Abstract
The surface of parallel transfer is a surface that admits the parametrization r(u,v) = U(u) + V(v). The surface of parallel transfer in E3 can be considered as a surface produced by parallel transfer of one line along the other.In this study the torus M is different from the classic torus T, which is obtained by rotating the circle along the axis. We consider the torus as the surface of parallel transfer obtained by parallel transfer of one circle along the other. The circles are located in mutually orthogonal intersecting planes.The closed curve on the torus M is defined with the 4n-periodic vector-function.Using this function, the equations of one-sided surfaces are obtained. In particular, the equation of the Mobius band with the mentioned curve being the boundary is found.The Klein bottle is cut into two Moebius band. Also, the cross-cap is considered.The inversion of torus T and the inversion of torus M is studied in this paper.The studied surfaces are constructed in Euclidean space E3 with the help of a mathematical software toolkit.DOI 10.14258/izvasu(2018)1-24
Highlights
В работе приводится пример тора M в E3, отличного от классического тора T, который получается при вращении окружности вокруг оси
The surface of parallel transfer in E3 can be considered as a surface produced by parallel transfer of one line along the other
In this study the torus M is different from the classic torus T, which is obtained by rotating the circle along the axis
Summary
Мы рассматриваем тор M как поверхность переноса, которая получается при параллельном переносе одной окружности вдоль другой, причем окружности расположены во взаимно ортогональных пересекающихся плоскостях. Ключевые слова: поверхность переноса, тор, периодическая функция, инверсия, лист Мебиуса, бутылка Клейна, скрещенный колпак. In this study the torus M is different from the classic torus T , which is obtained by rotating the circle along the axis. We consider the torus as the surface of parallel transfer obtained by parallel transfer of one circle along the other. The closed curve on the torus M is defined with the 4π-periodic vector-function. Using this function, the equations of one-sided surfaces are obtained. Кривые переноса тора Клиффорда есть окружности, расположенные во взаимно ортогональных непересекающихся плоскостях.
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