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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
2. Lane, N. D. and Scherk, Peter, Differentiate points in the conformai plane, Can. J. Math., 5 1953), 512–518.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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