Recognition of plane paths and plane curves under linear pseudo-similarity transformations

Abstract

$E^{2}_{1}$ be the 2-dimensional pseudo-Euclidean space of index 1, $G=Sim_{L}(E^{2}_{1})$ be the group of all linear pseudo-similarities of $E^{2}_{1}$ and $G=Sim_{L}^{+}(E^{2}_{1})$ be the group of all orientation-preserving linear pseudo-similarities of $E^{2}_{1}$. In this paper, groups $Sim_{GL}^{+}(E^{2}_{1})$ and $Sim_{GL}(E^{2}_{1})$ are defined. For the groups $G=Sim_{GL}^{+}(E^{2}_{1}),Sim_{GL}(E^{2}_{1})$, $Sim_{L}(E^{2}_{1}),$ $Sim_{L}^{+}(E^{2}_{1})$, G-invariants of paths in $E^{2}_{1}$ are investigated. Using hyperbolic numbers, a method to detect G-similarities of paths and curves is presented. We give an evident form of a path and a curve in terms of their G-invariants. For two paths and two curves with the common G-invariants, evident forms of all linear pseudo-similarity transformations, carrying the paths and the curves, are obtained.

1 Introduction

Let ${\mathbb {R}}$ be the field of real numbers, n and p are integers such that $0\le p < n$. Let $E^{n}_{p}$ be the n-dimensional pseudo-Euclidean space of index p and O(n, p) be the group of all pseudo-orthogonal transformations of $E^{n}_{p}$. Put $O^{+}(n,p)= \left\{ g\in O(n,p)\mid detg=1\right\}$. Clearly, $E^{n}_{1}$ and $E^{n}_{0}$ are known as the n-dimensional pseudo-Euclidean space of index 1 and the n-dimensional Euclidean space, respectively (see [22, 29]).

Let $Sim(E^{n}_{p})$ be the group of all pseudo-similarities of $E^{n}_{p}$, $Sim^{+}(E^{n}_{p})$ be the group of all orientation-preserving pseudo-similarities of $E^{n}_{p}$, $Sim_{L}(E^{n}_{p})$ be the group of all linear pseudo-similarities of $E^{n}_{p}$ and $Sim_{L}^{+}(E^{n}_{p})$ be the group of all orientation-preserving linear pseudo-similarities of $E^{n}_{p}$. The groups $Sim(E^{n}_{0})$ and $Sim(E^{n}_{1})$ are known as the fundamental groups of the Euclidean similarity geometry and Lorentzian similarity geometry, resp.(For these definitions, see [18, 21, 27]).

The theory of local invariants of curves in $E^{2}_{0}$ for the group $Sim(E^{2}_{0})$ is given without proofs by [11].

In [18], geometric invariants of curves in similarity geometry considered. Existence and rigidity theorems for a curve obtained only for the group $Sim^{+}(E^{n}_{0})$.

As a generalization of the paper [18], some characterizations of a curve $\alpha$ using the arc length parameters of its $V_{i}$ indicatrix and an arc length parameter of the curve $\alpha$ are given in [5]. Thus invariant theory of curves in the similarity geometry in $E^{n}_{0}$ was developed only for the group $Sim^{+}(E^{n}_{0})$.

As it is well known, global differential G-invariants are an important tool for many areas in sciences. Therefore, using complex numbers, the differential geometry of plane paths and plane curves under linear similarity transformations have been improved by [24].

Similarity transformations play an important role in mathematics and it has many applications in physics, mechanics and computer sciences (see some references [1, 2, 7,8,9, 14, 15, 19, 20, 25, 26, 30, 31] and references therein).

Lorentzian similarity(pseudo similarity) is also important for applications to relativistic fluids, relativistic quantum theory and cosmology. (see some references [3, 4, 6, 10, 13, 16, 17, 34, 35]).

In [21, 27], classical theory of curves in Lorentzian similarity geometry is investigated. Moreover, the existence and rigidity theorems are given for the group $Sim^{+}(E^{n}_{1})$ in [27].

The Lorentzian similarity of plane curves are investigated in [28], using the hyperbolic structure and spherical arc length parameter.

The hyperbolic numbers are related to 2-dimensional pseudo-Euclidean space of index 1 (see some references [12, 23, 28, 32, 33]).

The global differential invariants of curves in the Lorentzian similarity plane geometry and their applications have not been considered so far. Therefore, in this paper, the differential geometry of non-null plane paths and non-null plane curves under linear pseudo-similarity transformations are investigated using the hyperbolic numbers. Moreover, new groups entitled “generalized linear pseudo-similarity groups” are defined. These groups are denoted by $Sim_{GL}^{+}(E^{2}_{1})$ and $Sim_{GL}(E^{2}_{1})$. Above mentioned problems are also investigated for these groups.

This work is organized as follows:

Let $G=Sim_{GL}^{+}(E^{2}_{1}), Sim_{GL}(E^{2}_{1}), Sim_{L}(E^{2}_{1}), Sim_{L}^{+}(E^{2}_{1})$. In Sect. 2, hyperbolic numbers and groups of linear pseudo-similarity transformations are introduced. In Sect. 3, we define an LPR path and we give fundamental differential G-invariants of an LPR path. In Sect. 4, definitions of LPDR and LPNR I-paths are given. The conditions of the global G-similarity for the LPR, LPDR and LPNR paths are obtained. For two paths $\xi$ and $\eta$ with the common G-invariants, all evident forms of $g\in G$ are obtained. In Sect. 5, the existence and uniqueness theorems of paths for the groups G and general forms of a path in terms of its global differential G-invariants are found. In Sect. 6, global differential invariants of LPNR curves and the conditions of linear pseudo-similarity of these curves are given.

2 Hyperbolic numbers and groups $Sim_{GL}(E^{2}_{1})$, $Sim_{GL}^{+}(E^{2}_{1})$, $Sim_{L}(E^{2}_{1})$ and $Sim_{L}^{+}(E^{2}_{1})$

Let ${\mathbb {R}}$ be the field of real numbers and ${\mathbb {H}}$ be the algebra of hyperbolic numbers. Any hyperbolic number $z\in {\mathbb {H}}$ can be written in the form $z=x+jy$, where $x,y\in {\mathbb {R}}$, $j^{2}=1,j\ne \pm 1$. It follows that the multiplication of two hyperbolic numbers $z_{1}$ and $z_{2}$, denoted by $z_{1}z_{2}$, is defined by

$$\begin{aligned} z_{1}z_{2}=(x_{1}+jy_{1})(x_{2}+jy_{2})=(x_{1}x_{2}+y_{1}y_{2})+j(x_{1}y_{2}+x_{2}y_{1}). \end{aligned}$$

(1)

Consider the hyperbolic number $z=x+jy$ in the matrix form $z=\left( \begin{array}{c} x \\ y \end{array} \right)$.

Then, the equality (1) can be written in the form:

$$\begin{aligned} z_{1}z_{2}=\left( \begin{array}{c} x_{1}x_{2}+y_{1}y_{2} \\ x_{1}y_{2}+x_{2}y_{1} \end{array} \right) = \left( \begin{array}{cc} x_{1} &{} y_{1} \\ y_{1} &{} x_{1} \end{array} \right) \left( \begin{array}{c} x_{2} \\ y_{2} \end{array} \right) . \end{aligned}$$

(2)

Denote by $M_{z}$ the symmetric matrix $\left( \begin{array}{cc} x &{} y \\ y &{} x \end{array} \right)$. Then $M_{z}:{\mathbb {H}}\rightarrow {\mathbb {H}}$ is a transformation.

Here, the equality (2) can be written in the form:

$$\begin{aligned} z_{1}z_{2}=M_{z_{1}}z_{2}. \end{aligned}$$

(3)

for all $z_{1},z_{2}\in {\mathbb {H}}$.

The algebra of hyperbolic numbers ${\mathbb {H}}$ can be used to represent elements of $E^{2}_{1}$. For two arbitrary elements $z_{1}=(x_{1}, y_{1})$ and $z_{2}=(x_{2}, y_{2})$ in ${\mathbb {H}}$, the inner product of $z_{1}$ and $z_{2}$, denoted by $<z_{1},z_{2}>$, has the form $<z_{1},z_{2}>=x_{1}x_{2}-y_{1}y_{2}$ and $<z_{1},z_{1}>$ is a quadratic form on ${\mathbb {H}}$. Let $z=x+jy\in {\mathbb {H}}$, then the hyperbolic conjugate of z, denoted by ${\overline{z}}$, is defined as ${\overline{z}}=x-jy$. Clearly,

we have $<z,z>=z{\overline{z}}$ and $<z_{1},z_{2}>=<\overline{z_{1}},\overline{z_{2}}>$ for all $z,z_{1},z_{2}\in {\mathbb {H}}$. Moreover, the inverse of z exists if and only if $<z,z>\ne 0$. In this case, the inverse of z denote by $z^{-1}$. It is easy to see that $z^{-1}=\frac{1}{z}=\frac{{\overline{z}}}{<z,z>}$.

For hyperbolic numbers $z_{1}=x_{1}+j y_{1}$ and $z_{2}=x_{2}+j y_{2}$, the matrix $\left( \begin{matrix} x_{1}&{}x_{2}\\ y_{1}&{}y_{2} \end{matrix}\right)$ will be denoted by $\Vert z_{1}\;z_{2}\Vert$. Denote by $[z_{1}\;z_{2}]$ the determinant of the matrix $\Vert z_{1}\;z_{2}\Vert$.

The following proposition is given as Proposition 4.3 in [23].

Proposition 1

Let $z_{1}$ and $z_{2}$ be two hyperbolic numbers such that $<z_{1},z_{1}>\ne 0$. Then the hyperbolic number $z_{2}z_{1}^{-1}$ exists, the following equalities hold:

$$\begin{aligned} z_{2}z_{1}^{-1}=\frac{<z_{1}, z_{2}>}{<z_{1}, z_{1}>}+j\frac{[z_{1}\; z_{2}]}{<z_{1}, z_{1}>} \end{aligned}$$

and

$$\begin{aligned} M_{z_{2}z_{1}^{-1}}=\left( \begin{array}{cc} \frac{<z_{1}, z_{2}>}{<z_{1}, z_{1}>} &{} \frac{[z_{1}\; z_{2}]}{<z_{1}, z_{1}>} \\ \frac{[z_{1}\; z_{2}]}{<z_{1}, z_{1}>} &{} \frac{<z_{1}, z_{2}>}{<z_{1}, z_{1}>} \end{array}\right) . \end{aligned}$$

(4)

Put ${\mathbb {H}}^{*}=\left\{ z\in {\mathbb {H}}\mid <z,z>\ne 0\right\}$. ${\mathbb {H}}^{*}$ is a group with respect to the multiplication operation in the algebra ${\mathbb {H}}$. Let $W=\left( \begin{matrix} 1&{}0\\ 0&{}-1 \end{matrix} \right)$. Denote by $\mathbb {H^{*}}W$ the set of all matrices $\left\{ zW\vert z\in {\mathbb {H}}^{*}\right\}$, where zW is the multiplication of matrices z and W.

Put ${\mathbb {H}}_{+}^{*}=\left\{ z\in \mathbb {H^{*}}\mid <z,z> > 0\right\}$. ${\mathbb {H}}_{+}^{*}$ is a subgroup of the group of ${\mathbb {H}}^{*}$. We consider a subgroup of the group ${\mathbb {H}}^{*}$ defined by

$$\begin{aligned} S({\mathbb {H}}_{+}^{*})=\left\{ z\in {\mathbb {H}} \mid <z,z>=1\right\} =&\left\{ e^{j\varphi }=cosh(\varphi )+jsinh(\varphi )\mid \varphi \in {\mathbb {R}} \right\} . \end{aligned}$$

It is obvious that ${\mathbb {H}}_{+}^{*}=\left\{ \lambda z\mid \lambda \in {\mathbb {R}}^{+},z\in S({\mathbb {H}}_{+}^{*}) \right\}$.

For $z=x+jy\in \mathbb {H_{+}^{*}}$, we put $M^{+}_{z}=\left( \begin{array}{cc} \frac{x}{\sqrt{<z,z>}} &{} \frac{y}{\sqrt{<z,z>}} \\ \frac{y}{\sqrt{<z,z>}} &{} \frac{x}{\sqrt{<z,z>)}} \end{array}\right)$. The proposition is obtained from [24] and [23, Theorem3.7].

Proposition 2

Let $z=x+jy\in \mathbb {H_{+}^{*}}$. Then, $M_{z}=\sqrt{<z,z>}M^{+}_{z}$ and $M^{+}_{z}\in O^{+}(2,1)$.

Put $Sim_{GL}^{+}(E^{2}_{1})=\left\{ M_{z}|z\in \mathbb {H^{*}}\right\}$. $Sim_{GL}^{+}(E^{2}_{1})$ is a group under the matrix multiplication. Denote by $Sim_{GL}^{-}(E^{2}_{1})$ the set of all matrices $\left\{ M_{z}W\vert z\in \mathbb {H^{*}}\right\}$, where $M_{z}W$ is the multiplication of matrices $M_{z}$ and W. It is easy to see that $Sim_{GL}^{+}(E^{2}_{1})\cap Sim_{GL}^{-}(E^{2}_{1})=\emptyset$. Put $Sim_{GL}(E^{2}_{1})=Sim_{GL}^{+}(E^{2}_{1})\cup Sim_{GL}^{-}(E^{2}_{1})$. $Sim_{GL}(E^{2}_{1})$ is a group. $Sim_{GL}(E^{2}_{1})$ will be called the group of generalized linear pseudo-similarities of $E^{2}_{1}$. The group $Sim_{GL}^{+}(E^{2}_{1})$ is the subgroup of $Sim_{GL}(E^{2}_{1})$. The group $Sim_{GL}^{+}(E^{2}_{1})$ will be called the group of orientation-preserving generalized linear pseudo-similarities of $E^{2}_{1}$.

The group of all linear pseudo-similarities of $E^{2}_{1}$ is denoted by $Sim_{L}(E^{2}_{1})$. Let $Sim_{L}^{+}(E^{2}_{1})$ be the group of all orientation-preserving linear pseudo-similarities of $E^{2}_{1}$ and $Sim_{L}^{-}(E^{2}_{1})$ be set generated by all orientation-reversing linear pseudo-similarities of $E^{2}_{1}$, resp. Clearly, $Sim_{L}^{+}(E^{2}_{1})\cap Sim_{L}^{-}(E^{2}_{1})=\emptyset$ and $Sim_{L}(E^{2}_{1})=Sim_{L}^{+}(E^{2}_{1})\cup Sim_{L}^{-}(E^{2}_{1})$. Here, $Sim_{L}(E^{2}_{1})$ is a group. The group $Sim_{L}^{+}(E^{2}_{1})$ is the subgroup of $Sim_{L}(E^{2}_{1})$. It is easy to see that the group $Sim_{L}(E^{2}_{1})$ is the subgroup of $Sim_{GL}(E^{2}_{1})$.

Hence, the following proposition holds.

Proposition 3

(i):

$Sim_{GL}^{+}(E^{2}_{1})=\left\{ M_{z}|z\in \mathbb {H^{*}}\right\}$.

(ii):

$Sim_{GL}^{-}(E^{2}_{1})=\left\{ M_{z}W|z\in \mathbb {H^{*}},W=\left( \begin{matrix} 1&{}0\\ 0&{}-1 \end{matrix}\right) \right\}$.

(iii):

$Sim_{GL}(E^{2}_{1})=Sim_{GL}^{+}(E^{2}_{1})\cup Sim_{GL}^{-}(E^{2}_{1})$.

(iv):

$Sim_{L}^{+}(E^{2}_{1})=\left\{ M_{z}|z\in {\mathbb {H}}_{+}^{*}\right\}$.

(v):

$Sim_{L}^{-}(E^{2}_{1})=\left\{ M_{z}W|z\in {\mathbb {H}}_{+}^{*},W=\left( \begin{matrix} 1&{}0\\ 0&{}-1 \end{matrix}\right) \right\}$.

(vi):

$Sim_{L}(E^{2}_{1})=Sim_{L}^{+}(E^{2}_{1})\cup Sim_{L}^{-}(E^{2}_{1})$.

3 On G-invariant functions of a path for the groups $G=Sim_{GL}(E^{2}_{1})$, $Sim_{GL}^{+}(E^{2}_{1})$, $Sim_{L}(E^{2}_{1})$ and $Sim_{L}^{+}(E^{2}_{1})$

Let $\xi :I\rightarrow E^{2}_{1}$ be a function, where $I=(a,b)$ is an open interval in ${\mathbb {R}}$. We write $\xi (t)=(\xi _{1}(t),\xi _{2}(t))$, where $\xi _{1}(t)$ and $\xi _{2}(t)$ are an ordinary real-valued function of a real variable. We suppose that $\xi _{1}(t)$ and $\xi _{2}(t)$ are real $C^{(1)}$-functions on I.

Definition 1

A $C^{(1)}$-mapping $\xi :I\rightarrow E^{2}_{1}$ is called an I-path (parametrized curve) in $E^{2}_{1}$.

Remark 1

Throughout the paper, G is a one of the groups $Sim_{GL}(E^{2}_{1})$, $Sim_{GL}^{+}(E^{2}_{1})$, $Sim_{L}(E^{2}_{1})$ and $Sim_{L}^{+}(E^{2}_{1})$.

If $\xi (t)$ is an I-path then $F\xi (t)$ is an I-path in $E^{2}_{1}$ for all F in G and for all $t\in I$.

Denote the first derivative of $\xi (t)$ by $\xi ^{\prime }(t)=(\xi ^{\prime }_{1}(t), \xi ^{\prime }_{2}(t))$.

Definition 2

An L-pseudo-regular I-path(shortly, LPR path) is a $C^{(1)}$-mapping $\xi :I\rightarrow E^{2}_{1}$ such that $<\xi (t),\xi (t)>\ne 0$ for all $t\in I$.

For example, consider the ${\mathbb {R}}$-path $\xi (t)=(cosh(t),sinh(t))$. It is easy to see that $<\xi (t),\xi (t)>=cosh^{2}(t)-sinh^{2}(t)=1\ne 0$ for all $t\in {\mathbb {R}}$. That is, $\xi (t)$ is an LPR ${\mathbb {R}}$-path.

Definition 3

Two paths $\xi ,\eta :I\rightarrow E^{2}_{1}$ are called G- similar if there exists $F\in G$ such that $\eta (t)=F\xi (t)$ for all $t\in I$.

Let the I-paths $\zeta _{1}=\zeta _{1}(t), \zeta _{2}=\zeta _{2}(t), \ldots , \zeta _{n}=\zeta _{n}(t)$ in $E^{2}_{1}$ are defined on I.

Definition 4

A function $\psi (\zeta _{1}, \zeta _{2}, \ldots , \zeta _{n})$ is called G-invariant if $\psi (F\zeta _{1}, F\zeta _{2}, \ldots , F\zeta _{n})=\psi (\zeta _{1}, \zeta _{2}, \ldots , \zeta _{n}),\forall F\in G, \forall t\in I$.

For the derivative $\xi ^{\prime }(t)$ and the I-path $\xi (t)$, the determinant of the matrix $\Vert \xi (t)\;\xi ^{\prime }(t)\Vert =\left( \begin{matrix} \xi _{1}(t)&{} \xi _{1}^{\prime }(t) \\ \xi _{2}(t)&{} \xi _{2}^{\prime }(t) \end{matrix} \right)$ will be denoted by $[\xi (t)\; \xi ^{\prime }(t)]$.

Let $\xi (t)$ be an LPR I-path. Put the functions $f_{\xi }(t)=\frac{<\xi (t), \xi ^{\prime }(t)>}{<\xi (t),\xi (t)>}$, $g_{\xi }(t)=\frac{[ \xi (t)\; \xi ^{\prime }(t)]}{<\xi (t),\xi (t)>}$ and $h_{\xi }(t)=\frac{<\xi ^{\prime }(t), \xi ^{\prime }(t)>}{<\xi (t),\xi (t)>}$ for all $t\in I$.

Proposition 4

(i):

The functions $f_{\xi }(t),g_{\xi }^{2}(t)$ and $h_{\xi }(t)$ are $Sim_{GL}(E^{2}_{1})$- invariant.

(ii):

The function $g_{\xi }(t)$ is $Sim_{GL}^{+}(E^{2}_{1})$- invariant.

Proof

(i):

Firstly, we prove that $f_{\xi }(t)$ is $Sim_{GL}(E^{2}_{1})$-invariant. Assume that $F\in Sim_{GL}^{+}(E^{2}_{1})$. Then, by Proposition 3(i) , F has the form $F=M_{z}$ such that $F(\xi (t))=M_{z}\xi (t)=z\xi (t)$, where $z\in \mathbb {H^{*}}$ and $\forall \xi (t)\in E^{2}_{1}$. Hence, we obtain

$$\begin{aligned} \frac{<F\xi (t), (F\xi (t))^{\prime }>}{<F\xi (t), F\xi (t)>}=\frac{\left\langle M_{z}(\xi (t)), M_{z}(\xi ^{\prime }(t))\right\rangle }{\left\langle M_{z}(\xi (t)), M_{z}(\xi (t)) \right\rangle }= \frac{<\xi (t), \xi ^{\prime }(t)>}{<\xi (t),\xi (t) >} \end{aligned}$$

for all $t\in I$. Hence $\frac{<\xi (t), \xi ^{\prime }(t)>}{<\xi (t),\xi (t) >}$ is $Sim_{GL}^{+}(E^{2}_{1})$- invariant. Moreover,

$$\begin{aligned} \frac{<{\overline{\xi }}(t), {\overline{\xi }}^{\prime }(t)>}{<{\overline{\xi }}(t),{\overline{\xi }}(t)>}=\frac{<\xi (t), \xi ^{\prime }(t)>}{<\xi (t),\xi (t) >} \end{aligned}$$

for all $t\in I$. Thus $f_{\xi }(t)$ is $Sim_{GL}^{+}(E^{2}_{1})$- invariant and $Sim_{GL}^{-}(E^{2}_{1})$- invariant. Hence it is $Sim_{GL}(E^{2}_{1})$- invariant. Similarly, we are proved that $g_{\xi }^{2}(t)$ and $h_{\xi }(t)$ are $Sim_{GL}(E^{2}_{1})$- invariant.

(ii):

The proof is similar to the statement (i).

$\square$

These functions $f_{\xi }(t), g_{\xi }(t),g_{\xi }^{2}(t)$ and $h_{\xi }(t)$ will be called the global differential G-invariant functions for the groups $G=Sim_{GL}(E^{2}_{1})$ and $G=Sim_{GL}^{+}(E^{2}_{1})$.

4 Similarity of paths for the groups $Sim_{GL}(E^{2}_{1})$, $Sim_{GL}^{+}(E^{2}_{1})$, $Sim_{L}(E^{2}_{1})$ and $Sim_{L}^{+}(E^{2}_{1})$

In this section we will obtain the conditions of similarity of two paths in $E^{2}_{1}$ with respect to the groups $Sim_{GL}(E^{2}_{1})$, $Sim_{GL}^{+}(E^{2}_{1})$, $Sim_{L}(E^{2}_{1})$ and $Sim_{L}^{+}(E^{2}_{1})$.

Theorem 1

Let $\xi , \eta :I\rightarrow E^{2}_{1}$ are LPR paths. Then, $\xi (t){\mathop {\sim }\limits ^{Sim_{GL}^{+}(E^{2}_{1})}}\eta (t)$ $\Leftrightarrow$

$$\begin{aligned} \left\{ \begin{aligned} f_{\xi }(t)&=f_{\eta }(t), \\ g_{\xi }(t)&=g_{\eta }(t) \end{aligned} \right. \end{aligned}$$

(5)

for all $t\in I$.

In addition, if $\xi (t)$ and $\eta (t)$ are $Sim_{GL}^{+}(E^{2}_{1})$-similar, then the unique element F exists in $Sim_{GL}^{+}(E^{2}_{1})$ satisfying the equality $\eta (t)=F\xi (t), \forall t\in I$, and F can be written in the following matrix form

$$\begin{aligned} F=\left( \begin{array}{cc} \frac{<\xi (t),\eta (t)>}{<\xi (t),\xi (t)>}&{} \frac{[\xi (t)\; \eta (t)]}{<\xi (t),\xi (t)>}\\ \frac{[\xi (t)\; \eta (t)]}{<\xi (t),\xi (t)>}&{} \frac{<\xi (t),\eta (t)>}{<\xi (t),\xi (t)>} \end{array}\right) . \end{aligned}$$

(6)

Proof

$\Rightarrow :$ Let $\xi (t){\mathop {\sim }\limits ^{Sim_{GL}^{+}(E^{2}_{1})}}\eta (t)$. Then, by Proposition 3(i), there is $z\in \mathbb {H^{*}}$ such that $\eta (t)=z\xi (t), \forall t\in I$. Hence, we have $\eta ^{\prime }(t)=z\xi ^{\prime }(t), \forall t\in I$. Since $\xi (t)$ and $\eta (t)$ are LPR-paths, we obtain $<\xi (t),\xi (t)>\ne 0$ and $<\eta (t),\eta (t)>\ne 0$ for all $t\in I$. Then there are $(\xi )^{-1}(t)$ and $(\eta )^{-1}(t), \forall t\in I$. Putting $z_{1}=\xi (t), z_{2}=\xi ^{\prime }(t)$ in Proposition 1, we have

$$\begin{aligned} \frac{\xi ^{\prime }(t)}{\xi (t)}=\frac{<\xi (t), \xi ^{\prime }(t)>+j\left[ \xi (t)\; \xi ^{\prime }(t)\right] }{<\xi (t),\xi (t)>} \end{aligned}$$

(7)

, $\forall t\in I$. Similarly, putting $z_{1}=\eta (t), z_{2}=\eta ^{\prime }(t)$ in Proposition 1, we have

$$\begin{aligned} \frac{\eta ^{\prime }(t)}{\eta (t)}=\frac{<\eta (t), \eta ^{\prime }(t)>+j\left[ \eta (t)\; \eta ^{\prime }(t)\right] }{<\eta (t),\eta (t)>} \end{aligned}$$

(8)

for all $t\in I$. The equalities $\eta (t)=z\xi (t)$ and $\eta ^{\prime }(t)=z\xi ^{\prime }(t)$ imply $\frac{\eta ^{\prime }(t)}{\eta (t)}=\frac{\xi ^{\prime }(t)}{\xi (t)}$. From this equality, (7) and (8), we obtain (5).

$\Leftarrow :$ Assume that the equalities (5) hold. From (7), (8) and Proposition 1, we have $\eta ^{\prime }(t)(\eta )^{-1}(t)=\xi ^{\prime }(t)(\xi )^{-1}(t)$ for all $t\in I$. By taking derivative of the equality $\eta (t)(\xi )^{-1}(t)$, we obtain

$$\begin{aligned} \begin{aligned} \frac{d(\eta (t)(\xi )^{-1}(t))}{dt}&=\eta ^{\prime }(t)(\xi )^{-1}(t)-\eta (t)\xi ^{\prime }(t)(\xi )^{-2}(t)\\&=\eta (t)(\eta ^{\prime }(t)(\eta )^{-1}(t)-\xi ^{\prime }(t)(\xi )^{-1}(t))(\xi )^{-1}(t)=0 \end{aligned} \end{aligned}$$

for all $t\in I$. Hence the function $\eta (t)(\xi )^{-1}(t)$ is constant on I. Put $w=\eta (t)(\xi )^{-1}(t)$. We prove that $w\ne 0$. Assume that $w=0$. Then, we have $w=\eta (t)(\xi )^{-1}(t)=0$ for all $t\in I$. Since $<\xi (t),\xi (t)>\ne 0$ for all $t\in I$, we have the equality $\eta (t)(\xi )^{-1}(t)\xi (t)=0.\xi (t)$ for all $t\in I$. Hence, the equality $\eta (t)=0, \forall t\in I$. But, this is a contradiction by $<\eta (t),\eta (t)>\ne 0$ for all $t\in I$. That is, $w\ne 0$.

Since $\eta (t)=\eta (t)(\xi )^{-1}(t)\xi (t)=(\eta (t)(\xi )^{-1}(t))\xi (t)$ for all $t\in I$, we have $\eta (t)=w\xi (t)$ for all $t\in I$. Using (3), we obtain $\eta (t)=w\xi (t)=M_{w}\xi (t)$ for all $t\in I$. By $w=\eta (t)(\xi )^{-1}(t)=\frac{<\xi (t), \eta (t)>}{<\xi (t),\xi (t)>}+j\frac{[\xi (t)\; \eta (t)]}{<\xi (t),\xi (t)>}$ for all $t\in I$ and Proposition 1, $M_{w}$ has the form (5) and $M_{w}=F$. Since w is a constant. Then $\eta (t)=F\xi (t)=M_{w}\xi (t)$ for all $t\in I$. By Proposition 3(i), we obtain $F\in Sim_{GL}^{+}(E^{2}_{1})$. Hence $\xi (t){\mathop {\sim }\limits ^{Sim_{GL}^{+}(E^{2}_{1})}}\eta (t)$.

We prove the uniqueness of $F\in Sim_{GL}^{+}(E^{2}_{1})$ satisfying $\eta (t)=F\xi (t)$ for all $t\in I$. Assume that $\varphi \in Sim_{GL}^{+}(E^{2}_{1})$ exists such that $\eta (t)=\varphi \xi (t)$ for all $t\in I$. Then by the equality (3), Propositions 1 and 3(i), there is the unique $z\in \mathbb {H^{*}}$ such that $\varphi =M_{z}$. Hence we have $\eta (t)=M_{z}\xi (t)$ for all $t\in I$. By the equality (3), we obtain $\eta (t)=z\xi (t)$ for all $t\in I$. Since $<\xi (t),\xi (t)>\ne 0$ and $\eta (t)=z\xi (t)$ implies that $z=\eta (t)(\xi )^{-1}(t)=w$, $\forall t\in I$. Hence $M_{z}=M_{w}=F$. $\square$

Example 1

Let $\xi (t)=(t,t^2)$ and $\eta (t)=(3t+4t^2,4t+3t^2)$ be two $I=(1,\infty )$-paths. Clearly they are LPR paths and the equalities in (5) hold. Then by Theorem 1, $\xi (t){\mathop {\sim }\limits ^{Sim_{GL}^{+}(E^{2}_{1})}}\eta (t)$. Moreover, by Theorem 1, we obtain that $F=\left( \begin{array}{cc} 3 &{} 4\\ 4 &{} 3 \end{array}\right) .$

Definition 5

An I-path $\xi$ in $E^{2}_{1}$ is called completely L-pseudo-degenerate path( shortly, LPD path) if $\left[ \xi (t)\;\xi ^{\prime }(t)\right] =0$ for all $t\in I$.

An I-path $\xi$ in $E^{2}_{1}$ is called L-pseudo-non-degenerate path (shortly, LPN path) if $\left[ \xi (t)\;\xi ^{\prime }(t)\right] \ne 0$ for all $t\in I$.

Remark 2

Let $G=Sim_{GL}^{+}(E^{2}_{1})$ or $G=Sim_{GL}(E^{2}_{1})$. If $\xi (t)$ and $\eta (t)$ be I-paths in $E^{2}_{1}$ such that $\xi (t)$ is an LPD path and $\xi (t){\mathop {\sim }\limits ^{G}}\eta (t)$, then $\eta (t)$ is also LPD path. Similarly, it is easy to see that if $\xi (t)$ is an LPN path and $\xi (t){\mathop {\sim }\limits ^{G}}\eta (t)$, then $\eta (t)$ is also an LPN path.

Example 2

Let $\xi (t)=(t,\frac{t}{2})$ in $E^{2}_{1}$ be an $I=(0,+\infty )$-path. Then, $\left[ \xi (t)\; \xi ^{\prime }(t)\right] =0$ for all $t\in I$. That is, $\xi (t)$ is an LPD path. Since $<\xi (t),\xi (t)>=\frac{3t^2}{4}\ne 0$, $\forall t\in I$, we have $\xi (t)$ is a LPR path.

Example 3

Let $\xi (t)=(cosh(t),sinh(t))$ in $E^{2}_{1}$ be an ${\mathbb {R}}$-path. Then, $\left[ \xi (t)\; \xi ^{\prime }(t)\right] \ne 0$, $\forall t\in {\mathbb {R}}$. Hence, $\xi (t)$ is an LPN path.

Example 4

Let $\xi (t)=(t,t+2)$ in $E^{2}_{1}$ be an ${\mathbb {R}}$-path. Then, $\left[ \xi (t)\; \xi ^{\prime }(t)\right] \ne 0$, $\forall t\in {\mathbb {R}}$. Hence, $\xi (t)$ is an LPN path. Since $<\xi (t),\xi (t)>=0$ for $t=-1$, $\xi (t)$ is not an LPR path.

Example 5

Let $\xi (t)=(t^2,\frac{t^2}{2})$ in $E^{2}_{1}$ be an $I=(0,+\infty )$-path. Then, $\left[ \xi (t)\; \xi ^{\prime }(t)\right] =0$, $\forall t\in I$, but $<\xi (t),\xi (t)>\ne 0$, $\forall t\in I$. Hence, $\xi (t)$ is not an LPN path, but $\xi (t)$ is an LPR path.

Remark 3

In above examples, it is easy to see that an LPN path can be not an LPR path. Moreover, an LPD path can be not an LPR path, via versa. Therefore, the following definition are obvious.

Definition 6

An I-path $\xi (t)$ will be called L-pseudo-non-degenerate pseudo-regular path( shortly, LPNR path) if it is an LPN path and an LPR path.

An I-path $\xi (t)$ will be called completely L-pseudo-degenerate pseudo-regular path( shortly, LPDR path) if it is an LPD path and an LPR path.

Theorem 2

Let $\xi , \eta :I\rightarrow E^{2}_{1}$ are LPDR paths. Then, $\xi (t){\mathop {\sim }\limits ^{Sim_{GL}(E^{2}_{1})}}\eta (t)$ if and only if

$$\begin{aligned} f_{\xi }(t)=f_{\eta }(t). \end{aligned}$$

(9)

for all $t\in I$.

In addition, if $\xi (t)$ and $\eta (t)$ are $Sim_{GL}(E^{2}_{1})$-similar, then only two elements $F=F_{1}, F_{2}$ exist in $Sim_{GL}(E^{2}_{1})$ satisfying the equality$\eta (t)=F\xi (t), \forall t\in I$. In this case, $F_{1}\in Sim_{GL}^{+}(E^{2}_{1})$ and $F_{2}\in Sim_{GL}^{-}(E^{2}_{1})$. Here $F_{1}$ has the matrix form (6), and $F_{2}\in Sim_{GL}^{-}(E^{2}_{1})$ has the following matrix form

$$\begin{aligned} F_{2}=\left( \begin{array}{cc} \frac{<{\overline{\xi }}(t),\eta (t)>}{<\xi (t),\xi (t)>}&{} -\frac{[{\overline{\xi }}(t)\; \eta (t)]}{<\xi (t),\xi (t)>}\\ \frac{[{\overline{\xi }}(t)\; \eta (t)]}{<\xi (t),\xi (t)>}&{} -\frac{<{\overline{\xi }}(t),\eta (t)>}{<\xi (t),\xi (t)>} \end{array}\right) . \end{aligned}$$

(10)

Proof

$\Rightarrow :$ Let $\xi (t){\mathop {\sim }\limits ^{Sim_{GL}(E^{2}_{1})}}\eta (t)$. Since the function $f_{\xi }(t)$ is $Sim_{GL}(E^{2}_{1})$-invariant, we obtain the equality (9) holds.

$\Leftarrow :$ Let the equality (9) holds. Since $\xi (t)$ and $\eta (t)$ are LPDR-paths, the following equality obtain.

$$\begin{aligned} \left[ \xi (t)\;\xi ^{\prime }(t)\right] =\left[ \eta (t)\;\eta ^{\prime }(t)\right] =0 \end{aligned}$$

(11)

for all $t\in I$. From (10) and (11), we have (5). Then, by Theorem 1, there exists the unique element $F_{1}$ in $Sim_{GL}^{+}(E^{2}_{1})$ such that $\eta (t)=F_{1}\xi (t)$ for all $t\in I$. Here $F_{1}$ can be written in the form (6).

We consider the I-path ${\overline{\xi }}(t)$. Since functions $<\xi (t),\xi (t)>$ and $<\xi (t),\xi ^{\prime }(t)>$ are $Sim_{GL}^{-}(E^{2}_{1})$-invariant, we have $<{\overline{\xi }}(t),{\overline{\xi }}(t)>=<\xi (t),\xi (t)>$ and $<{\overline{\xi }}(t),{\overline{\xi }}^{\prime }(t)>= <\xi (t),\xi ^{\prime }(t)>$ for all $t\in I$. Similarly we obtain $<{\overline{\eta }}(t),{\overline{\eta }}(t)>=<\eta (t),\eta (t)>$ and $<{\overline{\eta }}(t),{\overline{\eta }}^{\prime }(t)>= <\eta (t), \eta ^{\prime }(t)>$ for all $t\in I$. Using (11) for all $t\in I$, we obtain

$$\begin{aligned} \left[ {\overline{\xi }}(t)\;{\overline{\xi }}^{\prime }(t)\right] = -\left[ \xi (t)\;\xi ^{\prime }(t)\right] =\left[ \eta (t)\;\eta ^{\prime }(t)\right] =0. \end{aligned}$$

Using the equalities $<{\overline{\xi }}(t),{\overline{\xi }}(t)>=<\xi (t),\xi (t)>$, $<{\overline{\xi }}(t),{\overline{\xi }}^{\prime }(t)>= <\xi (t),\xi ^{\prime }(t)>$, $\left[ {\overline{\xi }}(t)\;{\overline{\xi }}^{\prime }(t)\right] =\left[ \eta (t)\; \eta ^{\prime }(t)\right] =0$ and the equalities (9) ,(11), we obtain equalities:

$$\begin{aligned} \left\{ \begin{aligned} f_{{\overline{\xi }}}(t)&=f_{\eta }(t) \\ g_{{\overline{\xi }}}(t)&=g_{\eta }(t) \end{aligned} \right. \end{aligned}$$

for all $t\in I$. Hence, by Theorem 1, there is the unique element $F_{2}\in Sim_{GL}^{-}(E^{2}_{1})$ such that $\eta (t)=F_{2}\xi (t), \forall t\in I$. Here $F_{2}$ can be written in the form (10).

Let $F\in Sim_{L}(E^{2}_{1})$ satisfying the equality $\eta (t)=F\xi (t), \forall t\in I$. We will prove that $F\xi (t)=F_{1}\xi (t)$ or $F\xi (t)=F_{2}\xi (t)$ for all $t\in I$. Since $F\in Sim_{GL}(E^{2}_{1})$, we obtain $F\in Sim_{GL}^{+}(E^{2}_{1})$ or $F\in Sim_{GL}^{-}(E^{2}_{1})$. Assume that $F\in Sim_{GL}^{+}(E^{2}_{1})$. Then, using Theorem 1,we have $F=F_{1}$. Assume that $F\in Sim_{GL}^{-}(E^{2}_{1})$. Then $F_{2}$ can be written as $F_{2}=F_{3}W$, where $F_{3}\in Sim_{GL}^{+}(E^{2}_{1})$ and $W=\left( \begin{matrix} 1&{}0\\ 0&{}-1 \end{matrix}\right)$ $\in Sim_{GL}^{-}(E^{2}_{1})$. We have $\eta (t)=(F_{3}W)\xi (t)=F_{3}(W\xi (t))=F_{3}{\overline{\xi }}(t)$ for all $t\in I$. Hence paths $\eta (t)$ and ${\overline{\xi }}(t)$ are $Sim_{GL}^{+}(E^{2}_{1})$-similar. By Theorem 1, $F_{3}=F_{1}$. $\square$

Theorem 3

Let $\xi , \eta :I\rightarrow E^{2}_{1}$ are LPNR paths. Then, $\xi (t){\mathop {\sim }\limits ^{Sim_{GL}(E^{2}_{1})}}\eta (t)$ $\Leftrightarrow$

$$\begin{aligned} \left\{ \begin{aligned} f_{\xi }(t)&=f_{\eta }(t), \\ g_{\xi }^{2}(t)&=g_{\eta }^{2}(t) \end{aligned} \right. \end{aligned}$$

(12)

for all $t\in I$.

In addition, if $\xi (t)$ and $\eta (t)$ are $Sim_{GL}(E^{2}_{1})$-similar, then the unique element F exists in $Sim_{GL}(E^{2}_{1})$ satisfying the equality $\eta (t)=F\xi (t), \forall t\in I$. In this case, for all $t\in I$, there are only four cases:

(i):

$g_{\xi }(t)>0$ and $g_{\eta }(t)>0$,

(ii):

$g_{\xi }(t)<0$ and $g_{\eta }(t)<0$,

(iii):

$g_{\xi }(t)>0$ and $g_{\eta }(t)<0$,

(iv):

$g_{\xi }(t)<0$ and $g_{\eta }(t)>0$. For (i) and (ii), $F=F_{1}$, where $F_{1}\in Sim_{GL}^{+}(E^{2}_{1})$ and it has the form (6). For (iii) and (iv), $F=F_{2}$, where $F_{2}\in Sim_{GL}^{-}(E^{2}_{1})$ and it has the form (10).

Proof

$\Rightarrow :$ Let $\xi (t){\mathop {\sim }\limits ^{Sim_{GL}(E^{2}_{1})}}\eta (t)$. Since the functions $f_{\xi }(t)$ and $g_{\xi }^{2}(t)$ are $Sim_{GL}(E^{2}_{1})$-invariant for all $t\in I$, we have obtain the equalities (12) hold.

$\Leftarrow :$ Let the equalities (12) hold. Since $\xi (t)$ and $\eta (t)$ are LPNR paths, $g_{\xi }(t)\ne 0$ and $g_{\eta }(t)\ne 0$, $\forall t\in I$. Then only the conditions (i), (ii), (iii), (iv) in theorem exist.

For (i) and (ii), the equality $g_{\xi }^{2}(t)=g_{\eta }^{2}(t)$ in theorem imply the equality $g_{\xi }(t)=g_{\eta }(t)$ for all $t\in I$. The equality $g_{\xi }(t)=g_{\eta }(t), \forall t\in I$ and (12) imply (5). Hence, using Theorem 1, we obtain that there is the unique $F_{1}\in Sim_{GL}^{+}(E^{2}_{1})$ satisfying the equality $\eta (t)=F_{1}\xi (t),\forall t\in I$. Here $F_{1}$ can be written in the form (6). Clearly, $F=F_{1}$.

In cases (iii) and (iv), $g_{\xi }^{2}(t)=g_{\eta }^{2}(t)$ in theorem imply $g_{\xi }(t)=-g_{\eta }(t),\forall t\in I$.

Using this equality, the equality $<{\overline{\xi }}(t),{\overline{\xi }}(t)>=<\xi (t),\xi (t)>$ and $<\xi (t),\xi (t)>\ne 0$, we obtain $g_{{\overline{\xi }}}(t)=-g_{\xi }(t)=g_{\eta }(t)$ for all $t\in I$.

The equality $g_{{\overline{\xi }}}(t)=-g_{\xi }(t)=g_{\eta }(t), \forall t\in I$ and $f_{{\overline{\xi }}}(t)=f_{\xi }(t)=f_{\eta }(t)$ imply (5) for the paths ${\overline{\xi }}(t)$ and $\eta (t)$. By Theorem 1, there is the unique $F_{3}\in Sim_{GL}^{+}(E^{2}_{1})$ such that $\eta (t)=F_{3}{\overline{\xi }}=F_{3}(W\xi (t))$, where $F_{3}$ can be written in the form (10). Since $F_{3}\in Sim_{GL}^{+}(E^{2}_{1})$, we have $F=F_{2}=F_{3}W\in Sim_{GL}^{-}(E^{2}_{1})\subset Sim_{GL}(E^{2}_{1})$. $\square$

The following lemma is given in [29, Lemma 2].

Lemma 1

Let $w_{1}, w_{2}, z_{1}, z_{2}$ be vectors in $E^{2}_{1}$. Then $[w_{1}\;w_{2}][z_{1}\;z_{2}]=\left\langle w_{1}, z_{2}\right\rangle \left\langle w_{2}, z_{1}\right\rangle -\left\langle w_{1}, z_{1}\right\rangle \left\langle w_{2}, z_{2}\right\rangle$.

Theorem 4

Let $\xi , \eta :I\rightarrow E^{2}_{1}$ are LPNR paths. Then, $\xi (t){\mathop {\sim }\limits ^{Sim_{GL}(E^{2}_{1})}}\eta (t)$ $\Leftrightarrow$

$$\begin{aligned} \left\{ \begin{aligned} f_{\xi }(t)&=f_{\eta }(t), \\ h_{\xi }(t)&=h_{\eta }(t) \end{aligned} \right. \end{aligned}$$

(13)

for all $t\in I$.

In addition, if $\xi (t)$ and $\eta (t)$ are $Sim_{GL}(E^{2}_{1})$-similar, then the unique element F exists in $Sim_{GL}(E^{2}_{1})$ satisfying the equality $\eta (t)=F\xi (t), \forall t\in I$. In this case, for all $t\in I$, only the cases (i),(ii),(iii),(iv) in Theorem 3 are:

For (i) and (ii), $F=F_{1}$, where $F_{1}\in Sim_{GL}^{+}(E^{2}_{1})$ and it has the form (6).

In cases (iii) and (iv), $F=F_{2}$, where $F_{2}\in Sim_{GL}^{-}(E^{2}_{1})$ and it has the form (10).

Proof

$\Rightarrow :$ Let $\xi (t){\mathop {\sim }\limits ^{Sim_{GL}(E^{2}_{1})}}\eta (t)$. In Lemma 1, we put $w_{1}=z_{1}=\xi (t), w_{2}=z_{2}=\xi ^{\prime }(t)$. Then we obtain

$$\begin{aligned} \left[ \xi (t)\;\xi ^{\prime }(t)\right] ^{2}=<\xi (t), \xi ^{\prime }(t)>^{2}-<\xi (t), \xi (t)> <\xi ^{\prime }(t), \xi ^{\prime }(t)> \end{aligned}$$

for all $t\in I$. Using this equality and $<\xi (t), \xi (t)>\ne 0$ for all $t\in I$, we have

$$\begin{aligned} \frac{<\xi ^{\prime }(t), \xi ^{\prime }(t)>}{<\xi (t),\xi (t)>}=\frac{<\xi (t), \xi ^{\prime }(t)>^{2}}{<\xi (t),\xi (t)>^{2}}-\frac{\left[ \xi (t)\;\xi ^{\prime }(t)\right] ^{2}}{<\xi (t),\xi (t)>^{2}}. \end{aligned}$$

(14)

Since $f_{\xi }^{2}(t)$ and $g_{\xi }(t)$ are $Sim_{GL}(E^{2}_{1})$-invariants, the function $h_{\xi }(t)$ is $Sim_{GL}(E^{2}_{1})$-invariant by (14). Since $f_{\xi }(t)$ and $h_{\xi }(t)$ are $Sim_{GL}(E^{2}_{1})$-invariants, we have (13).

$\Leftarrow :$ Let the equalities (13) hold. Using (13) and (14) , we obtain the equalities (12). Hence the proof follows from Theorem 3.

$\square$

Theorem 5

Let $\xi , \eta :I\rightarrow E^{2}_{1}$ are LPR paths such that $sgn(<\xi (t),\xi (t)>)=sgn(<\eta (t),\eta (t)>)$ for all $t\in I$. Then, $\xi (t){\mathop {\sim }\limits ^{Sim_{L}^{+}(E^{2}_{1})}}\eta (t)$ if and only if

$$\begin{aligned} \left\{ \begin{aligned} f_{\xi }(t)&=f_{\eta }(t), \\ g_{\xi }(t)&=g_{\eta }(t) \end{aligned} \right. \end{aligned}$$

(15)

for all $t\in I$.

In addition, if $\xi (t)$ and $\eta (t)$ are $Sim_{L}^{+}(E^{2}_{1})$-similar, then the unique element F exists in $Sim_{L}^{+}(E^{2}_{1})$ satisfying the equality $\eta (t)=F\xi (t), \forall t\in I$, and F has the following matrix form

$$\begin{aligned} F=\left( \begin{array}{cc} \frac{<\xi (t),\eta (t)>}{<\xi (t),\xi (t)>}&{} \frac{[\xi (t)\; \eta (t)]}{<\xi (t),\xi (t)>}\\ \frac{[\xi (t)\; \eta (t)]}{<\xi (t),\xi (t)>}&{} \frac{<\xi (t),\eta (t)>}{<\xi (t),\xi (t)>} \end{array}\right) . \end{aligned}$$

(16)

Proof

$\Rightarrow :$ Let $\xi (t){\mathop {\sim }\limits ^{Sim_{L}^{+}(E^{2}_{1})}}\eta (t)$. Then, by Proposition 3(iv), there exists $z\in \mathbb {H_{+}^{*}}$ such that $\eta (t)=z\xi (t), \forall t\in I$. From this equality, we obtain $\eta ^{\prime }(t)=z\xi ^{\prime }(t)$ for all $t\in I$. Since $\xi (t)$ and $\eta (t)$ are LPR paths, we have $<\xi (t),\xi (t)>\ne 0$ and $<\eta (t),\eta (t)>\ne 0$ for all $t\in I$. Then $(\xi )^{-1}(t)$ and $(\eta )^{-1}(t)$ exist for all $t\in I$. We put $z_{1}=\xi (t), z_{2}=\xi ^{\prime }(t)$ in Proposition 1, then we obtain

$$\begin{aligned} \frac{\xi ^{\prime }(t)}{\xi (t)}=\frac{<\xi (t), \xi ^{\prime }(t)>+j\left[ \xi (t)\; \xi ^{\prime }(t)\right] }{<\xi (t),\xi (t)>} \end{aligned}$$

(17)

for all $t\in I$. Similarly, We put $z_{1}=\eta (t), z_{2}=\eta ^{\prime }(t)$ in Proposition 1, then we obtain

$$\begin{aligned} \frac{\eta ^{\prime }(t)}{\eta (t)}=\frac{<\eta (t), \eta ^{\prime }(t)>+j\left[ \eta (t)\; \eta ^{\prime }(t)\right] }{<\eta (t),\eta (t)>} \end{aligned}$$

(18)

for all $t\in I$.

The equalities $\eta (t)=z\xi (t)$ and $\eta ^{\prime }(t)=z\xi ^{\prime }(t)$ imply $\frac{\eta ^{\prime }(t)}{\eta (t)}=\frac{\xi ^{\prime }(t)}{\xi (t)}$ for all $t\in I$. From this equality with (17) and (18), we have (15).

$\Leftarrow :$ Let the equalities (15) hold. From Theorem 1, there exists $w\in Sim_{GL}^{+}(E^{2}_{1})$ such that $\eta (t)=w\xi (t)=M_{w}\xi (t)$ for all $t\in I$, where $M_{w}$ is an element of $Sim_{GL}^{+}(E^{2}_{1})$. That is, $\xi (t){\mathop {\sim }\limits ^{Sim_{GL}^{+}(E^{2}_{1})}}\eta (t)$. Now we prove that $w\in Sim_{L}^{+}(E^{2}_{1})$. Since $\eta (t)=w\xi (t)$ for all $t\in I$, we have $w=\eta (t)\xi ^{-1}(t)$. Since $<\xi (t),\xi (t)>\ne 0$ and $<\eta (t),\eta (t)>\ne 0$ for all $t\in I$, $<w,w>=<\eta (t)\xi ^{-1}(t),\eta (t)\xi ^{-1}(t)>=\eta (t)\xi ^{-1}(t)\overline{\eta (t)\xi ^{-1}(t)}= \eta (t)\xi ^{-1}(t)\overline{\eta (t)} \overline{\xi ^{-1}(t)}=\eta (t)\overline{\eta (t)} \xi ^{-1}(t)\overline{\xi ^{-1}(t)}=\frac{<\eta (t),\eta (t)>}{<\xi (t),\xi (t)>}\ne 0$. Since $<w,w>\ne 0$ and $sgn(<\xi (t),\xi (t)>)=sgn(<\eta (t),\eta (t)>)$ for all $t\in I$, we have $<w,w> >0$. That is, $w\in Sim_{L}^{+}(E^{2}_{1})$. The proof of uniqueness of $w\in Sim_{L}^{+}(E^{2}_{1})$ satisfying the equality $\eta (t)=w\xi (t)$ for all $t\in I$ is follows from Theorem 1. The proof is completed. $\square$

Remark 4

Let $G=Sim_{L}^{+}(E^{2}_{1})$ or $G=Sim_{L}(E^{2}_{1})$. If $\xi (t)$ and $\eta (t)$ be I-paths in $E^{2}_{1}$ such that $\xi (t)$ is an LPD path and $\xi (t){\mathop {\sim }\limits ^{G}}\eta (t)$, then $\eta (t)$ is also LPD path. Similarly, it is obvious that if $\xi (t)$ is an LPN path and $\xi (t){\mathop {\sim }\limits ^{G}}\eta (t)$, then $\eta (t)$ is also an LPN path.

Theorem 6

Let $\xi , \eta :I\rightarrow E^{2}_{1}$ are LPDR paths such that $sgn(<\xi (t),\xi (t)>)=sgn(<\eta (t),\eta (t)>)$ for all $t\in I$. Then, $\xi (t){\mathop {\sim }\limits ^{Sim_{L}(E^{2}_{1})}}\eta (t)$ if and only if

$$\begin{aligned} f_{\xi }(t)=f_{\eta }(t) \end{aligned}$$

(19)

for all $t\in I$.

In addition, if $\xi (t)$ and $\eta (t)$ are $Sim_{L}(E^{2}_{1})$-similar, then only two elements $F=F_{1}, F_{2}$ exist in $Sim_{L}(E^{2}_{1})$ satisfying the equality $\eta (t)=F\xi (t), \forall t\in I$. In this case, $F_{1}\in Sim_{L}^{+}(E^{2}_{1})$ and $F_{2}\in Sim_{L}^{-}(E^{2}_{1})$. Here $F_{1}$ can be written in the matrix form (16), and $F_{2}\in Sim_{L}^{-}(E^{2}_{1})$ can be written in the following matrix form

$$\begin{aligned} F_{2}=\left( \begin{array}{cc} \frac{<{\overline{\xi }}(t),\eta (t)>}{<\xi (t),\xi (t)>}&{} -\frac{[{\overline{\xi }}(t)\; \eta (t)]}{<\xi (t),\xi (t)>}\\ \frac{[{\overline{\xi }}(t)\; \eta (t)]}{<\xi (t),\xi (t)>}&{} -\frac{<{\overline{\xi }}(t),\eta (t)>}{<\xi (t),\xi (t)>} \end{array}\right) . \end{aligned}$$

(20)

Proof

It can be proved from Theorems 2 and 5. $\square$

Theorem 7

Let $\xi , \eta :I\rightarrow E^{2}_{1}$ are LPNR paths such that $sgn(<\xi (t),\xi (t)>)=sgn(<\eta (t),\eta (t)>)$ for all $t\in I$. Then, $\xi (t){\mathop {\sim }\limits ^{Sim_{L}(E^{2}_{1})}}\eta (t)$ $\Leftrightarrow$

$$\begin{aligned} \left\{ \begin{aligned} f_{\xi }(t)&=f_{\eta }(t), \\ g_{\xi }^{2}(t)&=g_{\eta }^{2}(t) \end{aligned} \right. \end{aligned}$$

(21)

for all $t\in I$.

In addition, if $\xi (t)$ and $\eta (t)$ are $Sim_{L}(E^{2}_{1})$-similar, then the unique element F exists in $Sim_{L}(E^{2}_{1})$ satisfying the equality $\eta (t)=F\xi (t), \forall t\in I$. In this case, for all $t\in I$, there are only the following cases:

(i):

$g_{\xi }(t)>0$ and $g_{\eta }(t)>0$,

(ii):

$g_{\xi }(t)<0$ and $g_{\eta }(t)<0$,

(iii):

$g_{\xi }(t)>0$ and $g_{\eta }(t)<0$,

(iv):

$g_{\xi }(t)<0$ and $g_{\eta }(t)>0$. For (i) and (ii), $F=F_{1}$, where $F_{1}\in Sim_{L}^{+}(E^{2}_{1})$ and it has the form (16). For (iii) and (iv), $F=F_{2}$, where $F_{2}\in Sim_{L}^{-}(E^{2}_{1})$ and it has the form (20).

Proof

It can be proved from Theorems 3 and Theorem 5. $\square$

Theorem 8

Let $\xi , \eta :I\rightarrow E^{2}_{1}$ are LPNR paths such that $sgn(<\xi (t),\xi (t)>)=sgn(<\eta (t),\eta (t)>)$ for all $t\in I$. Then, $\xi (t){\mathop {\sim }\limits ^{Sim_{L}(E^{2}_{1})}}\eta (t)$ $\Leftrightarrow$

$$\begin{aligned} \left\{ \begin{aligned} f_{\xi }(t)&=f_{\eta }(t), \\ h_{\xi }(t)&=h_{\eta }(t) \end{aligned} \right. \end{aligned}$$

(22)

for all $t\in I$.

In addition, if $\xi (t)$ and $\eta (t)$ are $Sim_{L}(E^{2}_{1})$-similar, then the unique element F exists in $Sim_{L}(E^{2}_{1})$ satisfying the equality $\eta (t)=F\xi (t), \forall t\in I$. In this case, for all $t\in I$, only the cases (i),(ii),(iii),(iv) in Theorem 7 exist:

For (i) and (ii), $F=F_{1}$, where $F_{1}\in Sim_{L}^{+}(E^{2}_{1})$ and it has the form (16).

In cases (iii) and (iv), $F=F_{2}$, where $F_{2}\in Sim_{L}^{-}(E^{2}_{1})$ and it has the form (20).

Proof

It can be proved from Theorems 4 and 5. $\square$

5 Determining a plane path from its global differential invariants

Theorem 9

Let $a_{1}, a_{2}:I\rightarrow {\mathbb {R}}$. Assume that $\xi :I\rightarrow E_{1}^{2}$ be an LPR path satisfying the following equalities

$$\begin{aligned} \left\{ \begin{aligned} f_{\xi }(t)&=a_{1}(t) \\ g_{\xi }(t)&=a_{2}(t). \end{aligned} \right. \end{aligned}$$

(23)

Then $\xi (t)$ has the form

$$\begin{aligned} \xi (t)=c e^{\int ^{t}_{t_{0}}(a_{1}(u)+ja_{2}(u))du}, \end{aligned}$$

(24)

where $c\in {\mathbb {H}}, c\ne 0$ and $t_{0}\in I$. Conversely, every I-path of the form (24) is an LPR path and satisfies (23).

Proof

Assume that $\xi (t)$ be an LPR I-path satisfies the equalities (23). From the identity $z_{2}z_{1}^{-1}=\frac{<z_{1}, z_{2}>}{<z_{1},z_{1}>}+j\frac{[z_{1}\; z_{2}]}{<z_{1},z_{1}>}$ in Proposition 1, for the path $\xi (t)$, we have :

$$\begin{aligned} \xi ^{\prime }(t)(\xi )^{-1}(t)=f_{\xi }(t)+jg_{\xi }(t). \end{aligned}$$

Using this identity and (23), we obtain

$$\begin{aligned} \xi ^{\prime }(t)=(a_{1}(t)+ja_{2}(t))\xi (t). \end{aligned}$$

General solution of this differential equation is

$$\begin{aligned} \xi (t)=c e^{\int ^{t}_{t_{0}}(a_{1}(u)+ja_{2}(u))du}, \end{aligned}$$

where $c\in {\mathbb {H}}$ and $t_{0}\in I$. Since $\xi (t)$ is an LPR I-path, we have $c\ne 0$.

Conversely, let $\xi (t)$ be an I-path in the form (24). Then we have $a_{1}(t)=f_{\xi }(t)$ and $a_{2}(t)=g_{\xi }(t)$. Moreover, $\xi (t)$ is an LPR I-path. $\square$

Example 6

Let $a_{1}(t)=\frac{1-2t^2}{t(1-t^2)}$ and $a_{2}(t)=\frac{1}{1-t^2}$ be two real continuous functions on $I=(1,+\infty )$. Then the solution of (23) has the form:

$$\begin{aligned} \begin{aligned} \xi (t)&=c e^{\int ^{t}_{2}(a_{1}(u)+ja_{2}(u))du}\\&=c e^{\int ^{t}_{2}a_{1}(u)du}\left( cosh\left( \int ^{t}_{2}a_{2}(u)du\right) +jsinh\left( \int ^{t}_{2}a_{2}(u)du\right) \right) \\&=c\frac{t\sqrt{t^2-1}}{2\sqrt{3}}\left( cosh\left( ln\left( \sqrt{\frac{t+1}{3(t-3)}}\right) \right) +jsinh\left( ln\left( \sqrt{\frac{t+1}{3(t-3)}}\right) \right) \right) \\&=c\frac{t\sqrt{t^2-1}}{2\sqrt{3}}\left( \left( \frac{2t-1}{\sqrt{3}\sqrt{t^2-1}}\right) +j\left( \frac{2-t}{\sqrt{3}\sqrt{t^2-1}}\right) \right) \\&=\frac{c}{6}\left( (2t^2-t)+j(2t-t^2)\right) \end{aligned} \end{aligned}$$

where $0\ne c\in {\mathbb {H}}$. It is obvious that $\xi (t)$ is an LPR path.

Corollary 1

Let $a_{1}:I\rightarrow {\mathbb {R}}$. Assume that $\xi :I\rightarrow E_{1}^{2}$ be an LPDR path satisfying the following equality

$$\begin{aligned} f_{\xi }(t)=a_{1}(t). \end{aligned}$$

(25)

Then $\xi (t)$ has the following form

$$\begin{aligned} \xi (t)=c e^{\int ^{t}_{t_{0}}a_{1}(u)du}, \end{aligned}$$

(26)

where $c\in {\mathbb {H}},c\ne 0$ and $t_{0}\in I$. Conversely, every I-path in the form (26) is an LPDR path and satisfies (25).

Proof

It can be proved as a special case of the proof of Theorem 9. $\square$

Corollary 2

Let $a_{1}, a_{3}:I\rightarrow {\mathbb {R}}$. Assume that $\xi :I\rightarrow E_{1}^{2}$ be an LPNR path satisfying equalities

$$\begin{aligned} \left\{ \begin{aligned} f_{\xi }(t)&=a_{1}(t) \\ g_{\xi }^{2}(t)&=a_{3}(t). \end{aligned} \right. \end{aligned}$$

(27)

Then,

(i):

$a_{3}(t)>0$, $\forall t\in I$.

(ii):

For $g_{\xi }(t)>0$, $\forall t\in I$, it has the following form

$$\begin{aligned} \xi (t)=c e^{\int ^{t}_{t_{0}}(a_{1}(u)+j\sqrt{a_{3}(u)})du}, \end{aligned}$$

(28)

where $c\in {\mathbb {H}},c\ne 0$ and $t_{0}\in I$.

(iii):

For $g_{\xi }(t)<0$,$\forall t\in I$, it has the following form

$$\begin{aligned} \xi (t)=c e^{\int ^{t}_{t_{0}}(a_{1}(u)-j\sqrt{a_{3}(u)})du}, \end{aligned}$$

(29)

where $c\in {\mathbb {H}}, c\ne 0$ and $t_{0}\in I$. Conversely, for $a_{3}(t)>0$, $\forall t\in I$, every I-path in the forms (28) and (29) in $E^{2}_{1}$ is an LPDR path and satisfies (27).

Proof

(i):

Let $\xi , \eta :I\rightarrow E^{2}_{1}$ are LPNR paths. Since $\xi (t)$ be an LPNR path in $E^{2}_{1}$, we have $\left[ \xi (t)\;\xi ^{\prime }(t)\right] \ne 0$ and $<\xi (t),\xi (t)>\ne 0$,$\forall t\in I$. Then, we obtain $g_{\xi }(t)=\frac{[ \xi (t)\; \xi ^{\prime }(t)]}{<\xi (t),\xi (t)>}\ne 0$, $\forall t\in I$. So $a_{3}(t)>0$, $\forall t\in I$. Now, since $g_{\xi }(t)=\frac{[ \xi (t)\; \xi ^{\prime }(t)]}{<\xi (t),\xi (t)>}\ne 0$, $\forall t\in I$, we obtain $g_{\xi }(t)=\frac{[ \xi (t)\; \xi ^{\prime }(t)]}{<\xi (t),\xi (t)>}> 0$ or $g_{\xi }(t)=\frac{[ \xi (t)\; \xi ^{\prime }(t)]}{<\xi (t),\xi (t)>}< 0$ for all $t\in I$. Explicitly, for $g_{\xi }(t)> 0$, the equality

$$\begin{aligned} g_{\xi }^{2}(t)=a_{3}(t) \end{aligned}$$

(30)

implies

$$\begin{aligned} g_{\xi }(t)=\sqrt{a_{3}(t)}. \end{aligned}$$

Hence the following system obtain:

$$\begin{aligned} \left\{ \begin{aligned} f_{\xi }(t)&=a_{1}(t), \\ g_{\xi }(t)&=\sqrt{a_{3}(t)}. \end{aligned} \right. \end{aligned}$$

(31)

By Theorem 9, a general solution of (31) is (28) . Similarly, for $g_{\xi }(t)< 0$, the equality (30) implies

$$\begin{aligned} g_{\xi }(t)=-\sqrt{a_{3}(t)}. \end{aligned}$$

Hence the following system obtain:

$$\begin{aligned} \left\{ \begin{aligned} f_{\xi }(t)&=a_{1}(t), \\ g_{\xi }(t)&=-\sqrt{a_{3}(t)}. \end{aligned} \right. \end{aligned}$$

(32)

By Theorem 9, a general solution of (32) is (29) . Conversely, let I-path $\xi (t)$ be in the forms (28) or (29). Then by simply calculations, we have (27). Here, since $a_{3}(t)>0$,$\forall t\in I$, we have $\xi (t)$ is an LPNR path.

$\square$

Corollary 3

Let $a_{1}, a_{4}:I\rightarrow {\mathbb {R}}$. Assume that $\xi :I\rightarrow E_{1}^{2}$ be an LPNR path satisfying equalities

$$\begin{aligned} \left\{ \begin{aligned} f_{\xi }(t)&=a_{1}(t) \\ h_{\xi }(t)&=a_{4}(t). \end{aligned} \right. \end{aligned}$$

(33)

Then,

(i):

$a_{1}^{2}(t)-a_{4}(t)> 0$, $\forall t\in I$.

(ii):

For $g_{\xi }(t)=\frac{[ \xi (t)\; \xi ^{\prime }(t)]}{<\xi (t),\xi (t)>}>0$, $\forall t\in I$, the path can be written in the following form

$$\begin{aligned} \xi (t)=c e^{\int ^{t}_{t_{0}}(a_{1}(u)+j\sqrt{a_{1}^{2}(u)-a_{4}(u)})du}, \end{aligned}$$

(34)

where $c\in {\mathbb {H}}, c\ne 0$ and $t_{0}\in I$.

(iii):

For $g_{\xi }(t)=\frac{[ \xi (t)\; \xi ^{\prime }(t)]}{<\xi (t),\xi (t)>}<0$, $\forall t\in I$, the path can be written in the following form

$$\begin{aligned} \xi (t)=c e^{\int ^{t}_{t_{0}}(a_{1}(u)-j\sqrt{a_{1}^{2}(u)-a_{4}(u)})du}, \end{aligned}$$

(35)

where $c\in {\mathbb {H}}, c\ne 0$ and $t_{0}\in I$. Conversely, for $a_{1}^{2}(t)-a_{4}(t)> 0$ for all $t\in I$, every I-path in the forms (34) and (35) in $E^{2}_{1}$ is an LPDR path and the satisfies (33).

Proof

It can be proved from Theorem 4 and Corollary 2. $\square$

6 Global differential invariants of LPNR curves and their conditions of linear pseudo-similarity

Let $I_{1}=(a, b)$ and $I_{2}=(c, d)$ be two intervals in ${\mathbb {R}}$. The following definition is known in [29].

Definition 7

Let $\xi (t)$ be an $I_{1}$-path and be $\eta (u)$ an $I_{2}$-path in $E^{2}_{1}$. If a $C^{(1)}$-diffeomorphism $\psi :I_{2}\rightarrow I_{1}$ exists such that $\psi ^{\prime } (u)>0$ and $\eta (u)=\xi (\psi (u))$, $\forall u\in I_{2}$, we say that the paths $\xi (t)$ and $\eta (u)$ are S-equivalent. A class of S-equivalent paths in $E^{2}_{1}$ will be called a curve (non-parametrized curve) in $E^{2}_{1}$ and denote it by $\varPhi$. Moreover, we say that a path $\xi \in \varPhi$ is called a parametrization of a curve $\varPhi$.

Let $G=Sim_{GL}^{+}(E^{2}_{1}),Sim_{GL}(E^{2}_{1}),Sim_{L}(E^{2}_{1}),Sim_{L}^{+}(E^{2}_{1})$ and $\varPhi =\{\nu _{\tau },\tau \in \varPi \}$ be a curve , where $\nu _{\tau }$ is a parametrization of $\varPhi$. Then, $F\varPhi =\{F\nu _{\tau },\tau \in \varPi \}$, $\forall F\in G$ is a curve.

Definition 8

Two curves $\varPhi$ and $\varPsi$ are called G -similar if there is some $F\in G$ such that $\varPsi =F\varPhi$.

Definition 9

A curve $\varPhi$ will be called LPNR curve if there is an LPNR path in $\varPhi$.

Proposition 5

Let $\varPhi$ be an LPNR curve and $\xi$ be a parametrizations of $\varPhi$. Then every $\xi$ is an LPNR path.

Proof

It is clear from [24, Proposition 37]. $\square$

Remark 5

In this section, we consider paths and curves which are LPNR.

The arc length of the LPNR I-path $\xi$ from $t=c$ to $t=d$ define as follows:

$\ell _{\xi }(c,d)={\int \limits _{c}^{d}} \vert \frac{[ \xi (t)\; \xi ^{\prime }(t)]}{<\xi (t),\xi (t)>}\vert dt$, for $c,d\in I=(a,b)\subseteq {\mathbb {R}}$ and $c<d$. Then there are the limits $\lim _{c \rightarrow a} \ell _{\xi }(c, d) \le +\infty$ and $\lim _{d \rightarrow b} \ell _{\xi }(c, d) \le +\infty$. These limits denoted by $\ell _{\xi }(a,d)$ and $\ell _{\xi }(c,b)$, resp.

Assume the limits $\ell _{\xi }(a,d)=\lim _{c \rightarrow a} \ell _{\xi }(c, d) \le +\infty$ and $\ell _{\xi }(c,b)=\lim _{d \rightarrow b} \ell _{\xi }(c, d) \le +\infty$ exist. Then the following statements exist:

$(\delta _{1})$:

$0<\ell _{\xi }(a,d)<+\infty , \quad 0<\ell _{\xi }(c,b)<+\infty .$

$(\delta _{2})$:

$0<\ell _{\xi }(a,d)<+\infty , \quad \ell _{\xi }(c,b)=+\infty .$

$(\delta _{3})$:

$\ell _{\xi }(a,d)=+\infty ,\qquad 0<\ell _{\xi }(c,b)<+\infty .$

$(\delta _{4})$:

$\ell _{\xi }(a,d)=+\infty ,\qquad \ell _{\xi }(c,b)=+\infty .$

Assume that the case $(\delta _{1})$ or $(\delta _{2})$ satisfies for some $c,d\in I$. Then $l=\ell _{\xi }(a,d)+\ell _{\xi }(c,b)-\ell _{\xi }(c,d)$, where $0\le l \le +\infty$. In this case, the linear pseudo-similarity type of the path $\xi$ is (0, l). The cases $(\delta _{3})$ and $(\delta _{4})$ are independent of the choice of c, d in I. In these cases, the linear pseudo-similarity type of the path $\xi$ is $(-\infty ,0)$ and $(-\infty ,+\infty )$, resp. All linear pseudo-similarity types of the paths are (0, l), where $l<+\infty$, $(0,+\infty )$, $\left( -\infty ,0\right)$, and $\left( -\infty ,+\infty \right)$. The linear pseudo-similarity type of a path $\xi$ will be denoted by $L_{\xi }$.

The proofs of the following propositions are similar to proofs of propositions in [24, Section 6].

Proposition 6

(i):

If $\xi$ and $\eta (t)$ are $Sim_{GL}(E^{2}_{1})$-similar, then $L_{\xi }=L_{\eta }$.

(ii):

If $\xi ,\eta \in \varPhi$, then $L_{\xi }=L_{\eta }$.

According to the group $Sim_{GL}(E^{2}_{1})$, the linear pseudo-similarity type of an LPNR path $\xi \in \varPhi$ is called the linear pseudo-similarity type of the LPNR curve $\varPhi$ and it denoted by $L_{\varPhi }$.

Proposition 7

If two curves $\varPhi$ and $\varPsi$ are $Sim_{GL}(E^{2}_{1})$-similar, then $L_{\varPhi }=L_{\varPsi }$.

For all linear pseudo-similarity types of the group $Sim_{GL}(E^{2}_{1})$, we define the function $s_{\xi }(t)$ for an LPNR I-path $\xi$, where $I=(a,b)$, as follows:

(i):

$s_{\xi }(t)=\ell _{\xi }(a, t)$ for $L_{\xi }=(0,l)$, where $l\le +\infty$.

(ii):

$s_{\xi }(t)=-\ell _{\xi }(t,b)$ for $L_{\xi }=(-\infty ,0)$.

(iii):

For every interval $I=(a, b)$ of the line ${\mathbb {R}}$, we choose a fixed point and denote it by $x_{I}$ . For $I=(-\infty ,+\infty )$, we choose $x_{I}=0$. We put $s_{\xi }(t)=\ell _{\xi }(x_{I},t)$ for the interval I.

Since $\xi$ is an LPNR path, $\dfrac{ds_{\xi }(t)}{dt}>0$. Clearly, the function $s_{\xi }(t)$ has an inverse function $t_{\xi }(s)$ and the domain of $t_{\xi }(s)$ is $L_{\xi }$.

The proofs of the following Propositions 8, 9 are similar to proofs of propositions in [29].

Proposition 8

Let $I=(a_{1},b_{1})$ and $J=(a_{2},b_{2})$. For I-path $\xi$ and for all $F\in Sim_{GL}(E^{2}_{1})$, the following statements hold:

(i):

$s_{F\xi }(t)=s_{\xi }(t)$ and $t_{F\xi }(s)=t_{\xi }(s)$ for all $t \in I$, for all $s\in L_{\xi }$ and all $F\in Sim_{GL}(E^{2}_{1})$.

(ii):

For any $C^{(1)}$-diffeomorphism with $\psi ^{\prime }(r) >0$ for all $r\in I$, the following equalities hold: $s_{\xi (\psi )}(r)=s_{\xi }(\psi (r))+a_{0}, \forall r\in I$, and $\psi (t_{\xi (\psi )}(s+a_{0}))=t_{\xi }(s), \forall s\in L_{\xi }$. Here, $a_{0}=0$ for $L_{\xi } \ne (-\infty ,+\infty )$ and $a_{0}=\ell _{\xi }(\psi (a_{J}),a_{I})$ for $L_{\xi }=(-\infty ,+\infty )$.

According to Proposition 8, we have $\xi (t_{\xi }(s))\in \varPhi$.

Definition 10

(see [29]) $\xi (t_{\xi }(s))\in \varPhi$ is called an invariant parametrization of $\varPhi$.

Denote $P_{\varPhi }$ by the set of all invariant parametrizations of $\varPhi$.

Proposition 9

Let $\xi \in \varPhi$ and $\xi$ be an LPNR I-path, where $I=L_{\varPhi }$. Then the followings are equivalent:

(i):

$\xi \in \varPhi$ is an invariant parametrization.

(ii):

$\vert g_{\xi }(s)\vert =\vert \frac{[ \xi (s)\; \xi ^{\prime }(s)]}{<\xi (s),\xi (s)>}\vert =1$, $\forall s\in L_{\varPhi }$.

(iii):

$s_{\xi }(s)=s$, $\forall s\in L_{\varPhi }$.

For $s_{\xi }(s)=s,\forall s\in L_{\varPhi }$, s will be called an invariant parameter of $\varPhi$.

Suppose I is one of the intervals $(0,l), l<+\infty$; $(0,+\infty )$, $\left( -\infty ,0\right)$ or $\left( -\infty ,+\infty \right)$.

Theorem 10

Let $\xi (s)\in P_{\varPhi }$. Then $\xi (s)$ can be written in the form

$$\begin{aligned} x(s)=ce^{\int ^{s}_{s_{0}}(a_{1}(u)+j)du}, \end{aligned}$$

(36)

or in the form

$$\begin{aligned} x(s)=ce^{\int ^{s}_{s_{0}}(a_{1}(u)-j)du} \end{aligned}$$

(37)

where $c \in {\mathbb {H}}, c\ne 0$ and $s_{0}\in I$ and $a_{1}(u)$ is a real continuous function on I.

Conversely, paths $\xi (s)$ of the forms (36) and (37) are invariant parametrizations of $\varPhi$ for $\forall c \in {\mathbb {H}}, c\ne 0$, $\forall s_{0}\in I$ and arbitrary $a_{1}(u)$ real continuous functions on I.

Proof

The proof follows Theorem 9 and Proposition 9(ii). $\square$

Proposition 10

For the type $L_{\varPhi }\ne (-\infty ,+\infty )$, there is the unique invariant parametrization of $\varPhi$.

Remark 6

For $L_{\varPhi }=(-\infty ,+\infty )$, $P_{\varPhi }$ is infinite and uncountable. Moreover, if $\xi (t)$ is a periodic path then $L_{\xi }=\left( -\infty ,+\infty \right)$.

Proposition 11

Let $\xi \in P_{\varPhi }$ and $L_{\varPhi }=(-\infty ,+\infty )$. Then

$P_{\varPhi }=\left\{ \eta :\eta (s)=\xi (s+u),u\in (-\infty ,+\infty ) \right\}$.

Theorem 11

Let $\varPhi$ and $\varPsi$ are LPNR curves and $\xi \in P_{\varPhi }, \eta \in P_{\varPsi }$ are invariant parametrizations.

(i):

For $L_{\varPhi }=L_{\varPsi }\ne (-\infty , +\infty )$, $\varPhi$ and $\varPsi$ are G-similar if and only if $\xi$ and $\eta$ are G-similar.

(ii):

For $L_{\varPhi }=L_{\varPsi }= (-\infty , +\infty )$, $\varPhi$ and $\varPsi$ are G-similar if and only if $\xi$ and $\eta (\psi _{x})$ are G-similar for some $x\in (-\infty , +\infty )$,where $\psi _{x}(s)=s+x$.

Proof

The proof of this theorem is similar to the proof of [29, Theorem 1] for the groups $G=Sim_{GL}(E^{2}_{1})$ and $G=Sim_{GL}^{+}(E^{2}_{1})$. $\square$

Definition 11

${\mathbb {R}}$-paths $\xi$ and $\eta$ are $\left[ G, (-\infty , +\infty )\right]$-similar provided there exist $g\in G$ and $d\in {\mathbb {R}}$ such that $\eta =g\xi (t+d)$ for all $t\in {\mathbb {R}}$.

Let $\varPhi$ and $\varPsi$ be two LPNR curves, where $L_{\varPhi }=L_{\varPsi }= (-\infty , +\infty )$. Then, Theorem 11 reduces the G-similarity of these curves to $\left[ G, (-\infty , +\infty )\right]$-similarity of paths.

Now, we will give the conditions of the global G-similarity of LPNR curves in terms of the linear pseudo-similarity type and global differential G-invariants of an LPNR curve for the groups $G=Sim_{GL}(E^{2}_{1})$ and $G=Sim_{GL}^{+}(E^{2}_{1})$.

By Theorem 11, G-similarity and uniqueness problems for LPNR curves are reduced to the same problems for invariant parametrizations of LPNR curves only for the case $L_{\varPhi }=L_{\varPsi }\ne (-\infty ,+\infty )$.

Let $\varPhi$ be LPNR curve and $\xi \in P_{\varPhi }$ be an invariant parametrization.

Then we denote the function $sgn(\frac{[ \xi (s)\; \xi ^{\prime }(s)]}{<\xi (s),\xi (s)>}))$ by $\rho _{\xi }(s)$.

Theorem 12

Let $\varPhi$ and $\varPsi$ are LPNR curves such that $L_{\varPhi }\ne (-\infty ,+\infty ), L_{\varPsi }\ne (-\infty ,+\infty )$ and $\xi \in P_{\varPhi }, \eta \in P_{\varPsi }$ are invariant parametrizations. Then

$$\begin{aligned} \varPhi {\mathop {\sim }\limits ^{Sim_{GL}^{+}(E^{2}_{1})}}\varPsi \Leftrightarrow \left\{ \begin{aligned} L_{\varPhi }&=L_{\varPsi } \\ f_{\xi }(s)&=f_{\eta }(s) \\ \rho _{\xi }(s)&=\rho _{\eta }(s) \end{aligned} \right. \end{aligned}$$

(38)

for all $s\in L_{\varPhi }$.

In addition, if $\varPhi$ and $\varPsi$ are $Sim_{GL}^{+}(E^{2}_{1})$-similar, then the unique element F exists in $Sim_{GL}^{+}(E^{2}_{1})$ satisfying the equalities (38), and F can be written in the matrix form (6).

Proof

$\Rightarrow :$ Let $\varPhi$ and $\varPsi$ be $Sim_{GL}^{+}(E^{2}_{1})$-similar. Using Proposition 7, we have $L_{\varPhi }=L_{\varPsi }$. Hence, by $L_{\varPhi }=L_{\varPsi }$ and Theorem 11(i), we have $\xi$ and $\eta$ are $Sim_{GL}^{+}(E^{2}_{1})$-similar. By Theorem 1, for all $s\in L_{\varPhi }$, we obtain $f_{\xi }(s)=f_{\eta }(s)$ and $g_{\xi }(s)=g_{\eta }(s)$.

Since $\xi (s)$ and $\eta (s)$ are LPNR paths, with using $g_{\xi }(s)=g_{\eta }(s)$, we obtain $\rho _{\xi }(s)=\rho _{\eta }(s)$. So the equalities (38) hold.

$\Leftarrow :$ Let $L_{\varPhi }=L_{\varPsi }, f_{\xi }(s)=f_{\eta }(s)$ and $\rho _{\xi }(s)=\rho _{\eta }(s)$ for all $s\in L_{\varPhi }$. Since $\xi \in P_{\varPhi }, \eta \in P_{\varPsi }$, by Proposition 9(ii), we have $\vert g_{\xi }(s)\vert =\vert g_{\eta }(s)\vert =1$ for all $s\in L_{\varPhi }$. Using this equality and $\rho _{\xi }(s)=\rho _{\eta }(s)$, we have $g_{\xi }(s)=g_{\eta }(s)$ for all $s\in L_{\varPhi }$. From $g_{\xi }(s)=g_{\eta }(s)$ and (38), we have (5). By Theorem 1, we obtain that $\xi (s)$ and $\eta (s)$ are $Sim_{GL}^{+}(E^{2}_{1})$-similar. Then, the unique element F exists in $Sim_{GL}^{+}(E^{2}_{1})$ satisfying the equality $\eta (s)=F\xi (s)$. Then F has the form (6). From $\xi \in P_{\varPhi }, \eta \in P_{\varPsi }$, Theorem 11(i) and $\eta (s)=F\xi (s)$, we have $\varPsi =F\varPhi$. $\square$

Theorem 13

Let $\varPhi$ and $\varPsi$ are LPNR curves such that $L_{\varPhi }=L_{\varPsi }= (-\infty ,+\infty )$ and $\xi \in P_{\varPhi }, \eta \in P_{\varPsi }$ are invariant parametrizations. Then $\varPhi {\mathop {\sim }\limits ^{Sim_{GL}^{+}(E^{2}_{1})}}\varPsi \Leftrightarrow$ there exists $s_{1}\in L_{\varPhi }$ satisfying the equalities

$$\begin{aligned} \left\{ \begin{aligned} f_{\xi }(s+s_{1})&=f_{\eta }(s) \\ \rho _{\xi }(s)&=\rho _{\eta }(s) \end{aligned} \right. \end{aligned}$$

(39)

for all $s\in L_{\varPhi }$.

In addition, the unique element F exists in $Sim_{GL}^{+}(E^{2}_{1})$ satisfying the equality $\varPsi =F\varPhi$, and F can be written in the following matrix form

$$\begin{aligned} F=\left( \begin{array}{cc} \frac{<\xi (s+s_{1}), \eta (s)>}{< \xi (s+s_{1}),\xi (s+s_{1})>} &{} \frac{[\xi (s+s_{1})\; \eta (s)]}{< \xi (s+s_{1}),\xi (s+s_{1})>}\\ \frac{[\xi (s+s_{1})\; \eta (s)]}{< \xi (s+s_{1}),\xi (s+s_{1})>} &{} \frac{<\xi (s+s_{1}), \eta (s)>}{< \xi (s+s_{1}),\xi (s+s_{1})>} \end{array}\right) \end{aligned}$$

(40)

.

Proof

$\Rightarrow :$ Let $\varPhi$ and $\varPsi$ be $Sim_{GL}^{+}(E^{2}_{1})$-similar. Using Proposition 7, we have $L_{\varPhi }=L_{\varPsi }$. Hence, by $L_{\varPhi }=L_{\varPsi }$ and Theorem 11(ii), there exits $s_{1}\in (-\infty ,+\infty )$ such that $\xi (s+s_{1})$ and $\eta (s)$ are $Sim_{GL}^{+}(E^{2}_{1})$-similar. By Theorem 1, for all $s\in L_{\varPhi }$, we obtain $f_{\xi }(s+s_{1})=f_{\eta }(s)$ and $g_{\xi }(s+s_{1})=g_{\eta }(s)$.

Since $\xi$ and $\eta$ are LPNR paths, with using $g_{\xi }(s+s_{1})=g_{\eta }(s)$, we obtain $\rho _{\xi }(s)=\rho _{\xi }(s+s_{1})=\rho _{\eta }(s)$. So the equalities (39) hold.

$\Leftarrow :$ Let $L_{\varPhi }=L_{\varPsi }, f_{\xi }(s+s_{1})=f_{\eta }(s)$ and $\rho _{\xi }(s+s_{1})=\rho _{\eta }(s)$ for all $s\in L_{\varPhi }$ and for some $s_{1}\in L_{\varPhi }$. Since $\xi \in P_{\varPhi }, \eta \in P_{\varPsi }$, by Proposition 9, we have $\vert g_{\xi }(s)\vert =\vert g_{\xi }(s+s_{1})\vert =\vert g_{\eta }(s)\vert =1$ for all $s\in L_{\varPhi }$. Using this equality and $\rho _{\xi }(s)=\rho _{\eta }(s)$, we have $g_{\xi }(s+s_{1})=g_{\eta }(s)$ for all $s\in L_{\varPhi }$. From $g_{\xi }(s+s_{1})=g_{\eta }(s)$ and (39), we have (5) for the functions $f_{\xi }(s s_{1}$ and $g_{\xi }(s+s_{1})$. By Theorem 1, we obtain that $\xi$ and $\eta$ are $Sim_{GL}^{+}(E^{2}_{1})$-similar. Then, the unique element F exists in $Sim_{GL}^{+}(E^{2}_{1})$ satisfying the equality $\eta (s)=F\xi (s+s_{1})$. From $\xi \in P_{\varPhi }, \eta \in P_{\varPsi }$, Theorem 11 and $\eta (s)=F\xi (s+s_{1})$, we have $\varPsi =F\varPhi$. $\square$

Theorem 14

Let $\varPhi$ and $\varPsi$ are LPNR curves such that $L_{\varPhi }\ne (-\infty ,+\infty ), L_{\varPsi }\ne (-\infty ,+\infty )$ and $\xi \in P_{\varPhi }, \eta \in P_{\varPsi }$ are invariant parametrizations. Then $\varPhi$ and $\varPsi$ are $Sim_{GL}(E^{2}_{1})$-similar if and only if

$$\begin{aligned} \left\{ \begin{aligned} L_{\varPhi }&=L_{\varPsi }\\ f_{\xi }(s)&=f_{\eta }(s) \end{aligned} \right. \end{aligned}$$

(41)

for all $s\in L_{\varPhi }$.

In addition, if $\varPhi$ and $\varPsi$ are $Sim_{GL}(E^{2}_{1})$-similar, then the unique element F exists in $Sim_{GL}(E^{2}_{1})$ satisfying the equality $\varPsi =F\varPhi$, and

(i):

For $\rho _{\xi }(s)=\rho _{\eta }(s)$, $F=F_{1}\in Sim_{GL}^{+}(E^{2}_{1})$ has the form (6).

(ii):

For $\rho _{\xi }(s)=-\rho _{\eta }(s)$, $F=F_{2}\in Sim_{GL}^{-}(E^{2}_{1})$ has the form (10) .

Proof

It is proved using Theorems 3, 11 and 12. $\square$

Theorem 15

Let $\varPhi$ and $\varPsi$ are LPNR curves such that $L_{\varPhi }=L_{\varPsi }=(-\infty ,+\infty )$ and $\xi \in P_{\varPhi }, \eta \in P_{\varPsi }$ are invariant parametrizations. Then $\varPhi$ and $\varPsi$ are $Sim_{GL}(E^{2}_{1})$-similar if and only if there exists $s_{1}\in L_{\varPhi }$ satisfying the equality

$$\begin{aligned} f_{\xi }(s+s_{1})&=f_{\eta }(s) \end{aligned}$$

(42)

for all $s\in L_{\varPhi }$.

In addition, if $\varPhi$ and $\varPsi$ are $Sim_{L}(E^{2}_{1})$-similar, then the unique element F exists in $Sim_{GL}(E^{2}_{1})$ satisfying the equality $\varPsi =F\varPhi$, then

(i):

For $\rho _{\xi }(s+s_{1})=\rho _{\eta }(s)$, $F=F_{1}\in Sim_{GL}^{+}(E^{2}_{1})$ has the form (40).

(ii):

For $\rho _{\xi }(s+s_{1})=-\rho _{\eta }(s)$, $F=F_{2}\in Sim_{GL}^{-}(E^{2}_{1})$ has the following matrix form

$$\begin{aligned} F_{2}=\left( \begin{array}{cc} \frac{<{\overline{\xi }}(s+s_{1}), \eta (s)>}{<\xi (s+s_{1}), \xi (s+s_{1})>} &{} -\frac{[{\overline{\xi }}(s+s_{1})\; \eta (s)]}{<\xi (s+s_{1}), \xi (s+s_{1})>}\\ \frac{[{\overline{\xi }}(s+s_{1})\; \eta (s)]}{<\xi (s+s_{1}), \xi (s+s_{1})>} &{} -\frac{<{\overline{\xi }}(s+s_{1}), \eta (s)>}{<\xi (s+s_{1}), \xi (s+s_{1})>} \end{array}\right) \end{aligned}$$

.

Proof

It is proved using Theorems 4, 11 and 13. $\square$