Pedal Curves of the Mixed-Type Curves in the Lorentz-Minkowski Plane

: In this paper, we consider the pedal curves of the mixed-type curves in the Lorentz– Minkowski plane R 21 . 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.


Introduction
As an important kind of submanifolds, curves in different spaces have attracted wide attention from mathematicians. Studies have focused on investigating not only regular curves, but also singular curves, and have made great achievements (see [1][2][3][4][5][6][7][8][9][10][11]). Because Lorentz space is strongly connected to the theory of general relativity, the investigation of submanifolds in Lorentz space and its subspaces has great significance. Scholars have shown interest in curves in Lorentz space and its subspace and have studied evolutes, involutes, parallels and some other associated curves in these spaces. There have been several relevant investigations in this area (see [3][4][5][6][7][12][13][14][15]). Having the appearance of a negative index, there are three types of vectors in Lorentz space. For a curve, the type of tangent vector at each point determines the type of point. As for non-lightlike curves in Lorentz space, we always select their arc-length parameters and adopt the Frenet-Serret frame to investigate them (see [8,16]).
In fact, curves in Lorentz space do not always consist of a single type of points, but rather can involve all three types of points. This is what we mean by mixed-type curves. As a more familiar condition, the investigation of mixed-type curves has important significance. Because the curvature at lightlike points cannot be defined, the classical Frenet-Serret frame does not work. Due to the lack of necessary tools for its research, almost no research has been conducted on this subject. In 2018, S. Izumiya, M. C. Romero Fuster, and M. Takahashi presented the lightcone frame and established the fundamental theory of mixed-type curves in R 2 1 in [17]. As an application of the theory, they studied the evolutes of regular mixed-type curves. In [18], T. Liu and the second author of this paper gave the lightcone frame in Lorentz 3-space and considered mixed-type curves in this space. Currently, the investigation of mixed-type curves in R 2 1 has not been completed. As the depth of their work, the (n, m)-cusp mixed-type curves in R 2 1 were investigated, as well as the evolutes of the (n, m)-cusp mixed-type curves, as presented by us in [19]. Later, we also considered the evolutoids of mixed-type curves in R 2 1 . The pedal curves is a kind of significant curves due to their geometric properties. In the Euclidean space R 2 , the pedal curve is always defined by the locus of the foot of the perpendicular from the given point to the tangent to the base curve. M. Božek and G. Foltán considered the relationship of singular points of regular curves' pedal curves and the inflections of the base curves in R 2 in [20]. Later, in [21], Y. Li and the second author of this paper studied the pedal curve of the given curves with singular points in R 2 . O. Ogulcan Tuncer et al. described the relationship of the pedal curves and contrapedal curves in R 2 in [22]. However, on the topic of pedal curves of mixed-type curves in R 2 1 , which is an interesting and worthy subject, there have not been relevant investigations.
Our purpose in this paper was to solve the problems related to the pedal curves of mixed-type curves in R 2 1 . In Section 2, we review some essential knowledge about R 2 1 and introduce the lightcone frame. Then, we define the pedal curves of mixed-type curves and investigate their properties in Section 3. We consider when the pedal curves of mixed-type curves have singular points and investigate the relationship of the types of points of the pedal curves and the base curves. Finally, in Section 4, for the purpose of showing the characteristics of the pedal curves of mixed-type curves, we present two examples.
If not specifically mentioned, all maps and manifolds in this paper are infinitely differentiable.

Preliminaries
Here, we introduce some essential knowledge about the Lorentz-Minkowski plane for the sake of convenience.
2} be a vector space of dimension 2. If R 2 is endowed with the metric which is induced by the pseudo-scalar product where x = (x 1 , x 2 ), y = (y 1 , y 2 ), and x, y ∈ R 2 , then we call (R 2 , , ) the Lorentz-Minkowski plane and denote it by R 2 1 . For a non-zero vector x ∈ R 2 1 , there are three types of vectors in R 2 1 . When x, x is positive, negative and vanishing, it is called spacelike, timelike or lightlike, respectively. A non-lightlike vector refers to a vector that is spacelike or timelike.
For a vector x ∈ R 2 1 , if there exists a vector y ∈ R 2 1 , which satisfies x, y = 0, we say y is pseudo-perpendicular to x.
We define the norm of x = (x 1 , x 2 ) ∈ R 2 1 by and the pseudo-orthogonal complement of x is given by x ⊥ = (x 2 , x 1 ). By definition, x and x ⊥ are pseudo-orthogonal to each other, and It is obvious that x ⊥ = ±x if and only if x is lightlike, and x ⊥ is timelike (resp. spacelike) if and only if x is spacelike (resp. timelike).
Moreover, we say a curve is non-lightlike if it is a spacelike or timelike curve and a point is non-lightlike if it is a spacelike or timelike point. If ρ(t) contains three types of points simultaneously, then it is exactly a mixed-type curve, which is the main research object in this paper.
This satisfies Definition 1.

Then, ρ(t 0 ) is an inflection of ρ if and only iḟ
1 be a regular mixed-type curve with the lightlike tangential date (α, β).
is called an ordinary inflection. In this paper, we only consider ordinary inflections of the mixed-type curves, and we call them inflections for short.

Pedal Curves of the Mixed-Type Curves in
The pedal curves of the regular curves in R 2 are widely studied. As for the regular curves in R 2 1 , the pedal curves of them are defined similarly. They are always given by the pseudo-orthogonal projection of a fixed point on the tangent lines of the base curves. Therefore, the definitions of pedal curves of the regular non-lightlike curves are given as follows.

Definition 2.
Let ρ : I → R 2 1 be a regular non-lightlike curve and Q be a point in R 2 1 . Then, the pedal curve Pe(ρ)(t) of the base curve ρ(t) is given by It is obvious that the pedal curve of a non-lightlike curve with the lightcone frame {L + , L − } and the lightlike tangential data (α, β) is Let ρ : I → R 2 1 be a regular mixed-type curve. Since ρ(t 0 ),ρ(t 0 ) = 0 when ρ(t 0 ) is a lightlike point, it is probably not always possible to define a pedal curve of a mixed type curve. In fact, if Q coincides with the lightlike point or Q is on the tangent line of the lightlike point, we can define the pedal curve Pe(ρ) : I → R 2 1 of ρ with the lightcone frame {L + , L − } and the lightlike tangential data (α, β) by Formula (3).
When ρ(t 0 ) is a non-lightlike point, Pe(ρ)(t 0 ) satisfies Formula (3), obviously. When ρ(t 0 ) is a lightlike point, α(t 0 )β(t 0 ) = 0, and we suppose that Q coincides with the lightlike point or Q is on the tangent line of the lightlike point. In these cases, Formula (3) also holds, and in the following, we discuss the specific forms of Pe(ρ)(t 0 ).
If ρ(t) is non-lightlike, by direct calculation, If ρ(t 0 ) is a lightlike point. Firstly, suppose that α(t 0 ) = 0 and β(t 0 ) = 0, then Q coincides with the lightlike point or Q is on the tangent line of the lightlike point is exactly Q − ρ(t 0 ), L + = 0. In this case, we define Then, we can find that .
By the above calculation, we can define Pe(ρ)(t 0 ) as lim t→t 0

Pe(ρ)(t). To sum up, if
Q coincides with ρ(t 0 ), then Q − ρ(t 0 ), L + = Q − ρ(t 0 ), L − = 0, Pe(ρ)(t 0 ) is given by If Q is on the tangent line of ρ(t 0 ), then Q − ρ(t 0 ), L − = 0, Pe(ρ)(t 0 ) is given by As for the condition of α(t 0 ) = 0 and β(t 0 ) = 0, in this case Q coincides with the lightlike point or Q is on the tangent line of the lightlike point refers to Q − ρ(t), L − = 0, similarly we can find that: If Q coincides with ρ(t 0 ), then Pe(ρ)(t 0 ) is given by If Q is on the tangent line of ρ(t 0 ), then Pe(ρ)(t 0 ) is given by nor on the tangent line of ρ(t 0 ), then when t approaches to t 0 , Considering when the pedal curves of the regular mixed-type curves have singular points, we have following conclusions. (i) ρ(t 0 ) is an inflection but Q is not coincides with ρ(t 0 ); (ii) ρ(t 0 ) is not an inflection, but Q coincides with ρ(t 0 ); (iii) ρ(t 0 ) is an inflection and Q coincides with ρ(t 0 ). (2) if ρ(t 0 ) is a lightlike point, and Q coincides with ρ(t 0 ) or Q is on the tangent line of ρ(t 0 ), then Pe(ρ)(t 0 ) is regular.
Proof. As the pedal curve of the mixed-type curve ρ(t) is given by the formula (3), by direct calculation, we can geṫ When is an inflection and Q coincides with ρ(t 0 ).
Following that, we consider the condition when ρ(t 0 ) is a lightlike point. First, we sup- When Q − ρ(t), L + = 0, we have known that Pe(ρ)(t 0 ) is asymptotic with lightlike line along the positive or negative direction of L + . So we consider the condition that Q − ρ(t 0 ), L + = 0.
First we suppose that ρ(t 0 ) is not an inflection of ρ(t), .

Proposition 1.
Let ρ : I → R 2 1 be a regular mixed-type curve and Q be a point in R 2 1 . Pe(ρ) : We have given the definition of (n, m)-cusp in [19]. According to the conclusion in [19], we can obtain Proposition 1 directly.

Proposition 2.
Let ρ : I → R 2 1 be a regular mixed-type curve and Q be a point in R 2 1 . Pe(ρ) : I → R 2 1 is the pedal curve of ρ. Suppose that Q is on the tangent line of ρ(t 0 ). (1) If ρ(t 0 ) is a non-lightlike point, then Pe(ρ)(t 0 ) coincides with Q; Proof. Since the pedal curve of the mixed-type curve ρ(t) is given by formula (3).
Suppose that Q is on the tangent line of ρ(t 0 ), then we have Q − ρ(t 0 ) and α(t 0 )L + + β(t 0 )L − are linearly dependent.
If ρ(t 0 ) is a non-lightlike point, then there exists λ ∈ R, such that We can obtain Therefore, Pe(ρ)(t 0 ) coincides with Q.
If ρ(t 0 ) is a lightlike point, we have know that when α(t 0 ) = 0 and β(t 0 ) = 0, when α(t 0 ) = 0 and β(t 0 ) = 0, Then, we investigate the type of points of the pedal curve of the mixed-type curve in R 2 1 and the following proposition can be obtained.
In this case, Pe(ρ)(t 0 ) is a spacelike point. See the red dashed curve in Figure 2.
When t 0 = 0, ρ(t 0 ) is a lightlike point and it is also an inflection. See the blue curve in Figure 3.
In this case Pe(ρ)(t 0 ) is asymptotic with lightlike line along the positive and negative direction of L + . See the green curve in Figure 3.
In this case Pe(ρ)(t 0 ) is a lightlike point. See the red dashed curve in Figure 3.