In this paper, we consider the pedal curves of the mixed-type curves in the Lorentz–Minkowski plane ${\mathbb{R}}_1^2$. The pedal curve is always given by the pseudo-orthogonal projection of a fixed point on the tangent lines of the base curve. For a mixed-type curve, the pedal curve at lightlike points cannot always be defined. Herein, we investigate when the pedal curves of a mixed-type curve can be defined and define the pedal curves of the mixed-type curve using the lightcone frame. Then, we consider when the pedal curves of the mixed-type curve have singular points. We also investigate the relationship of the type of the points on the pedal curves and the type of the points on the base curve.

The evolutoid of a regular curve in the Lorentz-Minkowski plane is the envelope of the lines between tangents and normals of the curve. It is regarded as the generalized caustic (evolute) of the curve. The evolutoid of a mixed-type curve has not been considered since the definition of the evolutoid at lightlike point can not be given naturally. In this paper, we devote ourselves to consider the evolutoids of the regular mixed-type curves in . As the angle of lightlike vector and nonlightlike vector can not be defined, we introduce the evolutoids of the nonlightlike regular curves in and give the conception of the -transform first. On this basis, we define the evolutoids of the regular mixed-type curves by using a lightcone frame. Then, we study when does the evolutoid of a mixed-type curve have singular points and discuss the relationship of the type of the points of the mixed-type curve and the type of the points of its evolutoid.

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.

Pedal and contrapedal curves are important study objects of plane curves. As for a mixed-type Minkowski plane curve, since the definitions of the pedal and contrapedal curves at lightlike points can not always be given, the investigation of them is difficult. We have done some research on the pedal curves of a mixed-type curve. In this paper, we discuss when the contrapedal curves of a mixed-type curve exist and give the definition of them when they exist. Then, we study when the contrapedal curves of the mixed-type curve have singular points. Meanwhile, we consider the types of the points on the contrapedal curves. Moreover, we investigate the relationship between the pedal and contrapedal curves of a mixed-type curve, as well as the relationship among them and the evolute of the base curve.