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Parabolic Differentiation

Published online by Cambridge University Press:  20 November 2018

N. D. Lane
N. D. Lane*
Affiliation:
McMaster University and the Summer Research Institute of the Canadian Mathematical Congress
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The purpose of this paper is the study of parabolically differentiable points of arcs in the real affine plane. In Section 2, two different definitions of convergence of a family of parabolas are given and it is observed (Theorem 1) that these are equivalent. In Section 3, tangent parabolas at a point p of an arc A are discussed and it is proved (Theorem 2) that all the non-degenerate non-tangent parabolas of A through p intersect A at p or that all of them support. In Section 4, osculating parabolas are introduced and the condition that an arc be twice parabolically differentiable at a point p is stated.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 15 , 1963 , pp. 546 - 562
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Lane, N. D. and Scherk, P.. Differentiable points in the conformai plane, Can. J. Math., 5 (1953), 512–518.Google Scholar