Abstract
In this work, a pair of dual associate null scrolls are defined from the Cartan Frenet frame of a null curve in Minkowski 3-space. The fundamental geometric properties of the dual associate null scrolls are investigated and they are related in terms of their Gauss maps, especially the generalized 1-type Gauss maps. At the same time, some representative examples are given and their graphs are plotted by the aid of a software programme.
Highlights
In differential geometry, the associate curves and associate surfaces such as the Bertrand curve, the Mannheim curve, evolute-involute pair, the parallel surfaces and the focal surfaces etc. compose a large class of fascinating subjects in the curve and surface theory in Euclidean space and in pseudo-Euclidean space, such as Minkowski space [1,2,3,4]
We always can choose a null curve as the base curve of a ruled surface with lightlike ruling and the normalization condition is satisfied at the same time, which is said to be a null scroll [7,8,9,10]
According to the definitions of generalized T-associate curve, generalized B-associate curve of a null curve and the definition of null scrolls, we want to construct a pair of null scrolls which satisfy the same normalization condition
Summary
The associate curves and associate surfaces such as the Bertrand curve, the Mannheim curve, evolute-involute pair, the parallel surfaces and the focal surfaces etc. compose a large class of fascinating subjects in the curve and surface theory in Euclidean space and in pseudo-Euclidean space, such as Minkowski space [1,2,3,4]. For the ruled surfaces with lightlike rulings, the base curves can be null curves or non-null curves obviously. We always can choose a null curve as the base curve of a ruled surface with lightlike ruling and the normalization condition is satisfied at the same time, which is said to be a null scroll [7,8,9,10]. Throughout this paper, all the geometric objects under consideration are smooth and all surfaces are connected unless otherwise stated
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