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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Total curvatures of a closed curve in Euclidean $n$-space


Authors: L. Hernández Encinas and J. Muñoz Masqué
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 2127-2132
MSC (2000): Primary 53A04; Secondary 28A75, 51M20
Published electronically: January 23, 2004
MathSciNet review: 2053986
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Abstract | References | Similar Articles | Additional Information

Abstract: A classical result by J. W. Milnor states that the total curvature of a closed curve $C$ in the Euclidean $n$-space is the limit of the total curvatures of polygons inscribed in $C$. In the present paper a similar geometric interpretation is given for all total curvatures $\int_{C}\vert\kappa _{r}\vert\mathrm{d}s$, $r=1,\ldots,n-1$.


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Additional Information

L. Hernández Encinas
Affiliation: Instituto de Física Aplicada, Consejo Superior de Investigaciones Cientificas, Calle Serrano 144, 28006-Madrid, Spain

J. Muñoz Masqué
Affiliation: Instituto de Física Aplicada, Consejo Superior de Investigaciones Cientificas, Calle Serrano 144, 28006-Madrid, Spain
Email: \luis, jaime\@iec.csic.es

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07310-1
PII: S 0002-9939(04)07310-1
Keywords: Curvatures of a curve, Fr\'enet frame, polygon, total curvature
Received by editor(s): February 26, 2003
Received by editor(s) in revised form: March 25, 2003
Published electronically: January 23, 2004
Additional Notes: This work was supported by Ministerio de Ciencia y Tecnología (Spain) under grants TIC2001–0586 and BFM2002–00141.
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2004 American Mathematical Society