Characteristic and order of differentiable points in the conformal plane

Abstract

Introduction.In [3], the authors discussed a definition of the conformal differentiability of an arc at a point in the conformal plane.It was based on the postulation of tangent circles and of an osculating circle.The intersection and support properties of all the circles through a differentiable point were studied, and by means of these properties the differentiable points were classified into various types.Each type was uniquely described by a certain triple of numbers, the characteristic of that type.In this paper, relationships between the characteristic of a differentiable point and its cyclic order are established.In this connection some-partly familiar-differentiability properties of arcs of order three will be discussed.Our main results are stated in five theorems; cf. 2.1, 3.4, 3.5, 5.1, 5.9.1. Prerequisites 1.1.In the following, P, Q, denote points in the real conformal plane, and C, C, denote oriented circles.Such a circle C decomposes the plane into two open regions, its interior C* and its exterior C*, the latter lying at its right.If C degenerates into a point, then C* is empty.The circle through three mutually distinct points P, Q, and R, will occasionally be de-