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

Abstract

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.

1. Introduction

The study of relationships among geometrically associated curves is a classical theme in differential geometry, often revealing deep structures that transcend individual curve properties. For instance, the interplay among a curve and its associated curves such as the involute, evolute, pedal curve, and contrapedal curve has been extensively studied in Euclidean geometry. These studies have revealed connections to various fields; for example, refs. [1,2] focus on mechanical linkages and pedal coordinates, pedal curves of fronts and their singularities are investigated in [3,4], and ref. [5] examines the relationship between pedal and contrapedal curves in the Euclidean plane. Among these associated curves, the pedal curve and contrapedal curve are defined via orthogonal projections onto the tangent and normal lines, respectively, while the evolute is the locus of centers of curvature. The relationships among these three constructions encapsulate fundamental duality principles. In this paper, we focus on these three associated curves in the setting of mixed-type Minkowski plane curves.

The Minkowski plane ${\mathbb{R}}_1^2$, equipped with the indefinite inner product, serves as the two-dimensional model of special relativity. The study of geometric objects in Minkowski space and its subspaces is very extensive, and the research results are abundant (see [6,7,8,9]). In this space, non-zero vectors are classified as spacelike, timelike, or lightlike according to the sign of their indefinite inner product. For curves, this leads to significant geometric subtleties. While non-lightlike curves can be studied using Frenet–Serret frames (see [10,11,12,13]), and lightlike curves require specialized tools such as Cartan frames (see [6,14,15,16]), the situation becomes more intricate when a curve contains points of all three types simultaneously. Such curves are called mixed-type curves. Systematic study became possible after the introduction of the lightcone frame by Izumiya, Romero Fuster and Takahashi in [17], which provides a unified approach to handle lightlike points. Since then, mixed-type curves have attracted increasing attention. In particular, lightcone framed curves and mixed-type curves are investigated in Minkowski 3-space in [18,19], respectively. More recently, framed curve approaches have been developed to study singular curves in Minkowski spaces (see [16,20]).

In previous works, we studied evolutes and pedal curves of mixed-type curves. In [21,22], we investigated evolutes using a modified Frenet–Serret type frame and a region division method. In [23], we studied pedal curves (which we call T-dual curves in this paper), discussing their existence at lightlike points, singularities, and point types. However, the contrapedal curve (which we call the N-dual curve) of a mixed-type curve has not yet been systematically studied. This gap is significant because without a complete understanding of the N-dual curve, the full picture of the relationships among the T-dual, N-dual, and evolute remains incomplete. In the Euclidean setting, the pedal and contrapedal curves are known to be dual to each other, and their relationship with the evolute is well documented (see [5]). In the Minkowski plane, the situation is more delicate due to lightlike points, where the normal line is not well-defined. To reveal the analogous dual relationships in the mixed-type setting, we first establish a rigorous theory for the N-dual curve. This paper aims to fill this gap by providing a systematic treatment of the N-dual curve, and based on this, exploring the geometric connections among the three associated curves. Specifically, we clarify the conditions under which the N-dual curve exists at lightlike points, analyze its singularities and point types, and establish the relations between the T-dual and N-dual curves as well as between the N-dual curve and the evolute. These results complete the theoretical framework for associated curves of mixed-type Minkowski plane curves and reveal a coherent duality structure that parallels the Euclidean case while exhibiting distinctive features arising from lightlike geometry.

To emphasize the dual relationship between the two projection constructions, one using the tangent direction and the other using the normal direction, we adopt the terminology T-dual curve and N-dual curve throughout this paper. Here, T and N stand for tangent and normal, respectively, and the term “dual” reflects the fact that these two curves are related by the pseudo-orthogonal complement. For clarity, we note that the T-dual curve corresponds to the classical pedal curve, and the N-dual curve corresponds to the classical contrapedal curve.

The structure of this paper is as follows. In Section 2, we review basic concepts of the Minkowski plane, the lightcone frame, and the definitions of T-dual curves and evolutes. In Section 3, we define the N-dual curve, discuss its existence at lightlike points, and analyze its singularities and point types. In Section 4, we investigate the relationships among the T-dual curve, N-dual curve and evolute. Finally, in Section 5, we conclude with a summary and discuss future directions.

All maps and submanifolds considered in this paper are assumed to be smooth unless otherwise stated.

2. Preliminaries

We now review some basic concepts about the Minkowski plane. For further details, please see [12,13,17].

Definition 1. 

Let ${\mathbb{R}}^2=\left\{(u_1,u_2)\right|u_i\in \mathbb{R},i=1,2\}$ be the vector space of dimension 2. If ${\mathbb{R}}^2$ is endowed with the metric induced by the indefinite inner product

$$\langle \mathit{u},\mathit{v}\rangle =-u_1{v}_1+u_2{v}_2,$$

where $\mathit{u}=(u_1,u_2)$, $\mathit{v}=(v_1,v_2)$, and $\mathit{u},\mathit{v}\in {\mathbb{R}}^2$. Then we call $({\mathbb{R}}^2,\langle ,\rangle )$ the Minkowski plane and denote it by ${\mathbb{R}}_1^2$.

Definition 2. 

For a non-zero vector $\mathit{u}\in {\mathbb{R}}_1^2$, it is called spacelike, timelike or lightlike, if $\langle \mathit{u},\mathit{u}\rangle >0$, $\langle \mathit{u},\mathit{u}\rangle <0$ or $\langle \mathit{u},\mathit{u}\rangle =0$, respectively. A spacelike or timelike vector is called a non-lightlike vector.

Definition 3. 

For a vector $\mathit{u}\in {\mathbb{R}}_1^2$, if there exists a vector $\mathit{v}\in {\mathbb{R}}_1^2$, such that $\langle \mathit{u},\mathit{v}\rangle =0$, we say $\mathit{v}$ is pseudo-orthogonal to $\mathit{u}$.

Definition 4. 

The Lorentzian norm of a vector $\mathit{u}=(u_1,u_2)\in {\mathbb{R}}_1^2$ is defined by

$$\parallel \mathit{u}\parallel =\sqrt{|\langle \mathit{u},\mathit{u}\rangle |}.$$

The pseudo-orthogonal vector of $\mathit{u}$ is given by ${\mathit{u}}^{\perp }=(u_2,u_1)$. By definition, $\mathit{u}$ and ${\mathit{u}}^{\perp }$ are pseudo-orthogonal to each other, and

$$\parallel \mathit{u}\parallel = \parallel {\mathit{u}}^{\perp }\parallel .$$

It is obvious that ${\mathit{u}}^{\perp }=\pm \mathit{u}$ if and only if $\mathit{u}$ is lightlike, and ${\mathit{u}}^{\perp }$ is timelike (resp. spacelike) if and only if $\mathit{u}$ is spacelike (resp. timelike).

Denote ${\mathbb{L}}^{+}=(1,1)$ and ${\mathbb{L}}^{-}=(1,-1)$. Then ${\mathbb{L}}^{+}$ and ${\mathbb{L}}^{-}$ are independent lightlike vectors, and $\langle {\mathbb{L}}^{+},{\mathbb{L}}^{-}\rangle =-2$. We call $\{{\mathbb{L}}^{+},{\mathbb{L}}^{-}\}$ a lightcone frame in ${\mathbb{R}}_1^2$. It is given by S. Izumiya, M. C. Romero Fuster and M. Takahashi in [17].

Let $\gamma :I\to {\mathbb{R}}_1^2$ be a regular mixed-type curve. There exists a smooth map $(\alpha ,\beta ):I\to {\mathbb{R}}^2\setminus \left\{\mathbf{0}\right\}$ such that

$$\begin{array}{c}\hfill \dot{\gamma }\left(t\right)=\alpha \left(t\right){\mathbb{L}}^{+}+\beta \left(t\right){\mathbb{L}}^{-},\end{array}$$

for all $t\in I$. We say that a regular curve $\gamma$ has the lightlike tangential data $(\alpha ,\beta )$ if the above equation holds. Then we have

$$\dot{\gamma }{\left(t\right)}^{\perp }=\alpha \left(t\right){\mathbb{L}}^{+}-\beta \left(t\right){\mathbb{L}}^{-}.$$

Since

$$\langle \dot{\gamma }\left(t\right),\dot{\gamma }\left(t\right)\rangle =-4\alpha \left(t\right)\beta \left(t\right),$$

$\gamma \left(t_0\right)$ is a spacelike (resp. lightlike or timelike) point if and only if $\alpha \left(t_0\right)\beta \left(t_0\right)<0$ (resp. $=0$ or $>0$).

Definition 5. 

Let $\gamma :I\to {\mathbb{R}}_1^2$ be a regular mixed-type curve. We call a point $\gamma \left(t_0\right)$ an inflection if $\langle \ddot{\gamma }\left(t_0\right),\dot{\gamma }{\left(t_0\right)}^{\perp }\rangle =0$.

On the basis of the above definition, if ${\langle \ddot{\gamma }\left(t_0\right),\dot{\gamma }{\left(t_0\right)}^{\perp }\rangle }^{\prime }\ne 0$, we call $\gamma \left(t_0\right)$ an ordinary inflection.

Proposition 1. 

If we choose the lightcone frame $\{{\mathbb{L}}^{+},{\mathbb{L}}^{-}\}$ and the lightlike tangential data $(\alpha ,\beta )$, then we can obtain that $\gamma \left(t_0\right)$ is an inflection point if and only if

$$\dot{\alpha }\left(t_0\right)\beta \left(t_0\right)-\alpha \left(t_0\right)\dot{\beta }\left(t_0\right)=0,$$

and $\gamma \left(t_0\right)$ being an ordinary inflection point means not only

$$\dot{\alpha }\left(t_0\right)\beta \left(t_0\right)-\alpha \left(t_0\right)\dot{\beta }\left(t_0\right)=0,$$

but also

$$\ddot{\alpha }\left(t_0\right)\beta \left(t_0\right)-\alpha \left(t_0\right)\ddot{\beta }\left(t_0\right)\ne 0.$$

Definition 6. 

Let $\gamma :I\to {\mathbb{R}}_1^2$ be a regular mixed-type curve without inflections. In [17], we have known that the evolute $Ev:I\to {\mathbb{R}}_1^2$ of γ with the lightlike data $(\alpha ,\beta )$ is defined as

$$Ev\left(\gamma \right)\left(t\right)=\gamma \left(t\right)-{\displaystyle \frac{2\alpha \left(t\right)\beta \left(t\right)}{\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right)}}(\alpha \left(t\right){\mathbb{L}}^{+}-\beta \left(t\right){\mathbb{L}}^{-}).$$

3. N-Dual Curves of the Mixed-Type Curves

For a regular curve in the Minkowski plane, its T-dual curve (pedal curve) is defined as the pseudo-orthogonal projection of a fixed point onto the tangent lines of the base curve, and its N-dual curve (contrapedal curve) is defined as the pseudo-orthogonal projection of a fixed point onto the normal lines of the base curve. We have investigated the T-dual curves of the mixed-type curves in [23]. Herein, we consider the N-dual curves of the mixed-type curves.

Similar to the definition of the T-dual curve of the regular mixed-type curve, we can define the N-dual curve of a regular mixed-type curve as follows.

Definition 7. 

Let $\gamma :I\to {\mathbb{R}}_1^2$ be a regular mixed-type curve and $\mathit{Q}$ be a point in ${\mathbb{R}}_1^2$. If $\mathit{Q}$ coincides with the lightlike point or $\mathit{Q}$ is on the tangent line of the lightlike point, then the N-dual curve $\mathcal{ND}\left(\gamma \right)\left(t\right)$ of the base curve $\gamma \left(t\right)$ is given by

$$\begin{array}{c}\hfill \mathcal{ND}\left(\gamma \right)\left(t\right)=\gamma \left(t\right)+{\displaystyle \frac{\langle \mathit{Q}-\gamma \left(t\right),\alpha \left(t\right){\mathbb{L}}^{+}-\beta \left(t\right){\mathbb{L}}^{-}\rangle }{4\alpha \left(t\right)\beta \left(t\right)}}(\alpha \left(t\right){\mathbb{L}}^{+}-\beta \left(t\right){\mathbb{L}}^{-}).\end{array}$$

(1)

If $\gamma \left(t_0\right)$ is a non-lightlike point, then $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ satisfies the above form obviously.

If $\gamma \left(t_0\right)$ is a lightlike point, then $\alpha \left(t_0\right)\beta \left(t_0\right)=0$, and we suppose that $\mathit{Q}$ coincides with the lightlike point or $\mathit{Q}$ is on the tangent line of the lightlike point. In this case we define $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ as ${\displaystyle \underset{t\to t_0}{lim}\mathcal{ND}\left(\gamma \right)\left(t\right)}$, and the specific forms of the N-dual curve at $\gamma \left(t_0\right)$ are as follows.

CASE I. Suppose that $\alpha \left(t_0\right)\ne 0$ and $\beta \left(t_0\right)=0$; the condition that $\mathit{Q}$ coincides with $\gamma \left(t_0\right)$ or lies on the tangent line at $\gamma \left(t_0\right)$ is equivalent to $\langle \mathit{Q}-\gamma \left(t\right),{\mathbb{L}}^{+}\rangle =0$.

If $\mathit{Q}$ coincides with $\gamma \left(t_0\right)$, then

$$\mathcal{ND}\left(\gamma \right)\left(t_0\right)=\gamma \left(t_0\right).$$

If $\mathit{Q}$ is on the tangent line of the lightlike point, then

$$\mathcal{ND}\left(\gamma \right)\left(t_0\right)=\gamma \left(t_0\right)-{\displaystyle \frac{1}{4}}\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{-}\rangle {\mathbb{L}}^{+}.$$

CASE II. Suppose that $\alpha \left(t_0\right)=0$ and $\beta \left(t_0\right)\ne 0$; the condition that $\mathit{Q}$ coincides with $\gamma \left(t_0\right)$ or lies on the tangent line at $\gamma \left(t_0\right)$ is equivalent to $\langle \mathit{Q}-\gamma \left(t\right),{\mathbb{L}}^{-}\rangle =0$.

If $\mathit{Q}$ coincides with $\gamma \left(t_0\right)$, then

$$\mathcal{ND}\left(\gamma \right)\left(t_0\right)=\gamma \left(t_0\right).$$

If $\mathit{Q}$ is on the tangent line of the lightlike point, then

$$\mathcal{ND}\left(\gamma \right)\left(t_0\right)=\gamma \left(t_0\right)-{\displaystyle \frac{1}{4}}\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{+}\rangle {\mathbb{L}}^{-}.$$

Remark 1. 

Let $\gamma :I\to {\mathbb{R}}_1^2$ be a regular mixed-type curve, let $\mathit{Q}\in {\mathbb{R}}_1^2$ be a fixed point, and let $\mathcal{ND}\left(\gamma \right):I\to {\mathbb{R}}_1^2$ be its the N-dual. Suppose $\gamma \left(t_0\right)$ is a lightlike point. Then either $\alpha \left(t_0\right)=0$ or $\beta \left(t_0\right)=0$. In addition, $\mathit{Q}$ is neither consistent with $\gamma \left(t_0\right)$ nor lies on the tangent line of $\gamma \left(t_0\right)$, then

$${\displaystyle \underset{t\to t_0}{lim}{\displaystyle \frac{\langle \mathit{Q}-\gamma \left(t\right),\alpha \left(t\right){\mathbb{L}}^{+}+\beta \left(t\right){\mathbb{L}}^{-}\rangle }{4\alpha \left(t\right)\beta \left(t\right)}}=\infty .}$$

Therefore, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is asymptotic to the lightlike line of ${\mathbb{L}}^{+}$ or ${\mathbb{L}}^{-}$. More precisely:

If $\langle \mathit{Q}-\gamma \left(t\right),{\mathbb{L}}^{+}\rangle \ne 0$, then $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is asymptotic with the lightlike line along the positive or negative direction of ${\mathbb{L}}^{+}.$

If $\langle \mathit{Q}-\gamma \left(t\right),{\mathbb{L}}^{-}\rangle \ne 0$, then $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is asymptotic with the lightlike line along the positive or negative direction of ${\mathbb{L}}^{-}.$

Next, we consider that the N-dual curve of a regular mixed-type curve has singular points and we have the following theorem.

Theorem 1. 

Let $\gamma :I\to {\mathbb{R}}_1^2$ be a regular mixed-type curve, $\mathit{Q}$ be a point in ${\mathbb{R}}_1^2$ and $\mathcal{ND}\left(\gamma \right):I\to {\mathbb{R}}_1^2$ be the N-dual curve of γ. Then,

(1) 

If $\gamma \left(t_0\right)$ is a non-lightlike point, then $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a singular point if and only if

$$\alpha \left(t_0\right)+{\displaystyle \frac{\dot{\alpha }\left(t_0\right)\beta \left(t_0\right)-\alpha \left(t_0\right)\dot{\beta }\left(t_0\right)}{4{\beta }^2\left(t_0\right)}}\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{+}\rangle =0$$

and

$$\beta \left(t_0\right)-{\displaystyle \frac{\dot{\alpha }\left(t_0\right)\beta \left(t_0\right)-\alpha \left(t_0\right)\dot{\beta }\left(t_0\right)}{4{\alpha }^2\left(t_0\right)}}\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{-}\rangle =0;$$

(2) 

If $\gamma \left(t_0\right)$ is a lightlike point, and $\mathit{Q}$ coincides with $\gamma \left(t_0\right)$ or $\mathit{Q}$ is on the tangent line of $\gamma \left(t_0\right)$, then $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is regular.

Proof. 

Since the N-dual curve of the mixed-type curve $\gamma \left(t\right)$ is given by the Formula (1), by calculations, we can obtain

$$\begin{array}{cc}\hfill \dot{\mathcal{ND}}\left(\gamma \right)\left(t\right)=& (\alpha \left(t\right)+{\displaystyle \frac{\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right)}{4{\beta }^2\left(t\right)}}\langle \mathit{Q}-\gamma \left(t\right),{\mathbb{L}}^{+}\rangle ){\mathbb{L}}^{+}\hfill \\ \hfill +& (\beta \left(t\right)-{\displaystyle \frac{\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right)}{4{\alpha }^2\left(t\right)}}\langle \mathit{Q}-\gamma \left(t\right),{\mathbb{L}}^{-}\rangle ){\mathbb{L}}^{-}.\hfill \end{array}$$

(2)

Let $\gamma \left(t_0\right)$ be a non-lightlike point. $\dot{\mathcal{ND}}\left(\gamma \right)\left(t_0\right)=0$ if and only if

$$\alpha \left(t_0\right)+{\displaystyle \frac{\dot{\alpha }\left(t_0\right)\beta \left(t_0\right)-\alpha \left(t_0\right)\dot{\beta }\left(t_0\right)}{4{\beta }^2\left(t_0\right)}}\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{+}\rangle =0$$

and

$$\beta \left(t_0\right)-{\displaystyle \frac{\dot{\alpha }\left(t_0\right)\beta \left(t_0\right)-\alpha \left(t_0\right)\dot{\beta }\left(t_0\right)}{4{\alpha }^2\left(t_0\right)}}\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{-}\rangle =0.$$

Let $\gamma \left(t_0\right)$ be a lightlike point with $\alpha \left(t_0\right)\ne 0,\beta \left(t_0\right)=0$ and $\langle Q-\gamma \left(t_0\right),{\mathbb{L}}^{+}\rangle =0$; a direct inspection of Formula (2) shows that $\dot{\mathcal{ND}}\left(\gamma \right)\left(t_0\right)\ne 0$. Hence $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is regular. The case $\alpha \left(t_0\right)=0,\beta \left(t_0\right)\ne 0$ is analogous. □

To analyze the point types of the N-dual curve, we introduce the following function $\mathrm{\Omega }\left(t\right)$. Its definition is chosen so that the Lorentzian squared norm of $\mathcal{ND}\left(\gamma \right)\left(t\right)$ simplifies to $-4\alpha \left(t\right)\beta \left(t\right)\mathrm{\Omega }\left(t\right)$, allowing the causal character of $\mathcal{ND}\left(\gamma \right)\left(t\right)$ to be read off directly from the signs of $\alpha \left(t\right)$, $\beta \left(t\right)$, and $\mathrm{\Omega }\left(t\right)$.

Specifically, $\mathrm{\Omega }\left(t\right)$ is defined as

$$\begin{array}{cc}\hfill \mathrm{\Omega }\left(t\right)=& 1+{\displaystyle \frac{\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right)}{4\alpha \left(t\right){\beta }^2\left(t\right)}}\langle \mathit{Q}-\gamma \left(t\right),{\mathbb{L}}^{+}\rangle -{\displaystyle \frac{\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right)}{4{\alpha }^2\left(t\right)\beta \left(t\right)}}\langle \mathit{Q}-\gamma \left(t\right),{\mathbb{L}}^{-}\rangle \hfill \\ \hfill & -{\displaystyle \frac{{(\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right))}^2}{16{\alpha }^3\left(t\right){\beta }^3\left(t\right)}}\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{+}\rangle \langle \mathit{Q}-\gamma \left(t\right),{\mathbb{L}}^{-}\rangle ,\hfill \end{array}$$

then the following proposition shows the type of points of the N-dual curve of a mixed-type curve.

Proposition 2. 

Let $\gamma :I\to {\mathbb{R}}_1^2$ be a regular mixed-type curve, $\mathit{Q}$ be a point in ${\mathbb{R}}_1^2$ and $\mathcal{ND}\left(\gamma \right):I\to {\mathbb{R}}_1^2$ be the N-dual curve of γ. For a regular point $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$, we have the following consequences.

(1) 

When $\gamma \left(t_0\right)$ is a non-lightlike point,

(i) 

if $\mathrm{\Omega }\left(t_0\right)>0$, then $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a spacelike (or timelike) point if and only if $\gamma \left(t_0\right)$ is a spacelike (or timelike) point;

(ii) 

if $\mathrm{\Omega }\left(t_0\right)<0$, then $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a spacelike (or timelike) point if and only if $\gamma \left(t_0\right)$ is a timelike (or spacelike) point;

(iii) 

if $\mathrm{\Omega }\left(t_0\right)=0$, then $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a lightlike point.

(2) 

When $\gamma \left(t_0\right)$ is a lightlike point, $\alpha \left(t_0\right)\ne 0$, $\beta \left(t_0\right)=0$ and $\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{+}\rangle =0$,

(i) 

suppose that $\gamma \left(t_0\right)$ is not the inflection of γ,

(a) 

$\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a lightlike point if and only if $\mathit{Q}$ coincides with $\gamma \left(t_0\right)$;

(b) 

$\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a non-lightlike point if and only if $\mathit{Q}$ is on the tangent line of $\gamma \left(t_0\right)$. Moreover, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a spacelike (or, timelike) point if and only if $\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{-}\rangle \dot{\beta }\left(t_0\right)<0$ (or, $\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{-}\rangle \dot{\beta }\left(t_0\right)>0$);

(ii) 

suppose that $\gamma \left(t_0\right)$ is an ordinary inflection of γ, then $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is always lightlike.

(3) 

When $\gamma \left(t_0\right)$ is a lightlike point, $\alpha \left(t_0\right)=0$, $\beta \left(t_0\right)\ne 0$ and $\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{-}\rangle =0$,

(i) 

suppose that $\gamma \left(t_0\right)$ is not the inflection of γ,

(a) 

$\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a lightlike point if and only if $\mathit{Q}$ coincides with $\gamma \left(t_0\right)$;

(b) 

$\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a non-lightlike point if and only if $\mathit{Q}$ is on the tangent line of $\gamma \left(t_0\right)$. Moreover, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a spacelike (or, timelike) point if and only if $\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{+}\rangle \dot{\alpha }\left(t_0\right)<0$ (or, $\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{+}\rangle \dot{\alpha }\left(t_0\right)>0$);

(ii) 

suppose that $\gamma \left(t_0\right)$ is an ordinary inflection of γ, then $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is always lightlike.

Proof. 

As $\dot{\mathcal{ND}}\left(\gamma \right)\left(t\right)$ is given by Formula (2), by calculations we can get

$$\begin{array}{cc}\hfill & \langle \dot{\mathcal{ND}}\left(\gamma \right)\left(t\right),\dot{\mathcal{ND}}\left(\gamma \right)\left(t\right)\rangle \hfill \\ \hfill =& -4\alpha \left(t\right)\beta \left(t\right)(1+{\displaystyle \frac{\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right)}{4\alpha \left(t\right){\beta }^2\left(t\right)}}\langle \mathit{Q}-\gamma \left(t\right),{\mathbb{L}}^{+}\rangle \hfill \\ \hfill & -{\displaystyle \frac{\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right)}{4{\alpha }^2\left(t\right)\beta \left(t\right)}}\langle \mathit{Q}-\gamma \left(t\right),{\mathbb{L}}^{-}\rangle \hfill \\ \hfill & -{\displaystyle \frac{{(\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right))}^2}{16{\alpha }^3\left(t\right){\beta }^3\left(t\right)}}\langle \mathit{Q}-\gamma \left(t\right),{\mathbb{L}}^{+}\rangle \langle \mathit{Q}-\gamma \left(t\right),{\mathbb{L}}^{-}\rangle )\hfill \\ \hfill =& -4\alpha \left(t\right)\beta \left(t\right)\mathrm{\Omega }\left(t\right).\hfill \end{array}$$

Then we can obtain the type of $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ easily. □

In the following, we give three examples to present the features of the N-dual curve of the regular mixed-type curve, especially at the lightlike point of the base curve.

Example 1. 

Let $\gamma :[0,2\pi )\to {\mathbb{R}}_1^2$ be a regular mixed-type curve, such that

$$\gamma \left(t\right)=\left(2cost,{\displaystyle \frac{2\sqrt{3}}{3}}sint\right).$$

When $t_0={\displaystyle \frac{5}{6}}\pi$, $\gamma \left(t_0\right)$ is a lightlike point. See the blue curve in Figure 1.

If $\mathit{Q}=(0,0)$, then $\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{+}\rangle \ne 0$, the N-dual curve of $\gamma \left(t\right)$ is

$$\mathcal{ND}\left(\gamma \right)\left(t\right)=\left(-{\displaystyle \frac{8{sin}^{2}tcost}{{cos}^{2}t-3{sin}^{2}t}},{\displaystyle \frac{8\sqrt{3}sint{cos}^{2}t}{3({cos}^{2}t-3{sin}^{2}t)}}\right).$$

In this case, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is asymptotic with the lightlike line along the positive and negative direction of ${\mathbb{L}}^{+}$. See the green curve in Figure 1.

If $\mathit{Q}=\left(-\sqrt{3},{\displaystyle \frac{\sqrt{3}}{3}}\right)$, then $\mathit{Q}$ coincides with $\gamma \left(t_0\right)$, the N-dual curve of $\gamma \left(t\right)$ is

$$\begin{array}{cc}\hfill & \mathcal{ND}\left(\gamma \right)\left(t\right)\hfill \\ \hfill =& \left({\displaystyle \frac{-\sqrt{3}{cos}^{2}t+sintcost-8{sin}^{2}tcost}{{cos}^{2}t-3{sin}^{2}t}},{\displaystyle \frac{-3\sqrt{3}{sin}^{2}t+9sintcost+8\sqrt{3}sint{cos}^{2}t}{3({cos}^{2}t-3{sin}^{2}t)}}\right).\hfill \end{array}$$

In this case, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a lightlike point. See the orange dashed curve in Figure 1.

If $\mathit{Q}=\left(0,{\displaystyle \frac{4\sqrt{3}}{3}}\right)$, then $\mathit{Q}$ is on the tangent line of $\gamma \left(t_0\right)$, the N-dual curve of $\gamma \left(t\right)$ is

$$\mathcal{ND}\left(\gamma \right)\left(t\right)=\left({\displaystyle \frac{-8{sin}^{2}tcost+4sintcost}{{cos}^{2}t-3{sin}^{2}t}},{\displaystyle \frac{-12\sqrt{3}{sin}^{2}t+8\sqrt{3}sint{cos}^{2}t}{3({cos}^{2}t-3{sin}^{2}t)}}\right).$$

In this case, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a spacelike point. See the red dashed curve in Figure 1.

Example 2. 

Let $\gamma :(-1,1)\to {\mathbb{R}}_1^2$ be a regular mixed-type curve, such that

$$\gamma \left(t\right)=(t,t^2).$$

When $t_0={\displaystyle \frac{1}{2}}$, $\gamma \left(t_0\right)$ is a lightlike point. See the blue curve in Figure 2. If $\mathit{Q}=(0,0)$, then $\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{+}\rangle \ne 0$, the N-dual curve of $\gamma \left(t\right)$ is

$$\mathcal{ND}\left(\gamma \right)\left(t\right)=\left({\displaystyle \frac{2t^3-t}{4t^2-1}},{\displaystyle \frac{4t^4-2t^2}{4t^2-1}}\right).$$

In this case, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is asymptotic with the lightlike line along the positive and negative direction of ${\mathbb{L}}^{+}$. See the green curve in Figure 2.

If $\mathit{Q}=\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{4}}\right)$, then $\mathit{Q}$ coincides with $\gamma \left(t_0\right)$, and the N-dual curve of $\gamma \left(t\right)$ is

$$\mathcal{ND}\left(\gamma \right)\left(t\right)=\left({\displaystyle \frac{2t^2+3t}{4t+2}},{\displaystyle \frac{8t^3+4t^2-2t+1}{8t+4}}\right).$$

In this case, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a lightlike point. See the orange dashed curve in Figure 2.

If $\mathit{Q}=\left(1,{\displaystyle \frac{3}{4}}\right)$, then $\mathit{Q}$ is on the tangent line of $\gamma \left(t_0\right)$, and the N-dual curve of $\gamma \left(t\right)$ is

$$\mathcal{ND}\left(\gamma \right)\left(t\right)=\left({\displaystyle \frac{2t^2+5t}{4t+2}},{\displaystyle \frac{8t^3+4t^2-2t+3}{8t+4}}\right).$$

In this case, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a timelike point. See the red dashed curve in Figure 2.

Example 3. 

Let $\gamma :(-1,1)\to {\mathbb{R}}_1^2$ be a regular mixed-type curve, such that

$$\gamma \left(t\right)=(t,t^3+t).$$

When $t_0=0$, $\gamma \left(t_0\right)$ is a lightlike point and it is an ordinary inflection. See the blue curve in Figure 3.

If $\mathit{Q}=(1,2)$, then $\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{+}\rangle \ne 0$, and the N-dual curve of $\gamma \left(t\right)$ is

$$\mathcal{ND}\left(\gamma \right)\left(t\right)=\left({\displaystyle \frac{3t^5+9t^4+4t^3+1}{9t^4+6t^2}},{\displaystyle \frac{9t^7+15t^5+4t^3+3t^2-1}{9t^4+6t^2}}\right).$$

In this case, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is asymptotic with the lightlike line along the positive and negative direction of ${\mathbb{L}}^{+}$. See the green curve in Figure 3.

If $\mathit{Q}=(0,0)$, then $\mathit{Q}$ coincides with $\gamma \left(t_0\right)$, and the N-dual curve of $\gamma \left(t\right)$ is

$$\mathcal{ND}\left(\gamma \right)\left(t\right)=\left({\displaystyle \frac{3t^3+4t}{9t^2+6}},{\displaystyle \frac{9t^5+15t^3-4t}{9t^2+6}}\right).$$

In this case, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a lightlike point. See the orange dashed curve in Figure 3.

If $\mathit{Q}=(1,1)$, then $\mathit{Q}$ is on the tangent line of $\gamma \left(t_0\right)$, and the N-dual curve of $\gamma \left(t\right)$ is

$$\mathcal{ND}\left(\gamma \right)\left(t\right)=\left({\displaystyle \frac{3t^3+9t^2+4t+3}{9t^2+6}},{\displaystyle \frac{9t^5+15t^3+4t+3}{9t^2+6}}\right).$$

In this case, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ is a lightlike point. See the red dashed curve in Figure 3.

4. Relationships Among T-Dual Curves, N-Dual Curves and Evolutes of Mixed-Type Curves

As the T-dual curves, N-dual curves and evolutes are all significant curves closely related to the base curve, we would like to investigate the relationships among them in this section. Firstly we study the relationship between the T-dual curves and N-dual curves of a mixed-type curve.

We consider $\mathcal{TD}\left(\gamma \right)\left(t_0\right)$ and $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ when $\gamma \left(t_0\right)$ is a lightlike point, and we have the following proposition.

Proposition 3. 

Let $\gamma :I\to {\mathbb{R}}_1^2$ be a regular mixed-type curve, $\mathit{Q}$ be a point in ${\mathbb{R}}_1^2$, $\mathcal{TD}\left(\gamma \right):I\to {\mathbb{R}}_1^2$ be the T-dual curve of γ and $\mathcal{ND}\left(\gamma \right):I\to {\mathbb{R}}_1^2$ be the N-dual curve of γ. Suppose that $\gamma \left(t_0\right)$ is a lightlike point. If $\mathit{Q}$ is on the tangent line of $\gamma \left(t_0\right)$, then

$$\mathcal{TD}\left(\gamma \right)\left(t_0\right)=\mathcal{ND}\left(\gamma \right)\left(t_0\right).$$

Proof. 

Suppose that $\gamma \left(t_0\right)$ is a lightlike point. When $\alpha \left(t_0\right)\ne 0$ and $\beta \left(t_0\right)=0$, according to the definitions of the T-dual curves and N-dual curves of a regular mixed-type curve, if $\mathit{Q}$ is on the tangent line of $\gamma \left(t_0\right)$, then

$$\mathcal{TD}\left(\gamma \right)\left(t_0\right)=\mathcal{ND}\left(\gamma \right)\left(t_0\right)=\gamma \left(t_0\right)-{\displaystyle \frac{1}{4}}\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{-}\rangle {\mathbb{L}}^{+}.$$

Similarly, when $\alpha \left(t_0\right)=0$ and $\beta \left(t_0\right)\ne 0$, if $\mathit{Q}$ is on the tangent line of $\gamma \left(t_0\right)$, then

$$\mathcal{TD}\left(\gamma \right)\left(t_0\right)=\mathcal{ND}\left(\gamma \right)\left(t_0\right)=\gamma \left(t_0\right)-{\displaystyle \frac{1}{4}}\langle \mathit{Q}-\gamma \left(t_0\right),{\mathbb{L}}^{+}\rangle {\mathbb{L}}^{-}.$$

Therefore, if $\gamma \left(t_0\right)$ is a lightlike point and $\mathit{Q}$ is on the tangent line of $\gamma \left(t_0\right)$, then $\mathcal{TD}\left(\gamma \right)\left(t_0\right)=\mathcal{ND}\left(\gamma \right)\left(t_0\right)$. □

The following example can show this property well.

Example 4. 

Let $\gamma :(-1,1)\to {\mathbb{R}}_1^2$ be a regular mixed-type curve, where

$$\gamma \left(t\right)=(t,t^3+t).$$

When $t_0=0$, $\gamma \left(t_0\right)$ is a lightlike point. See the blue curve in Figure 4.

Let $\mathit{Q}=(1,1)$, then $\mathit{Q}$ is on the tangent line of $\gamma \left(t_0\right)$. In this case, the T-dual curve of $\gamma \left(t\right)$ is

$$\mathcal{TD}\left(\gamma \right)\left(t\right)=\left({\displaystyle \frac{6t^3+2t+3}{9t^2+6}},{\displaystyle \frac{9t^2+2t+3}{9t^2+6}}\right).$$

See the red dashed curve in Figure 4. The N-dual curve of $\gamma \left(t\right)$ is

$$\mathcal{ND}\left(\gamma \right)\left(t\right)=\left({\displaystyle \frac{3t^3+9t^2+4t+3}{9t^2+6}},{\displaystyle \frac{9t^5+15t^3+4t+3}{9t^2+6}}\right).$$

See the orange dashed curve in Figure 4.

By calculations we can obtain that

$$\mathcal{TD}\left(\gamma \right)\left(t_0\right)=\mathcal{ND}\left(\gamma \right)\left(t_0\right)=\left({\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{2}}\right).$$

See the point $\mathit{P}$ in Figure 4.

Proposition 4. 

Let $\gamma :I\to {\mathbb{R}}_1^2$ be a regular mixed-type curve, $\mathit{Q}$ be a point in ${\mathbb{R}}_1^2$, $\mathcal{TD}\left(\gamma \right):I\to {\mathbb{R}}_1^2$ be the T-dual curve of γ and $\mathcal{ND}\left(\gamma \right):I\to {\mathbb{R}}_1^2$ be the N-dual curve of γ. Suppose that $\gamma \left(t_0\right)$ is a non-lightlike point.

(1) 

If $\mathit{Q}$ is on the tangent line of $\gamma \left(t_0\right)$, then $\mathcal{TD}\left(\gamma \right)\left(t_0\right)$ coincides with $\mathit{Q}$.

(2) 

If $\mathit{Q}$ is on the normal line of $\gamma \left(t_0\right)$, then $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ coincides with $\mathit{Q}$.

Proof. 

Firstly, suppose that $\mathit{Q}$ is on the tangent line of a non-lightlike point $\gamma \left(t_0\right)$, then we have that $\mathit{Q}-\gamma \left(t_0\right)$ and $\alpha \left(t_0\right){\mathbb{L}}^{+}+\beta \left(t_0\right){\mathbb{L}}^{-}$ are linearly dependent. Therefore, there exists $\lambda \in \mathbb{R}$, such that

$$\mathit{Q}-\gamma \left(t_0\right)=\lambda (\alpha \left(t_0\right){\mathbb{L}}^{+}+\beta \left(t_0\right){\mathbb{L}}^{-}).$$

Then, we have

$$\begin{array}{cc}\hfill & \mathcal{TD}\left(\gamma \right)\left(t_0\right)\hfill \\ \hfill =& \gamma \left(t_0\right)-\lambda {\displaystyle \frac{\langle \alpha \left(t_0\right){\mathbb{L}}^{+}+\beta \left(t_0\right){\mathbb{L}}^{-},\alpha \left(t_0\right){\mathbb{L}}^{+}+\beta \left(t_0\right){\mathbb{L}}^{-}\rangle }{4\alpha \left(t_0\right)\beta \left(t_0\right)}}(\alpha \left(t_0\right){\mathbb{L}}^{+}+\beta \left(t_0\right){\mathbb{L}}^{-})\hfill \\ \hfill =& \gamma \left(t_0\right)-\lambda {\displaystyle \frac{-4\alpha \left(t_0\right)\beta \left(t_0\right)}{4\alpha \left(t_0\right)\beta \left(t_0\right)}}(\alpha \left(t_0\right){\mathbb{L}}^{+}+\beta \left(t_0\right){\mathbb{L}}^{-})\hfill \\ \hfill =& \gamma \left(t_0\right)+\lambda (\alpha \left(t_0\right){\mathbb{L}}^{+}+\beta \left(t_0\right){\mathbb{L}}^{-})\hfill \\ \hfill =& \mathit{Q}.\hfill \end{array}$$

Thus, $\mathcal{TD}\left(\gamma \right)\left(t_0\right)$ coincides with $\mathit{Q}$.

Then, we consider that $\mathit{Q}$ is on the normal line of $\gamma \left(t_0\right)$. In this case, $\mathit{Q}-\gamma \left(t_0\right)$ and $\alpha \left(t_0\right){\mathbb{L}}^{+}-\beta \left(t_0\right){\mathbb{L}}^{-}$ are linearly dependent, and there exists $\eta \in \mathbb{R}$, such that

$$\mathit{Q}-\gamma \left(t_0\right)=\eta (\alpha \left(t_0\right){\mathbb{L}}^{+}-\beta \left(t_0\right){\mathbb{L}}^{-}).$$

Therefore,

$$\begin{array}{cc}\hfill & \mathcal{ND}\left(\gamma \right)\left(t_0\right)\hfill \\ \hfill =& \gamma \left(t\right)+\eta {\displaystyle \frac{\langle \alpha \left(t_0\right){\mathbb{L}}^{+}-\beta \left(t_0\right){\mathbb{L}}^{-},\alpha \left(t_0\right){\mathbb{L}}^{+}-\beta \left(t_0\right){\mathbb{L}}^{-}\rangle }{4\alpha \left(t_0\right)\beta \left(t_0\right)}}(\alpha \left(t_0\right){\mathbb{L}}^{+}-\beta \left(t_0\right){\mathbb{L}}^{-})\hfill \\ \hfill =& \gamma \left(t_0\right)-\eta {\displaystyle \frac{-4\alpha \left(t_0\right)\beta \left(t_0\right)}{4\alpha \left(t_0\right)\beta \left(t_0\right)}}(\alpha \left(t_0\right){\mathbb{L}}^{+}-\beta \left(t_0\right){\mathbb{L}}^{-})\hfill \\ \hfill =& \gamma \left(t_0\right)+\eta (\alpha \left(t_0\right){\mathbb{L}}^{+}-\beta \left(t_0\right){\mathbb{L}}^{-})\hfill \\ \hfill =& \mathit{Q}.\hfill \end{array}$$

Hence, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)$ coincides with $\mathit{Q}$. □

We will provide an example to explain the above proposition.

Example 5. 

Let $\gamma :(-1,1)\to {\mathbb{R}}_1^2$ be a regular mixed-type curve, where

$$\gamma \left(t\right)=(t,t^2).$$

When $t_0=1$, $\gamma \left(t_0\right)$ is a spacelike point. See the blue curve in Figure 5.

Let ${\mathit{Q}}_1=(2,3)$, then ${\mathit{Q}}_1$ is on the tangent line of $\gamma \left(t_0\right)$. The T-dual curve of $\gamma \left(t\right)$ is

$$\mathcal{TD}\left(\gamma \right)\left(t\right)=\left({\displaystyle \frac{2t^3+6t-2}{4t^2-1}},{\displaystyle \frac{13t^2-4t}{4t^2-1}}\right).$$

In this case, $\mathcal{TD}\left(\gamma \right)\left(t_0\right)=(2,3)$; it coincides with ${\mathit{Q}}_1$. See the green curve in Figure 5.

Let ${\mathit{Q}}_2=(3,2)$, then ${\mathit{Q}}_2$ is on the normal line of $\gamma \left(t_0\right)$. The N-dual curve of $\gamma \left(t\right)$ is

$$\mathcal{ND}\left(\gamma \right)\left(t\right)=\left({\displaystyle \frac{2t^3+12t^2-5t}{4t^2-1}},{\displaystyle \frac{4t^4-2t^2+6t-2}{4t^2-1}}\right).$$

In this case, $\mathcal{ND}\left(\gamma \right)\left(t_0\right)=(3,2)$; it coincides with ${\mathit{Q}}_2$. See the orange dashed curve in Figure 5.

Having established the relationships between T-dual and N-dual curves, we now turn to the connection among all three associated curves. The following theorem reveals a fundamental duality: the T-dual curve of the evolute coincides with the N-dual curve of the original curve.

Theorem 2. 

Let $\gamma :I\to {\mathbb{R}}_1^2$ be a regular mixed-type curve without inflection points, $\mathit{Q}$ be a point in ${\mathbb{R}}_1^2$, $\mathcal{TD}\left(\gamma \right):I\to {\mathbb{R}}_1^2$ be the T-dual curve of γ, $\mathcal{ND}\left(\gamma \right):I\to {\mathbb{R}}_1^2$ be the N-dual curve of γ and $Ev\left(\gamma \right):I\to {\mathbb{R}}_1^2$ be the evolute of γ. If $\mathit{Q}$ coincides with the lightlike point or $\mathit{Q}$ is on the tangent line of the lightlike point, then

$$\mathcal{TD}\left(Ev\right(\gamma \left)\right)\left(t\right)=\mathcal{ND}\left(\gamma \right)\left(t\right).$$

Proof. 

Since the evolute of a regular mixed-type curve without inflections is given by

$$Ev\left(\gamma \right)\left(t\right)=\gamma \left(t\right)-{\displaystyle \frac{2\alpha \left(t\right)\beta \left(t\right)}{\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right)}}(\alpha \left(t\right){\mathbb{L}}^{+}-\beta \left(t\right){\mathbb{L}}^{-}),$$

we have

$$\begin{array}{cc}\hfill & \dot{E}v\left(\gamma \right)\left(t\right)\hfill \\ \hfill =& \alpha \left(t\right){\mathbb{L}}^{+}+\beta \left(t\right){\mathbb{L}}^{-}\hfill \\ \hfill & -{\displaystyle \frac{2(\dot{\alpha }\left(t\right)\beta \left(t\right)+\alpha \left(t\right)\dot{\beta }\left(t\right))\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right)-2\alpha \left(t\right)\beta \left(t\right)\ddot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\ddot{\beta }\left(t\right)}{{(\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right))}^2}}\hfill \\ \hfill & (\alpha \left(t\right){\mathbb{L}}^{+}-\beta \left(t\right){\mathbb{L}}^{-})\hfill \\ \hfill & -{\displaystyle \frac{2\alpha \left(t\right)\beta \left(t\right)}{\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right)}}(\dot{\alpha }\left(t\right){\mathbb{L}}^{+}-\dot{\beta }\left(t\right){\mathbb{L}}^{-})\hfill \\ \hfill =& \alpha \left(t\right)(1-{\displaystyle \frac{2{\dot{\alpha }}^2\left(t\right){\beta }^2\left(t\right)-2{\alpha }^2\left(t\right)\dot{\beta }\left(t\right)-2\alpha \left(t\right)\beta \left(t\right)(\ddot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\ddot{\beta }\left(t\right))}{{(\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right))}^2}}\hfill \\ \hfill & -{\displaystyle \frac{2\dot{\alpha }\left(t\right)\beta \left(t\right)}{\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right)}}){\mathbb{L}}^{+}\hfill \\ \hfill & +\beta \left(t\right)(1+{\displaystyle \frac{2{\dot{\alpha }}^2\left(t\right){\beta }^2\left(t\right)-2{\alpha }^2\left(t\right)\dot{\beta }\left(t\right)-2\alpha \left(t\right)\beta \left(t\right)(\ddot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\ddot{\beta }\left(t\right))}{{(\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right))}^2}}\hfill \\ \hfill & +{\displaystyle \frac{2\dot{\alpha }\left(t\right)\beta \left(t\right)}{\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right)}}){\mathbb{L}}^{-}\hfill \\ \hfill =& \alpha \left(t\right){\displaystyle \frac{-3{\dot{\alpha }}^2\left(t\right){\beta }^2\left(t\right)+3\alpha \left(t\right){\dot{\beta }}^2\left(t\right)+2\alpha \left(t\right)\beta \left(t\right)(\ddot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\ddot{\beta }\left(t\right))}{{(\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right))}^2}}{\mathbb{L}}^{+}\hfill \\ \hfill & -\beta \left(t\right){\displaystyle \frac{-3{\dot{\alpha }}^2\left(t\right){\beta }^2\left(t\right)+3\alpha \left(t\right){\dot{\beta }}^2\left(t\right)+2\alpha \left(t\right)\beta \left(t\right)(\ddot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\ddot{\beta }\left(t\right))}{{(\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right))}^2}}{\mathbb{L}}^{-}.\hfill \end{array}$$

Writing

$${\alpha }_{Ev}\left(t\right)=\alpha \left(t\right){\displaystyle \frac{-3{\dot{\alpha }}^2\left(t\right){\beta }^2\left(t\right)+3\alpha \left(t\right){\dot{\beta }}^2\left(t\right)+2\alpha \left(t\right)\beta \left(t\right)(\ddot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\ddot{\beta }\left(t\right))}{{(\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right))}^2}}$$

and

$${\beta }_{Ev}\left(t\right)=-\beta \left(t\right){\displaystyle \frac{-3{\dot{\alpha }}^2\left(t\right){\beta }^2\left(t\right)+3\alpha \left(t\right){\dot{\beta }}^2\left(t\right)+2\alpha \left(t\right)\beta \left(t\right)(\ddot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\ddot{\beta }\left(t\right))}{{(\dot{\alpha }\left(t\right)\beta \left(t\right)-\alpha \left(t\right)\dot{\beta }\left(t\right))}^2}}.$$

Then,

$$\begin{array}{cc}\hfill & \mathcal{TD}\left(Ev\right(\gamma \left)\right)\left(t\right)\hfill \\ \hfill =& Ev\left(\gamma \right)\left(t\right)-{\displaystyle \frac{\langle \mathit{Q}-Ev\left(\gamma \right)\left(t\right),{\alpha }_{Ev}\left(t\right)\rangle {\mathbb{L}}^{+}+{\beta }_{Ev}\left(t\right){\mathbb{L}}^{-}}{4{\alpha }_{Ev}\left(t\right){\beta }_{Ev}\left(t\right)}}({\alpha }_{Ev}\left(t\right){\mathbb{L}}^{+}+{\beta }_{Ev}\left(t\right){\mathbb{L}}^{-})\hfill \\ \hfill =& \gamma \left(t\right)+{\displaystyle \frac{\langle \mathit{Q}-\gamma \left(t\right),\alpha \left(t\right){\mathbb{L}}^{+}-\beta \left(t\right){\mathbb{L}}^{-}\rangle }{4\alpha \left(t\right)\beta \left(t\right)}}(\alpha \left(t\right){\mathbb{L}}^{+}-\beta \left(t\right){\mathbb{L}}^{-})\hfill \\ \hfill =& \mathcal{ND}\left(\gamma \right)\left(t\right).\hfill \end{array}$$

□

Theorem 2 establishes a key duality among the three associated curves. This commutative relation reveals a coherent geometric framework linking the T-dual curve, N-dual curve and evolute of a mixed-type curve in ${\mathbb{R}}_1^2$.

5. Conclusions

In this paper, we have systematically studied the N-dual curve of mixed-type curves in the Minkowski plane, filling a gap in the existing literature. Our main contributions are threefold.

Firstly, we provided 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 analyzed its singularities and classified its point types (spacelike, timelike, and lightlike). Thirdly, based on these results, we established geometric relations among the three associated curves: the T-dual curve, the N-dual curve, and the evolute. In particular, we proved 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. These results reveal a coherent duality framework linking the three objects.

The present analysis is restricted to plane curves. Extension to curves in Minkowski 3-space remains an open problem and is a natural direction for future research. Other possible extensions include applications in geometric optics, where dual curves arise as wavefronts and caustics, as well as numerical computations of N-dual curves for given mixed-type curves.