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REGULARITY OF CURVES WITH A CONTINUOUS TANGENT LINE

Part of: Classical differential geometry Miscellaneous topics of analysis in the complex domain

Published online by Cambridge University Press:  01 February 2009

J. M. BURGUÉS  and
J. CUFÍ
J. M. BURGUÉS*
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain (email: josep@mat.uab.cat)
J. CUFÍ
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain (email: jcufi@mat.uab.cat)
*
✉For correspondence; e-mail: josep@mat.uab.cat
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Abstract

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This note contains a proof of the fact that a Jordan curve in ℝ2 with a continuous tangent line at each point admits a regular reparameterization. We extend the result both to more general curves in ℝn and to higher orders of differentiability.

Keywords

regular curvetangent lineJordan curveorder of differentiability

MSC classification

Secondary: 53A04: Curves in Euclidean space 30E99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 86 , Issue 1 , February 2009 , pp. 17 - 26
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

The authors have been partially supported by MEC grant no. MTM2005-08984-CO and by grant no. 2005SGR00611 of Generalitat de Catalunya.

References

[1]Garnett, J. and Marshall, D., Harmonic Measure, New Mathematical Monographs, 2 (Cambridge University Press, Cambridge, 2005).CrossRefGoogle Scholar
[2]Lindelöf, E., ‘Sur un principe général de l’analyse et ses applications à la théorie de la representation conforme’, Acta Soc. Sci. Fenn. A 46(4) (1915), 1–35.Google Scholar
[3]Pommerenke, Ch., Boundary Behaviour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften, 299 (Springer, Berlin, 1992).CrossRefGoogle Scholar