Direct differentiability, characteristic and order of curves (Q2739415)

This article, a survey of the research work covered in more than twenty papers related to direct differentiability of curves, presents a theory of curves which uses pure geometric methods, i.e. direct synthetic approach, hence the name direct differentiability, as contrast to analytic approach.NEWLINENEWLINENEWLINEA first concept is linear differentiation: a continuous arc \(A\) is differentiable at \(p\) if \(\displaystyle{\lim_{\substack{ q\to p\\ q\in A,q\neq p}}} l(p,q)\) exists, where \(l(p,q)\) denotes the line through \(p\) and \(q\).NEWLINENEWLINENEWLINEIn the context of conformal plane geometry, the natural concept which extends the Euclidean linear differentiation is the cyclic differentiability: An arc \(A\) (defined as the continuous image of a real line interval) is said to be once cyclically differentiable at \(p\in A\) if there exists a point \(Q\neq p\) such that if \(s\in A\), \(s\neq p\) then \(\lim_{s\to p} C(p,s,Q)\) exists. This limit circle is denoted by \(C(\tau, Q)\) and called a tangent circle of \(A\) at \(p\). An arc \(A\) is said to be cyclically differentiable at \(p\in A\) if for \(s\neq p\lim_{s\to p} C(\tau, s)\), \(s\in A\) exists.NEWLINENEWLINENEWLINEMoreover, the concepts of characteristic of a differentiable point and cyclic order are defined and the relations between these three concepts are given. A theory of direct differentiability in the projective plane, analogous to that in the conformal plane, with role of circles replaced by conics is given: that is the conical differentiation. In the affine plane the direct differentiation is studied by using parabolas as characteristic curves: that is the polynomial differentiation.NEWLINENEWLINENEWLINEFinally, the quasigraph differentiation is a general theory of differentiability which would include as particular cases the previous types of direct differentiations. In the last section the analytic equivalents of conical differentiability conditions are given.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00032].