Russian Federation
Moskva, Moscow, Russian Federation
Qualified presentation of the topic "Tangent Plane and Surface Normal" in terms of competence approach is possible with the proper level for students' attention focusing on both intra-subject and inter-subject relations of descriptive geometry. Intra-subject connections follow from the position that the contingence is a particular (limit) case of intersection. Therefore, the line of intersection of the tangent plane and the surface, or two touching surfaces, has a special point at the tangency point. It is known from differential geometry [1] that this point can be nodal, return, or isolated one. In turn, this point’s appearance depends on differential properties of the surface(s) in this point’s vicinity. That's why, for the competent solution of the considered positional problem account must be also taken of the inter-subject connections for descriptive and differential geometry. In the training courses of descriptive geometry tangent planes are built only to the simplest surfaces, containing, as a rule, the frames of straight lines and circles. Therefore, the tangent plane is defined by two tangents drawn at the tangency point to two such lines. In engineering practice, as such lines are used cross-sections a surface by planes parallel to any two coordinate planes. That is, from the standpoints for the course of higher mathematics, the problem is reduced to calculation for partial derivatives. Although this topic is studied after the course of descriptive geometry, it seems possible to give geometric explanation for computation of partial derivatives in a nutshell. It also seems that the study of this topic will be stimulated by a story about engineering problems, which solution is based on construction of the tangent plane and the normal to the technical surface. In this paper has been presented an example for the use of surface curvature lines for programming of milling processing for 3D-harness surfaces.
surface, tangent plane, normal, partial derivatives, surface curvature, Dupin indicatrix, milling.
Тема «Касательная плоскость и нормаль поверхности» в курсе начертательной геометрии является уникальной в том смысле, что:
• во-первых, в ее изложении следует четко выделить внутрипредметные связи (компетенции): касательная плоскость — это предельное положение секущей и как оно влияет на особенности сечения в точке касания, следовательно, характеризует дифференциальные свойства поверхности, в частности, как от них зависит ее развертываемость на плоскость;
• во-вторых, отмеченные выше внутрипредметные связи тесно переплетаются с межпредметными начертательной и дифференциальной геометрий, с одной стороны, и дифференциальным исчислением (частные производные), с другой стороны;
• в-третьих, эта тема имеет непосредственный выход в инженерную практику, в частности, в технологию фрезерной обработки поверхностей объемной оснастки на станках с числовым программным управлением.
Таким образом, квалифицированное изложение материала этой темы должно способствовать реальным подвижкам трансформирования традиционного курса начертательной геометрии в прикладную (инженерную) геометрию [14]. Для достижения этой цели в преподавании материала этой темы следует акцентировать внимание студентов на решении следующих задач:
1) построение сечения поверхности касательной плоскостью и построение линии пересечения двух соприкасающихся поверхностей, изучение вида (особенности) точки касания на этих линиях;
2) аналитическая реализация графического алгоритма построения касательной плоскости и нормали поверхности;
3) геометрическое толкование построения индикатрисы Дюпена и линий кривизны поверхности;
4) использование линий кривизны поверхности для программирования фрезерной обработки поверхностей объемной оснастки.
Далее обсудим решение перечисленных задач, обратив внимание на межпредметные связи начертательной геометрии и смежных разделов математики.
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