Basic properties of the metric of tangents
DOI:
https://doi.org/10.17721/1812-5409.2026/1.34Keywords:
generalized tangents, smooth metric, canonical decomposition, figure, metric of tangentsAbstract
The article studies a metric related to the boundaries of figures of a certain class. To properly define it, the class of figures, referred to as figures with convex-concave boundaries, is first introduced. For this, there must exist a finite set of points on the boundary of the figure, such that the arc between any two adjacent points is convex. This allows the boundary of the figure to be considered as a union of a finite number of convex arcs. Since the metric is defined using generalized tangents to the boundary of the figure, and for convex figures, a generalized tangent becomes a supporting line, this approach opens promising prospects for further study of figures of this class. A dense subset of this class is the collection of all polynomials. For practical applications, this serves as an efficient simplification, as figures are usually defined by a set of points on their boundary, and any approximation begins with approximating the figure by a polynomial.
For the given class of figures, a tangent metric is defined, which is determined through generalized tangents to the boundary of the figure. Previously, this metric was studied together with an initial term formed by the Hausdorff metric. This made it possible to avoid verifying the first axiom of the metric, as it followed from the Hausdorff metric. However, it was found that the Hausdorff metric is not a necessary component when defining the tangent metric, as demonstrated by the conducted research. In general, the presence of the Hausdorff metric stemmed from a certain similarity to the metric of the space of continuously differentiable functions, which has the metric of continuous functions as its first term. Therefore, this advancement, which enabled the elimination of the Hausdorff metric as a redundant component, proved to be pivotal. This will also streamline future algorithms for computing the tangent metric.
For further research, firstly, it will be necessary to narrow the class of functions to the set of all polynomials. As a result, studying the properties of the canonical decomposition of a figure within the class of all polynomials has become important. Additionally, for polynomials, constructing generalized tangents becomes a simpler problem.
Pages of the article in the issue: 265 - 272
Language of the article: English
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Copyright (c) 2026 Maryna Chigidina, Iryna Lazarenko, Dmytro Nomirovskii, Bogdan Rublyov, Vladimir Semenov, Kateryna Svitovenko
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