INTERPRETATION OF MUTUAL LOCATION OF POINTS OF METRIC SPACE BY HELP OF GRAPHIC MEANS
DOI:
https://doi.org/10.31110/2413-1571-2022-034-2-001Keywords:
distance between points, metric space, metric geometry, dynamic geometric environment, GeoGebra 3D, rectilinear placement of pointsAbstract
Formulation of the problem. This paper considers issues related to the method of studying the geometric properties of metric spaces. These questions necessarily arise when students learn the basic concepts of the theory of metric spaces. Difficulty in understanding these concepts arises due to the lack, in most cases, of their geometric interpretation, or appropriate visualization. To build a geometric interpretation of the concepts of rectilinear and flat placement of points of metric space, it is proposed to build appropriate analogs in two-dimensional and three-dimensional arithmetic Euclidean spaces. To visualize these concepts, it is proposed to use a dynamic geometric environment GeoGebra 3D. This approach allows us to demonstrate both the similarity of individual geometric concepts of metric space with the corresponding concepts of Euclidean geometry and to demonstrate cases of their "non-Euclidean".
Materials and methods. The study used the dynamic geometric environment GeoGebra 3D, a software tool for calculating the volume of a tetrahedron along the lengths of its edges, as well as graphical tools for constructing images.
Results. The examples of geometric interpretation and visualization of mutual placement of points of metric space given in this work promote deeper and more conscious perception and understanding by students of the basics of the theory of metric spaces.
Conclusions. Metric geometry makes it possible to consider Euclidean geometry and non-Euclidean geometries from one point of view. The analogy of individual relations between the points of metric space with the corresponding relations in Euclidean geometry makes it possible to trace the change in the characteristic geometric properties of space when its metric changes. The use of special graphical capabilities of the corresponding software allows not only to visualize the mutual location of the points of the metric space but also to track its change when changing the observation point of this location. Visualization of geometric properties of metric spaces contributes to a deeper and more conscious perception and understanding by students of the basics of the theory of metric spaces.
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Copyright (c) 2022 Катерина Валько, Валерій Кузьмич, Людмила Кузьмич, Олександр Савченко
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