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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.
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