Relationships of Associated Curves of Mixed-Type Curves and Their Singularities in Minkowski Plane

In this paper, we investigate the relationships and singularities of three associated curves, T-dual curves, N-dual curves and evolutes, for mixed-type curves in the Minkowski plane. While T-dual curves and evolutes have been studied in previous works, the N-dual curve has remained unexplored in the mixed-type setting. To fill this gap, this paper makes three main contributions. Firstly, we provide a rigorous definition of the N-dual curve, explicitly resolving the technical difficulties that arise at lightlike points where the normal line is not well-defined. Secondly, we analyze its singularities and classify its point types. Thirdly, based on these results, we establish new geometric relations among the T-dual curve, N-dual curve, and evolute. In particular, we prove that at lightlike points, the T-dual and N-dual curves coincide when the fixed point lies on the tangent line, and that the T-dual curve of the evolute coincides with the N-dual curve of the original curve under suitable conditions. These results reveal a coherent geometric framework linking the three objects. All theoretical findings are supported and validated by a variety of examples throughout the paper.