DEVELOPMENT OF AN INTERACTIVE GEOGEBRA APPLET FOR MODELING THE PROPERTIES OF SPACE CURVES AND EXPLORING THE FRENET FRAME

Abstract

The development of an interactive GeoGebra applet designed for modelling the properties of spatial curves defined parametrically, with the aim of exploring the Frenet frame and its components at each point of the curve, is presented. Within the framework of the study, a review of relevant scientific publications was carried out, focusing on the challenges of visualisation of complex spatial objects in the context of geometry education. The rationale for creating such a digital resource under conditions of distance and blended learning to enhance the effectiveness of teaching differential geometry to students of mathematics and physics-mathematics programmes was substantiated.

The developed applet enables the construction of a spatial curve based on parametric equations and facilitates interactive 3D exploration of the resulting model. The dynamic behaviour of the tangent, normal, and binormal vectors—as well as the corresponding planes—can be visualised at any point selected by the user through the applet’s interface. Curvature and torsion at the selected point are computed automatically.

The algorithm for applet development is described. It combines stages of mathematical modelling, integration of computational tools and pedagogical relevance. Special attention is given to the design of the applet interface, which integrates a 3D graphics for the dynamic visualisation of spatial elements with a 2D control panel and an output field for data. This approach to representing the dynamic motion of the Frenet frame along the trajectory supports deeper intuitive understanding of complex geometric concepts. The analysis of the applet’s functionality indicates its potential for supporting personalised learning by enabling students to formulate and test hypotheses through the manipulation of model parameters. Additionally, the applet serves as an effective visualisation tool that supports the development of analytical and spatial thinking.

Examples of interdisciplinary integration with mechanics, theoretical physics, computer science, and current research directions are provided. The results obtained may be applied to the modernisation of instructional resources in courses on differential geometry, mathematical modelling, computer graphics, and the development of interactive models for other spatial objects.

Key words: GeoGebra, differential geometry, Frenet frame, space curves, interactive visualization, educational applet, dynamic mathematics, distance learning.