# Evolutoids of the Mixed-Type Curves

Authors: Xin Zhao; Donghe Pei
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

The evolutoid of a regular curve in the Lorentz-Minkowski planeℝ12 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 ℝ12. As the angle of lightlike vector and nonlightlike vector can not be defined, we introduce the evolutoids of the nonlightlike regular curves in ℝ12 and give the conception of the σ-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.


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## Body

## 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 inℝ2 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 ℝ2 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 ℝ12.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 ℝ12, is interested by scholars. They take pains to study the curves in it. For the nonlightlike curves in ℝ12, 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 n,m-cusp mixed-type curves in ℝ12 were considered, as well as the evolutes of the n,m-cusp mixed-type curves were given. In [24], we investigated the pedal curves of mixed-type curves in ℝ12 and consider their properties. But as for the evolutoids of the mixed-type curves in ℝ12, 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 inℝ12 and study their propositions. In Section 2, we review some essential knowledge about ℝ12. 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 ℝ12. Meanwhile, we define what the evolutoids of the mixed-type curves is. Then, we consider when the evolutoids of the curves in ℝ12 have singular points. Also, we give the relationship of the types of the points on the evolutoid of a mixed-type curve in ℝ12 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 inℝ12

Some essential knowledge about the Lorentz-Minkowski plane is given here to provide convenience for the following research.Letℝ2=x1,x2∣x1,x2∈ℝ be a 2-dimensional vector space. If ℝ2 equipped with the metric which is induced by the pseudo-scalar product(1)x,y=−x1y1+x2y2,where x=x1,x2 and y=y1,y2. Then, we call ℝ12=ℝ2,, the Lorentz-Minkowski plane.There are three types of vectors inℝ12. For a nonzero vector x∈ℝ12, when the pseudoscalar product of x 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 vectorx∈ℝ12, if there exists a vector y∈ℝ12, which satisfies x,y=0, we say y is pseudo-orthogonal to x.We define thenorm of x=x1,x2∈ℝ12 by
(2)∥x∥=∣x,x∣,and the pseudo-orthogonal complement of x is given by x⊥=x2,x1. By definition, x and x⊥ are pseudo-orthogonal to each other, and
(3)∥x∥=∥x⊥∥.For a nonzero vectorx∈ℝ12, it is obvious that x⊥ is spacelike (resp., timelike) if and only if x is timelike (resp., spacelike) and x⊥=±x if and only if x is lightlike.Letρ:I⊆ℝ⟶ℝ12 be a regular curve. Denote ρ̇t=dρ/dtt; then, we say ρ is a spacelike (resp., timelike, lightlike) curve if the pseudoscalar of ρ̇t and itself is positive (resp., negative, vanishing) for any t∈I. Furthermore, the type of a point ρt (or t) is determined by the type of ρ̇t. For more details, see [10].Moreover, we say a curve isnonlightlike which means it is a spacelike or timelike curve and a point is nonlightlike if it is a spacelike or timelike point. If ρt 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 inℝ12 with centre p∈ℝ12 is defined as
(4)LC∗p,0=y∈ℝ12∣y−p,y−p=0.The lightcone centred at the origin inℝ12 is denoted by LC∗, and we denote the part of LC∗ in first quadrant, second quadrant, third quadrant, and forth quadrant by LC++∗, LC−+∗, LC+−∗, and LC−−∗, respectively. It is obvious that LC∗ divides ℝ12 into four parts. We write them as follows:
(5)Sl12+=y∈ℝ12∣y,y>0,y2>0,Sl12−=y∈ℝ12∣y,y>0,y2<0,Tl12+=y∈ℝ12∣y,y<0,y1>0,Tl12−=y∈ℝ12∣y,y<0,y1<0,where y=y1,y2∈ℝ12.Now, we introduce Lorentz motion inℝ12.Definition 1.
Letρ1 and ρ2:I⟶ℝ12 be two regular curves. ρ1 and ρ2 are called congruent through a Lorentz motion if there exist a matrix Q and a constant a∈ℝ12 such that ρ1t=Qρ2t+a for all t∈I, where Q is given by
(6)Q=coshσ−sinhσ−sinhσcoshσorQ=−coshσsinhσsinhσ−coshσ,for some σ∈ℝ.The type vector inℝ12 is unaltered through a Lorentz motion. More specifically, if a vector x=x1,x2 is in Sl12+ (resp. Sl12−, Tl12+, Tl12−, LC++∗, LC−+∗, LC+−∗, LC−−∗), when
(7)Qx=coshσx1−sinhσx2−sinhσx1+coshσx2,it is still in Sl12+ (resp. Sl12−, Tl12+, Tl12−, LC++∗, LC−+∗, LC+−∗, LC−−∗); when
(8)Qx=−coshσx1+sinhσx2sinhσx1−coshσx2,it is in Sl12− (resp. Sl12+, Tl12−, Tl12+, LC−+∗, LC++∗, LC−−∗, LC−+∗).Letρ:I⟶ℝ12 be a regular nonlightlike curve. We take s as the arc-length parameter of ρ. For arbitrary s∈I, ∥ρ′s∥=1, the unit tangent vector of ρs is given as ts=ρ′s and the unit normal vector of ρs is given as ns=−1ω+1ρ′s⊥; then, the orientation of ts,ns is anticlockwise, where
(9)ω=1,ifsistimelike,2,ifsisspacelike.So the Frenet-Serret formula is given by(10)t′sn′s=0κsκs0tsns,where κs=ρ′′s,ρ′s⊥ is the curvature of ρs.If taking not the arc-length parameters but the general parameter t of ρ, then, the unit tangent vector of ρt can be given as tt=ρ̇t/∥ρ̇t∥ and the unit normal vector of ρt can be given as nt=−1ω+1ρ̇t⊥/∥ρ̇t∥, and the orientation of tt,nt is anticlockwise.The Frenet-Serret formula is written as(11)ṫtṅt=∥ρ̇t∥0κtκt0ttnt,and the curvature of ρt is given by κt=ρ¨t,ρ̇t⊥/∥ρ̇t∥3.A pointt on a curve ρ is an inflection point if and only if ρ¨t,ρ̇t⊥=0. And for ρt, a regular nonlightlike curve without inflection points in ℝ12, its evolute is defined to be
(12)et=ρt−1κtnt.


### 2.2. Evolutoids of the Nonlightlike Curves inℝ12

Ifρ is a regular nonlightlike curve without inflection points in ℝ12, then, the family of the normal lines to ρ has an envelope, and so does the family of the tangent lines to ρ. As we all know, the envelope of a family of lines in ℝ12 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 ρ 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 ℝ12. To better carry out the following research, we think about them by another expression approach.In order to solve above question, we define theσ-transform first.Letρ:I⟶ℝ12 be a regular nonlightlike curve. Through Lorentz motion, the σ-transform of its unit tangent vector tt is defined by
(13)ttcoshσ+ntsinhσ,where σ∈ℝ. We discover that ttcoshσ+ntsinhσ rotates in the part which tt is in and the norm of ttcoshσ+ntsinhσ is the same as the norm of tt. The line obtained by ttcoshσ+ntsinhσ at t is written as l, and the envelope of lines l is denoted by τσ. It is obvious that when σ=0, τ0 is ρ itself. When σ≠0, τσ is so-called the evolutoid of ρ.Similarly, we can consider theσ-transform of nt. It is defined by ttsinhσ+ntcoshσ. The line obtained by ttsinhσ+ntcoshσ is written as l as well, and the envelope of lines l is also denoted by τσ. When σ=0, τ0 is the evolute of ρ, and when σ≠0, τσ is also called the evolutoid of ρ.Remark 2.
As the lightconeLC∗ divides ℝ12 into four parts and tt, nt, −tt, and −nt are in different parts, we shall consider the σ-transform of tt, nt, −tt, and −nt, respectively. In fact, the line l obtained by the σ-transform of tt and the σ-transform of −tt is the same, either the σ-transform of nt or the σ-transform of −nt. Therefore, we only need to consider the σ-transform of tt and nt.Letρ:I⟶ℝ12 be a regular nonlightlike curve without inflection points, and we consider the evolutoids of ρ. We suppose that l is obtained by the σ-transform of tt.As the direction ofl is ttcoshσ+ntsinhσ, the vector pseudo-orthogonal to l can be expressed as ttsinhσ+ntcoshσ, and for any u=u1,u2∈ℝ12, a vector equation of line L is Fu,t=0 such that
(14)Fu,t=u−ρt,ttsinhσ+ntcoshσ.If we fixσ, then Fu,t=0 describes a family of lines: for arbitrary t∈I, there is a line and when t varies the line moves in ℝ12. For a fixed σ, the envelope τσ of the family of the lines given by equation (14) is the set of points u=u1,u2 in ℝ12; then, there exists t∈I such that the following equation holds
(15)Fu,t=Ftu,t=0.For any fixedσ,
(16)Ftu,t=−ρ̇t,ttsinhσ+ntcoshσ+u−ρt,∥ρ̇t∥κtntsinhσ+∥ρ̇t∥κtttcoshσ=−∥ρ̇t∥tt,ttsinhσ+ntcoshσ+∥ρ̇t∥u−ρt,κtntsinhσ+κtttcoshσ=−∥ρ̇t∥−1ωsinhσ+∥ρ̇t∥u−ρt,κtntsinhσ+κtttcoshσ=0.Then,(17)u−ρt,κtntsinhσ+κtttcoshσ=−1ωsinhσ.Letu−ρt=ξ1tt+ξ2nt, where ξ1,ξ2∈ℝ, and it can denote any vectors in ℝ12. Applying u−ρt=ξ1tt+ξ2nt in F=Ft=0, we can obtain two equations about ξ1 and ξ2.
(18)ξ1sinhσ−ξ2coshσ=0,ξ1κtcoshσ−ξ2κtsinhσ=sinhσ.By direct calculation, we can get the evolutoidEvσρt of the regular nonlightlike curve without inflection points ρt in ℝ12 through the σ-transform of tt can be expressed by
(19)Evσρt=u=ρt+coshσsinhσκttt+sinh2σκtnt.Sinceρt is a regular curve without inflection points, κt≠0. If σ=0, then, the lines F=0 are tangent to ρ and the envelope is consistent with the original curve ρ, namely, Evσρt=ρt.Ifl is obtained by the σ-transform of nt, similar to the above method, we can get the evolutoid Evσρt of ρt through the σ-transform of nt expressed as follows:
(20)Evσρt=ρt−coshσsinhσκttt−cosh2σκtnt.Base on above all, for a regular nonlightlike curveρt in ℝ12, we can obtain the definition of its evolutoid.Definition 3.
Letρ:I⟶ℝ12 be a regular non-lightlike curve without inflection points in ℝ12, then the evolutoid of ρt is as follows:
(21)Evσρt=ρt+coshσsinhσκttt+sinh2σκtnt,iftheσ‐transformisontt,ρt−coshσsinhσκttt−cosh2σκtnt,iftheσ‐transformisonnt.Remark 4.
In fact, when theσ-transform is on tt (or nt), the evolutoid of ρt is consistent with ϕT-evolutoid (or ϕN-evolutoid) in [15]. But for the further research of the evolutoids of the mixed-type curves, we adopted the definition mode of σ-transform by our own.


## 2.1. Basic Conceptions inℝ12

Some essential knowledge about the Lorentz-Minkowski plane is given here to provide convenience for the following research.Letℝ2=x1,x2∣x1,x2∈ℝ be a 2-dimensional vector space. If ℝ2 equipped with the metric which is induced by the pseudo-scalar product(1)x,y=−x1y1+x2y2,where x=x1,x2 and y=y1,y2. Then, we call ℝ12=ℝ2,, the Lorentz-Minkowski plane.There are three types of vectors inℝ12. For a nonzero vector x∈ℝ12, when the pseudoscalar product of x 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 vectorx∈ℝ12, if there exists a vector y∈ℝ12, which satisfies x,y=0, we say y is pseudo-orthogonal to x.We define thenorm of x=x1,x2∈ℝ12 by
(2)∥x∥=∣x,x∣,and the pseudo-orthogonal complement of x is given by x⊥=x2,x1. By definition, x and x⊥ are pseudo-orthogonal to each other, and
(3)∥x∥=∥x⊥∥.For a nonzero vectorx∈ℝ12, it is obvious that x⊥ is spacelike (resp., timelike) if and only if x is timelike (resp., spacelike) and x⊥=±x if and only if x is lightlike.Letρ:I⊆ℝ⟶ℝ12 be a regular curve. Denote ρ̇t=dρ/dtt; then, we say ρ is a spacelike (resp., timelike, lightlike) curve if the pseudoscalar of ρ̇t and itself is positive (resp., negative, vanishing) for any t∈I. Furthermore, the type of a point ρt (or t) is determined by the type of ρ̇t. For more details, see [10].Moreover, we say a curve isnonlightlike which means it is a spacelike or timelike curve and a point is nonlightlike if it is a spacelike or timelike point. If ρt 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 inℝ12 with centre p∈ℝ12 is defined as
(4)LC∗p,0=y∈ℝ12∣y−p,y−p=0.The lightcone centred at the origin inℝ12 is denoted by LC∗, and we denote the part of LC∗ in first quadrant, second quadrant, third quadrant, and forth quadrant by LC++∗, LC−+∗, LC+−∗, and LC−−∗, respectively. It is obvious that LC∗ divides ℝ12 into four parts. We write them as follows:
(5)Sl12+=y∈ℝ12∣y,y>0,y2>0,Sl12−=y∈ℝ12∣y,y>0,y2<0,Tl12+=y∈ℝ12∣y,y<0,y1>0,Tl12−=y∈ℝ12∣y,y<0,y1<0,where y=y1,y2∈ℝ12.Now, we introduce Lorentz motion inℝ12.Definition 1.
Letρ1 and ρ2:I⟶ℝ12 be two regular curves. ρ1 and ρ2 are called congruent through a Lorentz motion if there exist a matrix Q and a constant a∈ℝ12 such that ρ1t=Qρ2t+a for all t∈I, where Q is given by
(6)Q=coshσ−sinhσ−sinhσcoshσorQ=−coshσsinhσsinhσ−coshσ,for some σ∈ℝ.The type vector inℝ12 is unaltered through a Lorentz motion. More specifically, if a vector x=x1,x2 is in Sl12+ (resp. Sl12−, Tl12+, Tl12−, LC++∗, LC−+∗, LC+−∗, LC−−∗), when
(7)Qx=coshσx1−sinhσx2−sinhσx1+coshσx2,it is still in Sl12+ (resp. Sl12−, Tl12+, Tl12−, LC++∗, LC−+∗, LC+−∗, LC−−∗); when
(8)Qx=−coshσx1+sinhσx2sinhσx1−coshσx2,it is in Sl12− (resp. Sl12+, Tl12−, Tl12+, LC−+∗, LC++∗, LC−−∗, LC−+∗).Letρ:I⟶ℝ12 be a regular nonlightlike curve. We take s as the arc-length parameter of ρ. For arbitrary s∈I, ∥ρ′s∥=1, the unit tangent vector of ρs is given as ts=ρ′s and the unit normal vector of ρs is given as ns=−1ω+1ρ′s⊥; then, the orientation of ts,ns is anticlockwise, where
(9)ω=1,ifsistimelike,2,ifsisspacelike.So the Frenet-Serret formula is given by(10)t′sn′s=0κsκs0tsns,where κs=ρ′′s,ρ′s⊥ is the curvature of ρs.If taking not the arc-length parameters but the general parameter t of ρ, then, the unit tangent vector of ρt can be given as tt=ρ̇t/∥ρ̇t∥ and the unit normal vector of ρt can be given as nt=−1ω+1ρ̇t⊥/∥ρ̇t∥, and the orientation of tt,nt is anticlockwise.The Frenet-Serret formula is written as(11)ṫtṅt=∥ρ̇t∥0κtκt0ttnt,and the curvature of ρt is given by κt=ρ¨t,ρ̇t⊥/∥ρ̇t∥3.A pointt on a curve ρ is an inflection point if and only if ρ¨t,ρ̇t⊥=0. And for ρt, a regular nonlightlike curve without inflection points in ℝ12, its evolute is defined to be
(12)et=ρt−1κtnt.


## 2.2. Evolutoids of the Nonlightlike Curves inℝ12

Ifρ is a regular nonlightlike curve without inflection points in ℝ12, then, the family of the normal lines to ρ has an envelope, and so does the family of the tangent lines to ρ. As we all know, the envelope of a family of lines in ℝ12 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 ρ 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 ℝ12. To better carry out the following research, we think about them by another expression approach.In order to solve above question, we define theσ-transform first.Letρ:I⟶ℝ12 be a regular nonlightlike curve. Through Lorentz motion, the σ-transform of its unit tangent vector tt is defined by
(13)ttcoshσ+ntsinhσ,where σ∈ℝ. We discover that ttcoshσ+ntsinhσ rotates in the part which tt is in and the norm of ttcoshσ+ntsinhσ is the same as the norm of tt. The line obtained by ttcoshσ+ntsinhσ at t is written as l, and the envelope of lines l is denoted by τσ. It is obvious that when σ=0, τ0 is ρ itself. When σ≠0, τσ is so-called the evolutoid of ρ.Similarly, we can consider theσ-transform of nt. It is defined by ttsinhσ+ntcoshσ. The line obtained by ttsinhσ+ntcoshσ is written as l as well, and the envelope of lines l is also denoted by τσ. When σ=0, τ0 is the evolute of ρ, and when σ≠0, τσ is also called the evolutoid of ρ.Remark 2.
As the lightconeLC∗ divides ℝ12 into four parts and tt, nt, −tt, and −nt are in different parts, we shall consider the σ-transform of tt, nt, −tt, and −nt, respectively. In fact, the line l obtained by the σ-transform of tt and the σ-transform of −tt is the same, either the σ-transform of nt or the σ-transform of −nt. Therefore, we only need to consider the σ-transform of tt and nt.Letρ:I⟶ℝ12 be a regular nonlightlike curve without inflection points, and we consider the evolutoids of ρ. We suppose that l is obtained by the σ-transform of tt.As the direction ofl is ttcoshσ+ntsinhσ, the vector pseudo-orthogonal to l can be expressed as ttsinhσ+ntcoshσ, and for any u=u1,u2∈ℝ12, a vector equation of line L is Fu,t=0 such that
(14)Fu,t=u−ρt,ttsinhσ+ntcoshσ.If we fixσ, then Fu,t=0 describes a family of lines: for arbitrary t∈I, there is a line and when t varies the line moves in ℝ12. For a fixed σ, the envelope τσ of the family of the lines given by equation (14) is the set of points u=u1,u2 in ℝ12; then, there exists t∈I such that the following equation holds
(15)Fu,t=Ftu,t=0.For any fixedσ,
(16)Ftu,t=−ρ̇t,ttsinhσ+ntcoshσ+u−ρt,∥ρ̇t∥κtntsinhσ+∥ρ̇t∥κtttcoshσ=−∥ρ̇t∥tt,ttsinhσ+ntcoshσ+∥ρ̇t∥u−ρt,κtntsinhσ+κtttcoshσ=−∥ρ̇t∥−1ωsinhσ+∥ρ̇t∥u−ρt,κtntsinhσ+κtttcoshσ=0.Then,(17)u−ρt,κtntsinhσ+κtttcoshσ=−1ωsinhσ.Letu−ρt=ξ1tt+ξ2nt, where ξ1,ξ2∈ℝ, and it can denote any vectors in ℝ12. Applying u−ρt=ξ1tt+ξ2nt in F=Ft=0, we can obtain two equations about ξ1 and ξ2.
(18)ξ1sinhσ−ξ2coshσ=0,ξ1κtcoshσ−ξ2κtsinhσ=sinhσ.By direct calculation, we can get the evolutoidEvσρt of the regular nonlightlike curve without inflection points ρt in ℝ12 through the σ-transform of tt can be expressed by
(19)Evσρt=u=ρt+coshσsinhσκttt+sinh2σκtnt.Sinceρt is a regular curve without inflection points, κt≠0. If σ=0, then, the lines F=0 are tangent to ρ and the envelope is consistent with the original curve ρ, namely, Evσρt=ρt.Ifl is obtained by the σ-transform of nt, similar to the above method, we can get the evolutoid Evσρt of ρt through the σ-transform of nt expressed as follows:
(20)Evσρt=ρt−coshσsinhσκttt−cosh2σκtnt.Base on above all, for a regular nonlightlike curveρt in ℝ12, we can obtain the definition of its evolutoid.Definition 3.
Letρ:I⟶ℝ12 be a regular non-lightlike curve without inflection points in ℝ12, then the evolutoid of ρt is as follows:
(21)Evσρt=ρt+coshσsinhσκttt+sinh2σκtnt,iftheσ‐transformisontt,ρt−coshσsinhσκttt−cosh2σκtnt,iftheσ‐transformisonnt.Remark 4.
In fact, when theσ-transform is on tt (or nt), the evolutoid of ρt is consistent with ϕT-evolutoid (or ϕN-evolutoid) in [15]. But for the further research of the evolutoids of the mixed-type curves, we adopted the definition mode of σ-transform by our own.


## 3. Evolutoids of the Mixed-Type Curves inℝ12

For a regular curve inℝ12, 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 ℝ12, 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].1,1 and 1,−1 are two lightlike vectors. Apparently, they are independent to each other. Denote 1,1 and 1,−1 by L+ and L−, respectively; then, the vector pair {L+, L−} is called a lightcone frame of ρt in ℝ12.Letρ:I⟶ℝ12 be a regular mixed-type curve. There exists a corresponding smooth map α,β:I⟶ℝ2\0, which satisfies
(22)ρ̇t=αtL++βtL−,∀t∈I.If equation (22) is established, α,β is called the lightlike tangential data of ρt. The pseudo-orthogonal complement of ρ̇t can be expressed by
(23)ρ̇t⊥=αtL+−βtL−.Since(24)ρ̇t,ρ̇t=−4αtβt,the type of ρt0 can be determined by αt0βt0.Letρ:I⟶ℝ12 be a regular mixed-type curve with the lightlike tangential date α,β, then, ρt0 is an inflection point of ρ if and only if
(25)α̇t0βt0−αt0β̇t0=0.Asρ̇t=αtL++βtL−, we use Lorentz motion on ρ̇t, and we can get
(26)MσαtL++NσβtL−or
(27)−MσαtL+−NσβtL−,where Mσ=coshσ−sinhσ and Nσ=coshσ+sinhσ. Formula (26) is called the σ-transform of αtL++βtL−, and formula (27) is called the σ-transform of −αtL++βtL−.Similarly, we can obtain that theσ-transform of αtL+−βtL− and −αtL+−βtL− is MσαtL+−NσβtL− and −MσαtL++NσβtL−, respectively.In fact, the linel obtained by MσαtL++NσβtL− and −MσαtL+−NσβtL− is the same, either MσαtL+−NσβtL− or −MσαtL++NσβtL−. Therefore, we only need to consider the σ-transform of αtL++βtL− and αtL+−βtL−. In addition, the type of MσαtL++NσβtL− is the same as αtL++βtL−, so does the type of MσαtL+−NσβtL− and αtL+−βtL−; thus, we can avoid the angle of lightlike vector and nonlightlike vector through σ-transform.The definition of the evolutoid of a regular mixed-type curve with the lightcone frameL+,L− and the lightlike tangential data α,β can be given as follows:Theorem 5.
Letρ:I⟶ℝ12 be a regular mixed-type curve without inflection points in ℝ12, then the evolutoid of ρt is as follows:
(28)Evσρt=ρt−2αtβtsinhσα̇tβt−αtβ̇tMσαtL++NσβtL−,iftheσ‐transformisonαtL++βtL−,ρt−2αtβtcoshσα̇tβt−αtβ̇tMσαtL+−NσβtL−,iftheσ‐transformisonαtL+−βtL−.Proof.
First, supposing that theσ-transform is on αtL++βtL−, then, the direction of l is
(29)MσαtL++NσβtL−.
The vector pseudo-orthogonal tol is
(30)MσαtL+−NσβtL−.
For anyu=u1,u2∈ℝ12, we have
(31)Fu,t=Ftu,t=0,where
(32)Fu,t=u−ρt,MσαtL+−NσβtL−and σ is fixed.
By direct calculation,(33)Ftu,t=−4αtβtsinhσ+u−ρt,Mσα̇tL+−Nσβ̇tL−.
Let(34)u−ρt=ζ1tL++ζ2tL−,where ζ1,ζ2:I⟶ℝ2\0 is a smooth map.
We have(35)2ζ1tNσβt−2ζ2tMσαt=0,2ζ1tNσβ̇t−2ζ2tMσα̇t=4αtβtsinhσ.
By direct calculation, we get(36)ζ1t=2α2tβtsinhσNσαtβ̇t−α̇tβt,ζ2t=2αtβ2tsinhσMσαtβ̇t−α̇tβt.
Thus,(37)u=ρt−2αtβtsinhσα̇tβt−αtβ̇tMσαtL++NσβtL−=Evσρt.
If theσ-transform is on αtL+−βtL−, similar to the above process, we can get the evolutoid Evσρt of the regular mixed-type curve ρt as follows:
(38)Evσρt=ρt−2αtβtcoshσα̇tβt−αtβ̇tMσαtL+−NσβtL−.Remark 6.
If theσ-transform is on αtL++βtL−, when σ0=0, Evσ0ρt is consistent with ρt. If the σ-transform is on αtL+−βtL−, when σ0=0, Evσ0ρt=ρt−2αtβt/α̇tβt−αtβ̇tαtL+−βtL−, it is exactly the evolute of ρt.As for a regular mixed-type curveρt in ℝ12, is its evolutoid always regular? If not, when does the evolutoid of ρt have singular points? We give the following theorem to answer this question.Theorem 7.
Letρ:I⟶ℝ12 be a regular mixed-type curve without inflection points in ℝ12, Evσρ:I⟶ℝ12 is the evolutoid of ρ.
(1)
Take the arc-length parameters, and suppose that ρs0 is a nonlightlike point; then, s0 is a singular point of Evσρs if and only if
(i)
κ2s0coshσ−κ′s0sinhσ=0, when the σ-transform is on ts;(ii)
−κ2s0sinhσ+κ′s0coshσ=0, when the σ-transform is on ns.(2)
Suppose thatρt0 is a lighlike point,(i)
When theσ-transform is on αtL++βtL−,
(a)
ifαt0=0, βt0≠0, t0 is a singular point of Evσρt if and only if σ=ln2,(b)
ifαt0≠0, βt0=0, t0 is a singular point of Evσρt if and only if σ=−ln2;(ii)
When theσ-transform is on αtL+−βtL−, Evσρt0 is always a regular point.Proof.
First, suppose thatρt0 is a non-lightlike point and the σ-transform is on tt. Using the Frenet-Serret formula we can get
(39)Ėvσρt0=∥ρ̇t0∥coshσ−sinhσκ′t0κ2t0coshtt0+sinhnt0.
Ėvσρt0=0 if and only if κ2t0∥ρ̇t0∥coshσ−κ′t0sinhσ=0. If we take the arc-length parameter s, then, we have
(40)κ2s0coshσ−κ′s0sinhσ=0.
Similarly, if theσ-transform is on nt, we have Ėvσρs0=0 if and only if −κ2s0sinhσ+κ′s0coshσ=0.
Next, suppose thatρt0 is a lightlike point and the σ-transform is on αtL++βtL−. By direct calculation, we can get
(41)Ėvσρt0=At0−α̇2t0β2t04sinhσMσ−1+α2t0β̇2t02sinhσMσ+1+2αt0α̇t0βt0β̇t0sinhσMσ−1+2αt0βt0α¨t0βt0−αt0β¨t0sinhσMσL++Bt0−α̇2t0β2t02sinhσNσ−1+α2t0β̇2t04sinhσNσ+1−2αt0α̇t0βt0β̇t0sinhσNσ+1+2αt0βt0α¨t0βt0−αt0β¨t0sinhσNσL−,where
(42)At0=αt0α̇t0βt0−αt0β̇t02,Bt0=βt0α̇t0βt0−αt0β̇t02.
Whenαt0=0, βt0≠0, Ėvσρt0=0 if and only if 2sinhσNσ−1=0, that is, σ=ln2. When αt0≠0, βt0=0, Ėvσρt0=0 if and only if 2sinhσMσ+1=0, that is, σ=−ln2.
Similarly, if theσ-transform is on αtL+−βtL−, we can get
(43)Ėvσρt0=At0−α̇2t0β2t04coshσMσ−1+α2t0β̇2t02coshσMσ+1+2αt0α̇t0βt0β̇t0coshσMσ−1+2αt0βt0α¨t0βt0−αt0β¨t0coshσMσL++Bt0α̇2t0β2t02coshσNσ+1−α2t0β̇2t04coshσNσ−1+2αt0α̇t0βt0β̇t0coshσMσ−1+2αt0βt0α¨t0βt0−αt0β¨t0coshσMσL−.
Whenαt0=0, βt0≠0, Ėvσρt0=0 if and only if 2coshσNσ+1=0, that is, e2σ=−2. We can not work out such σ, so there not exist t0 such that Evσρt0 is singular point. When αt0≠0 and βt0=0, there not exist t0 such that Evσρt0 is a singular point. Thus, Evσρt0 is always a regular pointIn [10], we know that for a point ρt0, if it is a lightlike point, then the corresponding point on the evolute Evρt0 is always a regular point. Here, we can conclude that for a lightlike point ρt0, its evolutoid Evσρt0 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 evolutoidEvσρt0 of a mixed-type curve ρt and the types of the points ρt0. By Theorem 7, the following proposition can be obtained.Proposition 8.
Letρ:I⟶ℝ12 be a regular mixed-type curve without inflection points in ℝ12, Evσρ:I⟶ℝ12 is the evolutoid of ρt.
(1)
Take the arc-length parameters, and suppose that ρs0 is a non-lightlike point; then,
(i)
when theσ-transform is on ts and κ2s0coshσ−κ′s0sinhσ≠0, then, Evσρs0 is spacelike (resp., timelike) if and only if ρs0 is spacelike (resp., timelike);(ii)
when theσ-transform is on ns and −κ2s0sinhσ+κ′s0coshσ≠0, then, Evσρs0 is timelike (resp., spacelike) if and only if ρs0 is spacelike (resp., timelike).(2)
Suppose thatρt0 is a lighlike point; then,(i)
when theσ-transform is on αtL++βtL−,(a)
ifαt0=0, βt0≠0, and σ≠ln2, then Evσρt0 is lightlike;(b)
ifαt0≠0, βt0=0, and σ≠−ln2, then Evσρt0 is lightlike;(ii)
when theσ-transform is on αtL+−βtL−, Evσρt0 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.
Letρ:−1,1⟶ℝ12, ρt=t,t2 be a regular mixed-type curve without inflections. See the blue curve in Figure 1.Figure 1
The mixed-type curve (blue) and the evolutoid (evolute) of it.We can get that the evolute ofρt is
(44)Evρt=4t2,3t3−12.See the orange dashed curve in Figure1.If theσ-transform is on αtL+−βtL− and σ=ln2, the evolutoid of ρt is
(45)Evσρt=129t3−32t2−14t+38,−3t3+132t2+34t−98.See the red dashed curve in Figure1.If theσ-transform is on αtL++βtL− and σ=−ln2, the evolutoid of ρt is
(46)Evσρt=12−t3−32t2+94t+38,−3t3+32t2+34t+18.Whent0=1/2, ρt0 is a lightlike point, and Evσρt0 is a singular point. See the green curve in Figure 1.Example 10.
Letρ:0,2π⟶ℝ12, ρt=cost,sint be a regular mixed-type curve without inflections. See the blue curve in Figure 2.Figure 2
The mixed-type curve (blue) and the evolutoid (evolute) of it.We can get that the evolute ofρt is
(47)Evρt=2cos3t,sin3t.See the orange dashed curve in Figure2.If theσ-transform is on αtL+−βtL− and σ=ln2, the evolutoid of ρt is
(48)Evσρt=cost+381−2sin2t3cost+sint,sint−381−2sin2tcost+3sint.See the red dashed curve in Figure2.If theσ-transform is on αtL++βtL− and σ=ln2, the evolutoid of ρt is
(49)Evσρt=cost−181−2sin2tcost+3sint,sint+181−2sin2t3cost+sint.Whent0=π/4 or 5π/4, ρt0 is a lightlike point, and Evσρt0 is a singular point. See the green curve in Figure 2.



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