Geodesic
Union and Extension of Arcs of Cyclic Order Three
Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 629-638

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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 A 3 (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). 
Singh, K. D.; Lane, N. D. Union and Extension of Arcs of Cyclic Order Three. Canadian journal of mathematics, Tome 20 (1968) no. 1, pp. 629-638. doi: 10.4153/CJM-1968-061-4
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[1] 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] 2. Lane, N. D. and Scherk, Peter, Differentiate points in the conformai plane, Can. J. Math., 5 1953), 512–518. Google Scholar

[3] 3. Lane, N. D. and Scherk, Peter, Characteristic and order of differentiable points in the conformai plane, Trans. Amer. Math. Soc, 81 (1956), 358–378. Google Scholar

[4] 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

[5] 5. Sauter, I., Zur Théorie der Bogen n-ter (Realitdts) Ordnung im projectiven Rn. I, Math. Z., 41 (1936), 507–536. Google Scholar

[6] 6. Sauter, I., II, Math. Z., 42 (1937), 580–592.10.1007/BF01160096 Google Scholar | DOI

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