408. MR 1509016 |
| Reference:
|
[7] Martinez-Maure, Y.: Sommets et normales concourantes des courbes convexes de largeur constante et singularités des hérissons.Arch. Math. 79 (2002), 489–498. Zbl 1025.52004, MR 1967267, 10.1007/BF02638386 |
| Reference:
|
[9] Santaló, L. A.: Integral Geometry and Geometric Probability.Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976, With a foreword by Mark Kac, Encyclopedia of Mathematics and its Applications, Vol. 1. Zbl 0342.53049, MR 0433364 |
| Reference:
|
[10] Spivak, M.: A Comprehensive Introduction to Differential Geometry.Publish or Perish, Inc. Berkeley, 1979, 2a ed., 5 v. Zbl 0439.53005 |
| . |
