Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-22T05:10:00.103Z Has data issue: false hasContentIssue false

Monotony of the Osculating Circles of Arcs of Cyclic Order Three

Published online by Cambridge University Press:  20 November 2018

N. D. Lane
Affiliation:
McMaster University and University of Toronto
K. D. Singh
Affiliation:
McMaster University and University of Toronto
P. Scherk
Affiliation:
McMaster University and University of Toronto
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is well-known in elementary calculus that if a differentiable function has a monotone increasing curvature, then its curvature is continuous and the circles of curvature at distinct points have no points in common. In particular, two one-sided osculating circles at distinct points of an arc A3 of cyclic order three have no points in common; cf. [l], [2], [3]. The conformai proof given here that any two general osculating circles at distinct points of A3 are disjoint (Theorem 1), may be of interest. We also prove that all but a countable number of points of A3 are strongly conformally differentiable (Theorem 2).

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Hjelmslev, J., Die graphische Geometrie, Attőnde Skandinav. Mat-Kong. Fortand 1 (Stockholm, 1934).Google Scholar
2. Haller, H., Ueber die K3-Schmieggebiide der ebenen Bogen von der K3-Ordnung drei, S.- B. Phys. - Med. Soz. Erlangen, 69 (1937), 1518.Google Scholar
3. Haupt, O., Zur geometrischen Kennzeichnung der Scheitel ebener Kurven, Archiv der Mathematik (1948), 102105.Google Scholar
4. Lane, N. D. and Scherk, Peter, Differentiable points in the conformai plane, Can. J. Math. 5(1953), 512518.Google Scholar
5. Lane, N. D. and Scherk, Peter, Characteristic and order of differentiable points in the conformai plane, Trans. Amer. Math. Soc., 81 (1956), 353378.Google Scholar
6. Marchaud, A., Sur les continus d' ordre borné, Acta Math. 55 (1930), 57115.Google Scholar