Density of an Oriented Curve
Abstract Generated abstract
This note defines a density for parametrized oriented curves in a metric space using the asymptotic behavior of the maximal number of disjoint arcs of diameter at least a given scale. After formalizing components and points of a parametrized curve, it introduces a scale-ratio function and derives from it a local metric invariant associated with each point. The resulting density function is proposed as an analogue of curvature, with stated bounds in three cases: rectifiable neighborhoods of finite nonzero length give density 1, nonrectifiable neighborhoods give density between 1 and infinity, and certain disconnected rectifiable neighborhoods give density between one half and 1.
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DENSITY OF AN ORIENTED CURVE
L. E. PORTNOV
(Presented in May 1966 at the mathematical section of the scientific-technical conference of the AVTF, devoted to the 70th anniversary of the TPI)
By a parametrized oriented curve of a metric space \(R\) we shall mean a mapping
\[ f(t), \quad t \in E \subset [0,1], \quad \text{into } R . \]
Definition 1. By a component of the mapping \(f(t)\), \(t \in E \subset [0,1]\), into \(R\) we shall mean any \(F \subset E\) possessing the following properties:
1) from \(t_1, t_2 \in F\) it follows that \(f(t_1)=f(t_2)\);
2) any point \(t \in E\) is strictly majorized by the points of the set \(F\), either from below or from above.
Definition 2. A mapping \(f(t)\), \(t \in F \subset [0,1]\), into the metric space \(R\) will be called a point of the parametrized curve determined by the mapping \(f(t)\), \(t \in E \subset [0,1]\), if \(F\) is a component of the mapping \(f(t)\), \(t \in E \subset [0,1]\).
To an oriented curve \(L\) we associate the function
\[ V_L(\varepsilon), \quad \varepsilon \in (0,\infty), \]
equal to the maximal number of nonintersecting arcs of the curve \(L\), each of which has diameter not less than \(\varepsilon\).
Consider the function
\[ K_L(\lambda)=\lim_{\varepsilon \to 0}\frac{V_L(\lambda \varepsilon)}{V_L(\varepsilon)}, \quad \lambda \in [0,1]. \tag{1} \]
Using passage to the limit, from (1) one may uniquely associate with each point of the curve \(L\) a function
\[ \chi_x(\lambda), \quad \lambda \in (0,1), \]
which is a local metric invariant of the curve \(L\).
We introduce the density function of the curve \(L\) at a point
\[ p(x)=\frac{1}{2\displaystyle\int_0^1 \chi_x(\lambda)\,d\lambda}, \quad x \in L, \quad \lambda \in [0,1], \]
which is an analogue of the curvature of a curve.
1. If in every neighborhood of the point \(x\) the curve \(L\) has finite nonzero ordinary length, then
\[ p(x)=1. \]
- If every neighborhood of the point \(x\) is nonrectifiable, then
\[ 1 \leqslant p(x) < \infty . \]
- If in every neighborhood \(U_x\) of a point \(x \in L\) there is a neighborhood \(V_x \subset U_x\) in which the curve \(L\) is disconnected and rectifiable, then
\[ \frac{1}{2} \leqslant p(x) \leqslant 1 . \]