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Union and Extension of Arcs of Cyclic Order Three

Published online by Cambridge University Press:  20 November 2018

K. D. Singh  and
N. D. Lane
K. D. Singh
Affiliation:
Lucknow University, Lucknow, India
N. D. Lane
Affiliation:
McMaster University, Hamilton, Ontario
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In (2) Lane and Scherk discussed differentiate points of arcs in the conformai (inversive) plane. Arcs A3 of cyclic order three were discussed in (3; 4). In the present note we give necessary and sufficient conditions for the union of two A3's to be an A3 (Theorem 1), and for an A3 to be extensible to a larger one (Theorem 2). The related problem of extending arcs in projective n-space was dealt with by Haupt in (1) and Sauter in (5; 6).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 20 , 1968 , pp. 629 - 638
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Haupt, O., Tiber die Erweiterung eines beliebigen Bogens dritter Ordnung, insbesondere zu einer Raumkurve dritter Ordnung, J. Reine Angew. Math., 170 (1933), 154–167.Google Scholar
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4. Lane, N. D., Singh, K. D., and Scherk, P., Monotony of the osculating circles of arcs of cyclic order three, Can. Math. Bull., 7 (1964), 265–271.Google Scholar
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