#### Abstract

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.

#### 1. Introduction

The caustic, also called evolute, is an important study object in physics and nonlinear sciences. Compared with the neighbouring spaces, the field strength on caustics increases sharply; thus, many interesting phenomena emerged. The caustics have wide applications in many other research fields, such as optics, mechanics, and electromagnetism; thus, they have drawn the attention of many scientists. In [1], in order to confirm the location of caustic regression points, the researchers gave a new geometric variational criterion. Besides, computing parametric equations of the caustic were solved, in whatever Cartesian coordinate system. The light focuses are not on caustic as usually what is said in catastrophe optics but on regions that are defined as second order evolutes was proofed in [2]. When a particle is compelled to slide on tautochrone curves, it is isochronal to the gained oscillation is. If two guides are incorporated for the pendulum string to make its length short forcing the particle to follow the tautochrone, these guides were exactly connected with the evolute of the pendulum trajectory. In [3], a special isochronous pendulum that is asymmetric, in which a part of the trajectory accords with the nonisochronous simple pendulum and the rest of the part corresponds to a peculiar trajectory, was considered. It is in the singular points of the evolute consistent with the limitations of the edge segments and in the points where some walls meet that the limitations of domain walls appears (see [4]). Because of the importance of the caustics, more and deeper studies of them are necessary.

From the geometric point of view, the evolute of a curve in the Euclidean plane is usually defined by the locus of centres of osculating circles of the base curve or the envelope of the normal lines of the base curve. As it is an important object of studying in classical differential geometry, it is widely considered by many scholars (see [5, 6]). Since the singular curves are more familiar conditions in the natural world, there are more and more studies about the evolutes of the curves with singular points (see [5, 7, 8]). Also, the concept of evolutes is generalized to hyperbolic, pseudosphere, lightcone, Lorentz space, and some other spaces, to show the application about the singularity theory (see [9–12]).

As we all know, in a plane, it is a curve tangent to all of the family of lines the envelope of them. The evolute of a curve can be regarded as the envelope of the normal lines, and the envelope of the tangent lines is the curve itself. But how about the condition between the envelope of tangents and the envelope of normals? In fact, it is precisely the evolutoid of the curve. As a generalization of the evolutes, the studies of the evolutoids have great significance. There have been some studies about the evolutoids of the curves. The evolutoids of the regular curves in were considered, and the family of parallels associated with the evolutoids was studied by Giblin and Warder [13]. In [14], Izumiya and Takeuchi considered the condition when the curves have singular points. They introduced the evolutoids of frontals in and gave the relationship of the evolutoids and pedaloids of frontals. In [15], Aydın Şekerci and Izumiya studied the evolutoids of nonlightlike curves including not only regular condition but also the condition with singular points in .

In addition, because the Lorentz space is in connection with general relativity theory strongly, studying the Lorentz space has great significance, of course also including its subspaces. Plentiful relevant studies have appeared (see [9–11, 16–22]). As a subspace of the Lorentz space, the Lorentz-Minkowski plane, always denoted by , is interested by scholars. They take pains to study the curves in it. For the nonlightlike curves in , we always select their arc-length parameter and adopt Frenet-Serret frame to study them (see [23]). However, the mixed-type curves, which are consisted by three types of points, are more familiar conditions. But with regard to the mixed-type curves, since the appearance of lightlike points, the Frenet-Serret frame does not work and we can not study them through the old method. Until 2018, Izumiya et al. gave the lightcone frame and established the fundamental theory of the mixed-type curves in the Lorentz-Minkowski plane in [10]. As an application of the theory, they studied the evolutes of the regular mixed-type curves. The mixed-type curves with singular points were studied by us in [12]. The -cusp mixed-type curves in were considered, as well as the evolutes of the -cusp mixed-type curves were given. In [24], we investigated the pedal curves of mixed-type curves in and consider their properties. But as for the evolutoids of the mixed-type curves in , which is an interesting and worthy subject, nobody has been studying them.

In this paper, we focus on solving this question. We consider the evolutoids of the regular mixed-type curves in and study their propositions. In Section 2, we review some essential knowledge about . In addition, to achieve the goal that defining the evolutoids of mixed-type curves, the evolutoids of the nonlightlike curves are introduced by us in this section first. In Section 3, we introduce a suitable frame for the mixed-type curves which is called lightcone frame in . Meanwhile, we define what the evolutoids of the mixed-type curves is. Then, we consider when the evolutoids of the curves in have singular points. Also, we give the relationship of the types of the points on the evolutoid of a mixed-type curve in and the types of the points on this base curve. Finally, for the purpose of showing the characteristics of the evolutoids of the mixed-type curves, we give two examples in Section 4.

If not specifically mentioned, all maps and manifolds in this paper are infinitely differentiable.

#### 2. Preliminaries

##### 2.1. Basic Conceptions in

Some essential knowledge about the Lorentz-Minkowski plane is given here to provide convenience for the following research.

Let be a 2-dimensional vector space. If equipped with the metric which is induced by the *pseudo-scalar product*where and . Then, we call the Lorentz-Minkowski plane.

There are three types of vectors in . For a nonzero vector , when the pseudoscalar product of and itself is positive, negative, and vanishing, it is called *spacelike*, *timelike*, or *lightlike*, respectively. A *nonlightlike* vector refers to the vector is spacelike or timelike.

For a vector , if there exists a vector , which satisfies , we say is pseudo-orthogonal to .

We define the *norm* of by
and the pseudo-orthogonal complement of is given by . By definition, and are pseudo-orthogonal to each other, and

For a nonzero vector , it is obvious that is spacelike (resp., timelike) if and only if is timelike (resp., spacelike) and if and only if is lightlike.

Let be a regular curve. Denote ; then, we say is a *spacelike* (resp., *timelike*, *lightlike*) curve if the pseudoscalar of and itself is positive (resp., negative, vanishing) for any . Furthermore, the type of a point (or ) is determined by the type of . For more details, see [10].

Moreover, we say a curve is *nonlightlike* which means it is a spacelike or timelike curve and a point is *nonlightlike* if it is a spacelike or timelike point. If contains three types of points simultaneously, then, it is exactly a mixed-type curve, which is the main research object in this paper.

The lightcone in with centre is defined as

The lightcone centred at the origin in is denoted by , and we denote the part of in first quadrant, second quadrant, third quadrant, and forth quadrant by , , , and , respectively. It is obvious that divides into four parts. We write them as follows: where .

Now, we introduce Lorentz motion in .

*Definition 1. *Let and be two regular curves. and are called congruent through a Lorentz motion if there exist a matrix and a constant such that for all , where is given by
for some .

The type vector in is unaltered through a Lorentz motion. More specifically, if a vector is in (resp. , , , , , , ), when it is still in (resp. , , , , , , ); when it is in (resp. , , , , , , ).

Let be a regular nonlightlike curve. We take as the arc-length parameter of . For arbitrary , , the unit tangent vector of is given as and the unit normal vector of is given as ; then, the orientation of is anticlockwise, where

So the Frenet-Serret formula is given by where is the curvature of .

If taking not the arc-length parameter but the general parameter of , then, the unit tangent vector of can be given as and the unit normal vector of can be given as , and the orientation of is anticlockwise.

The Frenet-Serret formula is written as and the curvature of is given by .

A point on a curve is an inflection point if and only if . And for , a regular nonlightlike curve without inflection points in , its evolute is defined to be

##### 2.2. Evolutoids of the Nonlightlike Curves in

If is a regular nonlightlike curve without inflection points in , then, the family of the normal lines to has an envelope, and so does the family of the tangent lines to . As we all know, the envelope of a family of lines in is a curve tangent to all of them. The envelope of the normal lines is the evolute of the curve, and the envelope of the tangent lines is itself. But what happens between the envelope of tangents and the envelope of normals. Here, we shall consider this question. The condition in the Euclidean plane was studied by Giblin and Warder [13]. In [15], Aydın Şekerci and Izumiya have introduced the evolutoids of nonlightlike curves in . To better carry out the following research, we think about them by another expression approach.

In order to solve above question, we define the -transform first.

Let be a regular nonlightlike curve. Through Lorentz motion, the -transform of its unit tangent vector is defined by where . We discover that rotates in the part which is in and the norm of is the same as the norm of . The line obtained by at is written as , and the envelope of lines is denoted by . It is obvious that when , is itself. When , is so-called the evolutoid of .

Similarly, we can consider the -transform of . It is defined by . The line obtained by is written as as well, and the envelope of lines is also denoted by . When , is the evolute of , and when , is also called the evolutoid of .

*Remark 2. *As the lightcone divides into four parts and , , , and are in different parts, we shall consider the -transform of , , , and , respectively. In fact, the line obtained by the -transform of and the -transform of is the same, either the -transform of or the -transform of . Therefore, we only need to consider the -transform of and .

Let be a regular nonlightlike curve without inflection points, and we consider the evolutoids of . We suppose that is obtained by the -transform of .

As the direction of is , the vector pseudo-orthogonal to can be expressed as , and for any , a vector equation of line L is such that

If we fix , then describes a family of lines: for arbitrary , there is a line and when varies the line moves in . For a fixed , the envelope of the family of the lines given by equation (14) is the set of points in ; then, there exists such that the following equation holds

For any fixed ,

Then,

Let , where , and it can denote any vectors in . Applying in , we can obtain two equations about and .

By direct calculation, we can get the evolutoid of the regular nonlightlike curve without inflection points in through the -transform of can be expressed by

Since is a regular curve without inflection points, . If , then, the lines are tangent to and the envelope is consistent with the original curve , namely, .

If is obtained by the -transform of , similar to the above method, we can get the evolutoid of through the -transform of expressed as follows:

Base on above all, for a regular nonlightlike curve in , we can obtain the definition of its evolutoid.

*Definition 3. *Let be a regular non-lightlike curve without inflection points in , then the evolutoid of is as follows:

*Remark 4. *In fact, when the -transform is on (or ), the evolutoid of is consistent with *-*evolutoid (or *-*evolutoid) in [15]. But for the further research of the evolutoids of the mixed-type curves, we adopted the definition mode of -transform by our own.

#### 3. Evolutoids of the Mixed-Type Curves in

For a regular curve in , when it contains spacelike regions and timelike regions simultaneously, the lightlike point appears between them. This kind of curve is the mixed-type curve that we said. There are many interesting phenomena because of the appearance of lightlike points. In this section, we would like to consider the evolutoids of the regular mixed-type curves in , which is the focus of our work. But it is badly observed that as the curvature cannot be defined very properly at lightlike point, the Frenet-Serret frame is not applicative for the mixed-type curves. Actually, following frame will be introduced by us which is pretty suitable for a mixed-type curve. The frame is given by Izumiya et al. [10].

and are two lightlike vectors. Apparently, they are independent to each other. Denote and by and , respectively; then, the vector pair {, } is called a *lightcone frame* of in .

Let be a regular mixed-type curve. There exists a corresponding smooth map , which satisfies

If equation (22) is established, is called *the lightlike tangential data* of . The pseudo-orthogonal complement of can be expressed by

Since the type of can be determined by .

Let be a regular mixed-type curve with the lightlike tangential date , then, is an inflection point of if and only if

As , we use Lorentz motion on , and we can get or where and . Formula (26) is called the -transform of , and formula (27) is called the -transform of .

Similarly, we can obtain that the -transform of and is and , respectively.

In fact, the line obtained by and is the same, either or . Therefore, we only need to consider the -transform of and . In addition, the type of is the same as , so does the type of and ; thus, we can avoid the angle of lightlike vector and nonlightlike vector through -transform.

The definition of the evolutoid of a regular mixed-type curve with the lightcone frame and the lightlike tangential data can be given as follows:

Theorem 5. *Let be a regular mixed-type curve without inflection points in , then the evolutoid of is as follows:
*

*Proof. *First, supposing that the -transform is on , then, the direction of is
The vector pseudo-orthogonal to is
For any , we have
where
and is fixed.

By direct calculation,
Let
where is a smooth map.

We have
By direct calculation, we get
Thus,
If the -transform is on , similar to the above process, we can get the evolutoid of the regular mixed-type curve as follows:

*Remark 6. *If the -transform is on , when , is consistent with . If the -transform is on , when , , it is exactly the evolute of .

As for a regular mixed-type curve in , is its evolutoid always regular? If not, when does the evolutoid of have singular points? We give the following theorem to answer this question.

Theorem 7. *Let be a regular mixed-type curve without inflection points in , is the evolutoid of .
*(1)*Take the arc-length parameter , and suppose that is a nonlightlike point; then, is a singular point of if and only if
(i) , when the -transform is on ;(ii), when the -transform is on .*(2)

*Suppose that is a lighlike point,*(i)

*When the -transform is on , (a)*(ii)

*if , , is a singular point of if and only if ,*(b)*if , , is a singular point of if and only if ;**When the -transform is on , is always a regular point.*

*Proof. *First, suppose that is a non-lightlike point and the -transform is on . Using the Frenet-Serret formula we can get
if and only if . If we take the arc-length parameter , then, we have
Similarly, if the -transform is on , we have if and only if .

Next, suppose that is a lightlike point and the -transform is on . By direct calculation, we can get
where
When , , if and only if , that is, . When , , if and only if , that is, .

Similarly, if the -transform is on , we can get
When , , if and only if , that is, . We can not work out such , so there not exist such that is singular point. When and , there not exist such that is a singular point. Thus, is always a regular point

In [10], we know that for a point , if it is a lightlike point, then the corresponding point on the evolute is always a regular point. Here, we can conclude that for a lightlike point , its evolutoid may be singular, which is a different phenomenon from the evolute.

We would like to consider the relationship of the types of the points on the evolutoid of a mixed-type curve and the types of the points . By Theorem 7, the following proposition can be obtained.

Proposition 8. *Let be a regular mixed-type curve without inflection points in , is the evolutoid of .
*(1)*Take the arc-length parameter , and suppose that is a non-lightlike point; then,
(i) when the -transform is on and , then, is spacelike (resp., timelike) if and only if is spacelike (resp., timelike);(ii)when the -transform is on and , then, is timelike (resp., spacelike) if and only if is spacelike (resp., timelike).*(2)

*Suppose that is a lighlike point; then,*(i)

*when the -transform is on ,*(a)

*if , , and , then is lightlike;*(b)

*if , , and , then is lightlike;*(ii)

*when the -transform is on , is always lightlike.*

#### 4. Examples

We would like to present the characteristics of the evolutoids of the regular mixed-type curves without inflections by the following two examples.

*Example 9. *Let , be a regular mixed-type curve without inflections. See the blue curve in Figure 1.

We can get that the evolute of is

See the orange dashed curve in Figure 1.

If the -transform is on and , the evolutoid of is

See the red dashed curve in Figure 1.

If the -transform is on and , the evolutoid of is

When , is a lightlike point, and is a singular point. See the green curve in Figure 1.

*Example 10. *Let , be a regular mixed-type curve without inflections. See the blue curve in Figure 2.

We can get that the evolute of is

See the orange dashed curve in Figure 2.

If the -transform is on and , the evolutoid of is

See the red dashed curve in Figure 2.

If the -transform is on and , the evolutoid of is

When or , is a lightlike point, and is a singular point. See the green curve in Figure 2.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interests in this work.

#### Acknowledgments

This research was funded by the National Natural Science Foundation of China (grant number 11671070).