FORMATION OF THE CONCEPT OF FLAT ARRANGEMENT OF POINTS USING MEANS OF METRIC GEOMETRY IN THE STUDY OF METRIC SPACES
DOI:
https://doi.org/10.31110/2413-1571-2023-038-5-001Keywords:
point, distance, metrics, metric space, angle, angular characteristic, rectilinear and flat arrangement of pointsAbstract
Formulation of the problem. When studying metric spaces, higher education students often need help understanding these spaces' basic concepts and properties. It, to a large extent, is a consequence of the significant formalization of such concepts on the one hand and the preservation of the corresponding formulations and names familiar to students from a school mathematics course. The most straightforward concepts of mutual placement of points of metric space, for example, the rectilinearity of their arrangement, can acquire different properties in different spaces. Sometimes, these properties do not agree with the corresponding properties in Euclidean spaces. It is advisable to use geometric interpretation and visualization methods of these properties to overcome these difficulties. At the same time, it is appropriate to use elements of metric geometry. Its methods make it possible to interpret the geometric features of the mutual placement of points of metric space in Cartesian (rectangular) coordinate systems known to students. Moreover, it becomes possible to visualize these features with the help of graphic editors since they, as a rule, use numerical values of the coordinates of points to visualize them. The paper gives examples of visualization of the property of the flat arrangement of four points of non-Euclidean metric space in a rectangular three-dimensional coordinate system.
Materials and methods. The results of the work were obtained by analyzing existing higher mathematics textbooks for higher education institutions and scientific publications. They were tested while reading the corresponding special course for students of the specialty "014.04 Secondary education (mathematics)" of the master's level of higher education. The dynamic geometric environment GeoGebra 3D was used to obtain images.
Results. Based on the definition of an angle as an ordered trio of points of an arbitrary metric space and the angular characteristic of this angle, the fact of the flat arrangement of four points of a non-Euclidean metric space is established, with using the dynamic geometric environment GeoGebra 3D examples of digital visualization of this arrangement are given. Such a visualization makes it possible to familiarize students with higher education with the most straightforward features of non-Euclidean geometries.
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Copyright (c) 2023 Катерина Валько, Валерій Кузьмич, Людмила Кузьмич, Олександр Савченко
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