Measurements of curvilineal angles (Q1909619)

The essential aim of this paper is to define a nontrivial measure for the angle between tangent curves. A directed ``curvilineal'' angle \(\alpha\) is a pair of curves in the Euclidean plane, emanating from the same vertex. A measure \(\varphi=\varphi(\varepsilon)\) of \(\alpha\) can be established in a geometric manner as follows: draw a (small) circle of radius \(\varepsilon (> 0)\) with center the vertex of the angle; call \(l(\varepsilon)\) the length of the arc cut out of the circle by the two ``legs'' of \(\alpha\); then put \(\varphi (\varepsilon) : = l (\varepsilon)/ \varepsilon\). This measure \((= : \mu_\infty)\) which in general depends on the choice of \(\varepsilon\) (being independent of \(\varepsilon\) if the two legs of \(\alpha\) are straight lines) could be replaced by a measure \(\mu_n\) \((n \in \mathbb{N})\) which consists of the Taylor polynomial of degree \(n\) of the power series of \(\varphi (\varepsilon)/ \varepsilon\). Moreover, the authors' investigations about curvilineal (plane) angles deal with: historical remarks; axiomatic requirements concerning the measurement of angles; examples (angle circle/circle or curve/tangent); bisectrix (curve) of a curvilineal angle.