In this paper,we define the evolute and focal surface of a spacelike framed curve with lightlike components in Minkowski 3-space.It is a generalization of the previous results of regular spacelike curves,since singula...In this paper,we define the evolute and focal surface of a spacelike framed curve with lightlike components in Minkowski 3-space.It is a generalization of the previous results of regular spacelike curves,since singularities are allowed in the original spacelike curves studied by spacelike framed curves with lightlike components.Meanwhile,we show a new geometric invariant to characterise singularities of the focal surface.Then,the classification theorem and recognition theorem for the singularities of the focal surface in generic are also given.展开更多
In the light of great importance of curves and their frames in many differentbranches of science, especially differential geometry as well as geometric properties andthe uses in various fields, we are interested here ...In the light of great importance of curves and their frames in many differentbranches of science, especially differential geometry as well as geometric properties andthe uses in various fields, we are interested here to study a special kind of curves calledSmarandache curves in Lorentz 3-space. Then, we present some characterizations for thesecurves and calculate their Darboux invariants. Moreover, we classify TP, TU, PU andTPU-Smarandache curves of a spacelike curve according to the causal character of thevector, curve and surface used in the study. Besides, we give some of differential geometricproperties and important relations between that curves. Finally, to demonstrate ourtheoretical results a computational example is given with graph.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.11671070)。
文摘In this paper,we define the evolute and focal surface of a spacelike framed curve with lightlike components in Minkowski 3-space.It is a generalization of the previous results of regular spacelike curves,since singularities are allowed in the original spacelike curves studied by spacelike framed curves with lightlike components.Meanwhile,we show a new geometric invariant to characterise singularities of the focal surface.Then,the classification theorem and recognition theorem for the singularities of the focal surface in generic are also given.
文摘In the light of great importance of curves and their frames in many differentbranches of science, especially differential geometry as well as geometric properties andthe uses in various fields, we are interested here to study a special kind of curves calledSmarandache curves in Lorentz 3-space. Then, we present some characterizations for thesecurves and calculate their Darboux invariants. Moreover, we classify TP, TU, PU andTPU-Smarandache curves of a spacelike curve according to the causal character of thevector, curve and surface used in the study. Besides, we give some of differential geometricproperties and important relations between that curves. Finally, to demonstrate ourtheoretical results a computational example is given with graph.
