Cyclic order and dissection order of certain arcs

Jackson, S. B.

doi:10.1090/S0002-9939-1973-0310818-5

Cyclic order and dissection order of certain arcs
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by S. B. Jackson

Proc. Amer. Math. Soc. 38 (1973), 609-613

DOI: https://doi.org/10.1090/S0002-9939-1973-0310818-5

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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.

References

Johann Haller, Über ordnungsminimale Bogen bzw. Kurven in der Ebene und ihre $k$-Paratingenten, Bayer. Akad. Wiss. Math.-Natur. Kl. S.-B. 1963 (1963), no. Abt. II, 15–25 (1964) (German). MR 172189
Otto Haupt and Hermann Künneth, Geometrische Ordnungen, Die Grundlehren der mathematischen Wissenschaften, Band 133, Springer-Verlag, Berlin-New York, 1967 (German). MR 0227907

Bemerkungen zu Sätzen von Herrn S. B. Jackson über ebene Bogen von der zyklischen Ordnung drei

1971

N. D. Lane and P. Scherk, Characteristic and order of differentiable points in the conformal plane, Trans. Amer. Math. Soc. 81 (1956), 358–378. MR 79293, DOI 10.1090/S0002-9947-1956-0079293-7
N. D. Lane, K. K. Singh, and P. Scherk, Monotony of the osculating circles of arcs of cyclic order three, Canad. Math. Bull. 7 (1964), 265–271. MR 162190, DOI 10.4153/CMB-1964-024-0
S. B. Jackson, A note on arcs of finite cyclic order, Proc. Amer. Math. Soc. 12 (1961), 364–368. MR 133037, DOI 10.1090/S0002-9939-1961-0133037-6
S. B. Jackson, Arcs of local cyclic order three, Math. Scand. 28 (1971), 355–365 (1972). MR 307138, DOI 10.7146/math.scand.a-11030

Arcs of local cyclic order three with prescribed end conditions

A. Marchaud, Sur les continus d’ordre borné, Acta Math. 55 (1930), no. 1, 67–115 (French). MR 1555315, DOI 10.1007/BF02546510