ON TRANSFORMATION OF CONIC SURFACES INTERCEPTION LINES IN SPECIAL CASES
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Abstract
Abstract. The question is considered in what cases and under what conditions certain combinations of second-order curves are obtained by varying the angle of rotation and taper of intersecting conical surfaces circumscribed around a sphere. The statement of the research problem is described on the basis of the Monge theorem and the conditions determined by it. Taking into account the practical significance of Monge's theorem, the importance of considering the transformation of the lines of intersection of conical surfaces is emphasized not only when the angle between their axes changes, but also depending on the conicity of the surfaces. For the convenience of visualizing the transformation of the type of lines of intersection of conical surfaces, the use of a pie chart is proposed. Particular cases of intersection of conical surfaces circumscribed around a sphere are considered. Examples of visualization of the process of transformation of the lines of intersection of conical surfaces using the proposed circular diagrams at various taper angles and angles between the axes of the surfaces are given. Possible variants of the line of mutual intersection of conical surfaces with different taper are analyzed, when the angle between the axes of the surfaces changes. A variant of using the research results in the educational process by analyzing the problem using the proposed diagrams, followed by computer three-dimensional modeling of intersecting surfaces and generative creation of a flat drawing is proposed. An example of a completed training task on the creation of lines of intersection of two conical surfaces described around a sphere by means of computer simulation is given.
Subject: a special case of intersection of surfaces of the second-order.
Materials and methods: geometric algorithm for modeling lines of mutual intersection of conical surfaces – curves of the second-order – using the method of spherical mediators.
Results: it is shown that the form of curves of the second order, when crossing conical surfaces in particular cases, depends not only on the angle between their axes, but also on the values of their taper. The use of a pie chart is proposed for the convenience of visualizing the process of transformation of the type of lines of intersection of conical surfaces.
Conclusions: the proposed pie chart allows you to systematize and visualize the options for transforming the type of line of intersection of conical surfaces. The results of the study can be used in the educational process to accelerate the perception of educational material and a comprehensive understanding of the Monge theorem.
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References
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