Vol. 28, No. 3, 1969

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Strong cyclic, parabolic and conical differentiability

Norman D. Lane and Kamla Devi Singh

Vol. 28 (1969), No. 3, 583–597

An arc A in the real inversive plane is strongly conformally differentiable at a point p of A if the circle through a fixed point different from p and two variable points of A converges as these points tend to p, and the circle through three points of A which tend to p converges. Analogous definitions of strong parabolic differentiability in the real affine plane, using parabolas and three conditions, and of strong conical differentiability in the real projective plane, using conics and four conditions, may be formulated.

Relations among the strong differentiability conditions in each of the above three geometries can be summarized as follows:

Let N = II in the inversive case; N = II or III in the affine case; and N = II, III or IV in the projective case. Then the N-th strong differentiability condition implies the K-th strong condition for 0 < K < N 1, but it does not imply the (N 1)-th strong condition. The (N 1)-th strong condition will also hold in each of the three geometries, however, if, in addition to the N-th strong condition, any one of the following is assumed:

(i) The limit characteristic curve of the N-th strong condition is nondegenerate,

(ii) The point p is an end-point of the arc A,

(iii) The arc A satisfies a weaker condition (N 1) at p.

Mathematical Subject Classification
Primary: 53.95
Received: 28 November 1967
Published: 1 March 1969
Norman D. Lane
Kamla Devi Singh