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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Cyclic order and dissection order of certain arcs


Author: S. B. Jackson
Journal: Proc. Amer. Math. Soc. 38 (1973), 609-613
MSC: Primary 53C75
DOI: https://doi.org/10.1090/S0002-9939-1973-0310818-5
MathSciNet review: 0310818
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Abstract: Let arc $ A$ in the conformal plane or on the sphere have local cyclic order three and cyclic order $ t$. It can be decomposed into a finite number of subarcs of cyclic order three. Let the disection order of $ A$ be the minimum number of arcs in such a decomposition. The principal result of this paper is that the cyclic order $ t$ and dissection order $ d$ of $ A$ satisfy the relation $ d + 2 \leqq t \leqq 3d$. In establishing this result it is proved that a necessary and sufficient condition, that an arc of local cyclic order three shall be of global cyclic order three, is that there exists a circle meeting it only at the endpoints.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0310818-5
Keywords: Cyclic order, local cyclic order three, dissection order, general tangent circle, osculating circle
Article copyright: © Copyright 1973 American Mathematical Society