Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Rotation and winding numbers for planar polygons and curves
HTML articles powered by AMS MathViewer

by Branko Grünbaum and G. C. Shephard PDF
Trans. Amer. Math. Soc. 322 (1990), 169-187 Request permission

Abstract:

The winding and rotation numbers for closed plane polygons and curves appear in various contexts. Here alternative definitions are presented, and relations between these characteristics and several other integer-valued functions are investigated. In particular, a point-dependent "tangent number" is defined, and it is shown that the sum of the winding and tangent numbers is independent of the point with respect to which they are taken, and equals the rotation number.
References
    Brückner [1900] M. Brückner, Vielecke und Vielflache, Teubner, Leipzig 1900. Hermes [1866] O. Hermes, Ausdehnung der Jacobischen Regel zur Bestimmung des Inhalts der Sternpolygone für den Fall vielfacher Punkte, J. Reine Angew. Math. 65 (1866), 177-179. Hess [1874] E. Hess, Ueber gleicheckige und gleichkantige Polygone. Schriften der Gesellschaft zur Befàrderung der gesammten Naturwissenschaften zu Marburg, vol. 10, no. 2, Th. Kay, Cassel, 1874, pp. 611-743 + plates. Jacobi [1866] C. G. J. Jacobi, Regel zur Bestimmung des Inhalts der Sternpolygone. J. Reine Angew. Math. 65 (1866), 173-176; Gesammelte Werke, Bd. 7, Reimer, Berlin, 1891, pp. 40-41. Meister [1769] A. L. F. Meister, Generalia de genesi figurarum planarum et inde pendentibus earum affectionibus, Novi Comm. Soc. Reg. Scient. Gotting. 1 (1769/70), pp. 144-180+9 plates. Màbius [1865] A. F. Màbius, Ueber die Bestimmung des Inhaltes eines Polyëders, Ber. Verh. Kànigl. Sächs. Ges. Wiss. Math.-Nat. Kl. 17 (1865), 31-68 (Gesammelte Werke, vol. 2, Hirzel, Leipzig, 1886, pp. 473-512). Steinitz [1922] E. Steinitz, Polyeder und Raumeinteilungen, Enzykl. Math. Wiss., vol. 3 (Geometrie), Part 3AB12, 1922, pp. 1-139.
  • Hassler Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937), 276–284. MR 1556973
    • Wiener [1865] C. Wiener, Über Vielecke und Vielflache, Teubner, Leipzig, 1864.
Similar Articles
Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 322 (1990), 169-187
  • MSC: Primary 52A99; Secondary 26B15, 51M99, 57M99
  • DOI: https://doi.org/10.1090/S0002-9947-1990-1024774-2
  • MathSciNet review: 1024774