On the Inclination of a Parameterized Curve

Given a plane curve \(\Gamma \) parameterized by arclength s on an open interval \(I \subset {\mathbb R}\) by a function \(\gamma : I \rightarrow {\mathbb R}^2\) with twice continuously differentiable component functions and an initial inclination angle \(\theta _0\in {\mathbb R}\) satisfying \(\dot{\gamma }(s_0) = (\cos \theta _0,\sin \theta _0)\) for some \(s_0 \in I\) , we show there exists a unique function \(\psi \in C^1(I)\) with \(\dot{\gamma }(s) = (\cos \psi (s), \sin \psi (s))\) for all \(s\in I\) and \(\psi (s_0) = \theta _0\) . Similar results holding for a parameterized curve defined on a compact interval are stated in many differential geometry texts. These results are usually based on a path lifting result for continuous maps into the circle \({\mathbb S}^1\) . Our result differs from these treatments both in that the interval I is taken to be open and that the techniques used to obtain the result are via a direct treatment of a system of ordinary differential equations. The system of ordinary differential equations differs from those usually considered in that it contains first order equations of singular type. Such systems seem to have independent interest and the approach presented should have broader application. We give one other related example of a similar singular system of ordinary differential equations, and we strongly suspect the development of a general axiomatic theory of such singular systems should be possible, though we are unaware of such a development. We also discuss the topological approach and offer a version of the path lifting lemma for paths defined on open interval (or any interval). Finally, we discuss applications of our result to the construction, classification, and analysis of plane curves and in relation to structure theorems for plane curves.