The λ-Point Map between Two Legendre Plane Curves

Abstract

The $\lambda$-point map between two Legendre plane curves, which is a map from the plane into the plane, is introduced. The singularity of this map is studied through this paper and many known plane map singularities are realized as special cases of this construction. Precisely, the corank one and corank two singularities of the $\lambda$-point map between two Legendre plane curves are investigated and the geometric conditions for this map to have corank one singularities, such as fold, cusp, swallowtail, lips, and beaks are obtained. Additionally, the geometric conditions for the $\lambda$-point map to have a sharksfin singularity, which is a corank two singularity, are obtained.

1. Introduction

The singularity theory is useful for studying the differential geometry of curves and surfaces and lots of geometric features can be studied from the singularity theory viewpoint (cf. [1,2,3]). One of the main subjects in the singularity theory of smooth maps is the classifications of the singularities of maps germs from the plane into the plane. This is because of its applications in several areas. For the applications of plane maps we refer the reader to [1,2,3,4]. In 1955, Whitney proved that, in general, maps from the plane into the plane have fold and cusp singularities. The classification of maps germ $({\mathbb{R}}^2,0)\to ({\mathbb{R}}^2,0)$ with a corank one singularities was studied by J.H. Rieger in [5]. Some of these singularities are shown in

Table 1. Classification of $({\mathbb{R}}^2,0)\to ({\mathbb{R}}^2,0)$.

Name	Normal Form
fold	$(x,y^2)$
cusp	$(x,xy+y^3)$
lips	$(x,x^{2}y+y^3)$
beaks	$(x,x^{2}y-y^3)$
swallowtail	$(x,xy+y^4)$

. In 2010, K. Saji obtained the criteria for lips, beaks, and swallowtail singularities of smooth maps germ $({\mathbb{R}}^2,0)\to ({\mathbb{R}}^2,0)$ with a corank one singularities.

The criteria for sharksfin and deltoid singularities, which are corank two singularities, of maps germ from the plane into the plane was investigated by Kabata and Saji [6]. In this paper, we introduce the $\lambda$-point map between two Legendre plane curves (Definition 6). Additionally, we study the classification of corank one (respect corank two) singularities of this map. In the beginning, we review some basic definitions and results through the second section which will be used in this paper. In the third section, we give the geometric conditions for the $\lambda$-point map between two Legendre plane curves to have fold, cusp, lips, beaks and swallowtail singularities when ${\gamma }_1$ (respect ${\gamma }_2$) is regular at $s_1=0$ (respect $s_2=0$) (Theorem 2). Additionally, we give the geometric conditions for the $\lambda$-point map between two Legendre plane curves to have fold and beaks singularities when one of the two curves is singular (Theorem 3). In the forth section, we give the geometric conditions for the $\lambda$-point map between two Legendre plane curves with a corank two singularity to have sharksfin singularity (Theorem 6). In the final section, we give three examples to illustrate some obtained results in this research. Precisely, for corank one singularity we give two examples for the $\lambda$-point map between two Legendre plane curves to have fold and beaks singularities and the third example deals with the sharksfin, which is a corank two, singularity of this map.

Throughout this paper, the definitions and results are provided for smooth maps.

2. Preliminaries

In this section, we review some definitions and results for Legendre plane curves and the singularity of maps from the plane into the plane. Additionally, we introduce the $\lambda$-point map between two Legendre plane curves.

Definition 1.

Let I be an interval of $\mathbb{R}$. The map $(\gamma ,\omega ):I\to {\mathbb{R}}^2\times S^1$ is called a Legendre curve if ${\gamma }^{\prime }\left(s\right)·\omega \left(s\right)=0$ for all $s\in I$, where $S^1$ is the unit circle and $\omega :I\to S^1$ is a smooth unit vector field.

The Frenet formula of a Legendre plane curve is given by

$${\omega }^{\prime }\left(s\right)=\ell \left(s\right)\mu \left(s\right),$$

$${\mu }^{\prime }\left(s\right)=-\ell \left(s\right)\omega \left(s\right),$$

where prime is the derivative with respect to the parameter s, $\ell \left(s\right)={\omega }^{\prime }\left(s\right)·\mu \left(s\right)$ and $\mu \left(s\right)=J\left(\omega \right(s\left)\right)$, such that J is the counterclockwise rotation by ${\displaystyle \frac{\pi }{2}}$. We call the pair $\left\{\omega \right(s),\mu (s\left)\right\}$ a moving frame of a Legendre plane curve $\gamma$. Furthermore, there exists a smooth function $\beta \left(s\right)$, such that $\beta \left(s\right)={\gamma }^{\prime }\left(s\right)·\mu \left(s\right)$. We call the pair $(\ell (s),\beta (s\left)\right)$ the curvature of this curve. For more information about the Legendre plane curves, we refer the reader to [7,8,9,10,11,12].

Definition 2.

A singular point of a map germ $h:(U\subseteq {\mathbb{R}}^l,0)\to ({\mathbb{R}}^m,0)$ is a point $p\in U$ which satisfies that $rank\left(dh\right)\left(p\right)<min(l,m)$, where $dh$ is the Jacobin matrix of h.

The set of singular points of h is denoted by $S\left(h\right)\subset {\mathbb{R}}^l$. We say that $q\in S\left(h\right)$ is of corank $\alpha$ if the rank of the Jacobin matrix of h at q is equal to $min(l,m)-\alpha$.

Definition 3.

Two map germs $h_1,h_2:({\mathbb{R}}^l,0)\to ({\mathbb{R}}^m,0)$ are said to be $\mathcal{A}$-equivalent if there exist smooth diffeomorphisms ${\phi }_1:({\mathbb{R}}^l,0)\to ({\mathbb{R}}^l,0)$ and ${\phi }_2:({\mathbb{R}}^m,0)\to ({\mathbb{R}}^m,0)$, such that the following diagram commutes.

In other words, $h_2\circ {\phi }_1={\phi }_2\circ h_1$ holds.

Definition 4

([13]). For a positive integer n, the n-jet of a differentiable map $\mathcal{F}$ at point p is the Taylor expansion at p truncated to the degree n which is denoted by $j^n\mathcal{F}$.

Definition 5

([14]). A map germ $h:(U\subseteq {\mathbb{R}}^l,q)\to ({\mathbb{R}}^m,0)$ is said to be n-determined whenever $j^{n}h\left(q\right)=j^{n}k\left(q\right)$ for any $k:(U\subseteq {\mathbb{R}}^l,q)\to ({\mathbb{R}}^m,0)$, then k is $\mathcal{A}$-equivalent to h.

For example, the lips and beaks are three-determined, whereas swallowtail is four-determined (see [5]). Let $h:(U\subseteq {\mathbb{R}}^2,q)\to ({\mathbb{R}}^2,0)$ be a map germ with a corank one singularity at a point $q\in U$. Then there exist a neighborhood C of q and a non-zero vector field (null vector field) $\rho$, such that $dh\left(\rho \right)\left(q\right)=0$ holds for any $q\in S\left(h\right)\cap C$. Let $(s_1,s_2)$ be coordinates of U. We define the discriminant function $\Omega$ of h by $\Omega (s_1,s_2)=det\left({\displaystyle \frac{\partial h}{\partial s_1}},{\displaystyle \frac{\partial h}{\partial s_2}}\right)(s_1,s_2).$ A singular point $q\in S\left(h\right)$ is a non-degenerate if $d\Omega \left(q\right)\ne 0$ and it is a degenerate if $d\Omega \left(q\right)=0$. Note that a non-degenerate singular point is of corank one. The normal forms of some simple generic singularities of corank one of maps from the plane into the plane are shown in Table 1.

We end this section by introducing the $\lambda$-point map between two Legendre plane curves.

Definition 6.

Let${\gamma }_i:I_i\subseteq \mathbb{R}\to {\mathbb{R}}^2\times {\mathbb{S}}^1$ $(i=1,2)$be two Legendre plane curves. The λ-point map between${\gamma }_1$ and ${\gamma }_2$ is a map $M:U\subseteq {\mathbb{R}}^2\to {\mathbb{R}}^2$ defined by

$$M(s_1,s_2)=(1-\lambda ){\gamma }_1\left(s_1\right)+\lambda {\gamma }_2\left(s_2\right),$$

where $U=I_1\times I_2$ and $\lambda \in (0,1).$

Note that M in the above definition is more general than the midpoint map of a smooth plane curve $\gamma$ which is defined by $M(s_1,s_2)={\displaystyle \frac{1}{2}}({\gamma }_1\left(s_1\right)+{\gamma }_2\left(s_2\right))$, where ${\gamma }_1$ and ${\gamma }_2$ are two smooth parts of $\gamma$ parametrized by $s_1$ and $s_2$, respectively. For more details on the midpoint map, we refer the reader to [15].

3. Classification of Corank One Singularities of $\mathbf{\lambda }$-Point Map between Two Legendre Plane Curves

The classification of corank one singularities of the $\lambda$-point map between two Legendre curves in plane breaks naturally into two cases depending on the regularity of ${\gamma }_1$ and ${\gamma }_2$.

Lemma 1.

Let M be the λ-point map between two Legendre plane curves ${\gamma }_1$ and ${\gamma }_2$. Then M is parametrized by a corank one singularity at $(s_{1_0},s_{2_0})$ if, and only if, one of the following cases holds:

• ${\beta }_1\left(s_{1_0}\right)\ne 0$, ${\beta }_2\left(s_{2_0}\right)\ne 0$ and ${\omega }_1\left(s_{1_0}\right)=\pm {\omega }_2\left(s_{2_0}\right)$.
• ${\beta }_1\left(s_{1_0}\right)\ne 0$ or ${\beta }_2\left(s_{2_0}\right)\ne 0$.

First, we will review the criteria for the fold, the cusp, the beaks, the lips and the swallowtail singularities which are the generic singularities of corank one of maps from the plane into the plane.

Theorem 1

([14]). Let $h:(U\subseteq {\mathbb{R}}^2,q)\to ({\mathbb{R}}^2,0)$ be a map germ and $q\in S\left(h\right)$. Then at q

• h is $\mathcal{A}$-equivalent to fold if, and only if, $\rho \Omega \left(q\right)\ne 0$.
• h is $\mathcal{A}$-equivalent to cusp if, and only if, q is non-degenerate, $\rho \Omega \left(q\right)=0$ and ${\rho }^2\Omega \left(q\right)\ne 0$.
• h is $\mathcal{A}$-equivalent to lips if, and only if, q is of corank one, $d\Omega \left(q\right)=0$ and Ω has a Morse type critical point of index 0 or 2 at q, namely $det\left({\mathcal{H}}_{\Omega }\left(q\right)\right)>0$.
• h is $\mathcal{A}$-equivalent to beaks if, and only if, q is of corank one, $d\Omega \left(q\right)=0$ and Ω has a Morse type critical point of index 1 at q, namely $det\left({\mathcal{H}}_{\Omega }\left(q\right)\right)<0$ and ${\rho }^2\Omega \left(q\right)\ne 0$.
• h is $\mathcal{A}$-equivalent to swallowtail if, and only if, $d\Omega \left(q\right)\ne 0$, $\rho \Omega \left(q\right)={\rho }^2\Omega \left(q\right)=0$ and ${\rho }^3\Omega \left(q\right)\ne 0$.

The expression $\rho \Omega$ means the directional derivative of $\Omega$ in the direction of the vector field $\rho$ and ${\mathcal{H}}_{\Omega }$ is the Hessian matrix of $\Omega$.

3.1. The Case When ${\mathbf{\beta }}_{\mathbf{1}}\left(\mathbf{0}\right)\ne \mathbf{0}$, ${\mathbf{\beta }}_{\mathbf{2}}\left(\mathbf{0}\right)\ne \mathbf{0}$ and ${\mathbf{\omega }}_{\mathbf{1}}\left(\mathbf{0}\right)=\pm {\mathbf{\omega }}_{\mathbf{2}}\left(\mathbf{0}\right)$

In this section, we study the corank one singularity of the $\lambda$-point map between two Legendre plane curves when ${\gamma }_1$ (respect ${\gamma }_2$) is regular at $s_1=0$ (respect $s_2=0$), that means ${\beta }_1\left(0\right)\ne 0,$${\beta }_2\left(0\right)\ne 0$, and ${\omega }_1\left(0\right)=\pm {\omega }_2\left(0\right)$.

Lemma 2.

Let M be the λ-point map between two Legendre plane curves ${\gamma }_1$ and ${\gamma }_2$, such that ${\beta }_1\left(0\right)\ne 0,$${\beta }_2\left(0\right)\ne 0$ and ${\omega }_1\left(0\right)=\pm {\omega }_2\left(0\right)$. The singular point $(0,0)$ is a non-degenerate if, and only if, ${\ell }_i\left(0\right)\ne 0$, $i=1,2$.

Proof. 

The proof of this lemma is obvious. □

We now give the main result of this section.

Theorem 2.

Let M be the λ-point map between two Legendre plane curves ${\gamma }_1$ and ${\gamma }_2$. Suppose that ${\beta }_1\left(0\right)\ne 0$, ${\beta }_2\left(0\right)\ne 0$ and ${\omega }_1\left(0\right)=\pm {\omega }_2\left(0\right)$. Then at $(0,0)$

• M is $\mathcal{A}$-equivalent to fold if, and only if, $\left({\displaystyle \frac{\lambda }{1-\lambda }}\right){\displaystyle \frac{{\beta }_2\left(0\right)}{{\beta }_1\left(0\right)}}{\ell }_1\left(0\right)\ne \mp {\ell }_2\left(0\right)$.
• M is $\mathcal{A}$-equivalent to cusp if, and only if, $\left({\displaystyle \frac{\lambda }{1-\lambda }}\right){\displaystyle \frac{{\beta }_2\left(0\right)}{{\beta }_1\left(0\right)}}{\ell }_1\left(0\right)=\mp {\ell }_2\left(0\right)$ and
  ${\beta }_2{\left({\displaystyle \frac{{\beta }_1}{{\ell }_1}}\right)}^{\prime }\ne {\beta }_1{\left({\displaystyle \frac{{\beta }_2}{{\ell }_2}}\right)}^{\prime }$ at $(0,0)$.
• M is $\mathcal{A}$-equivalent to lips if, and only if, ${\ell }_i\left(0\right)=0$, $i=1,2$, and ${\ell }_1^{\prime }\left(0\right){\ell }_2^{\prime }\left(0\right)<0$.
• M is $\mathcal{A}$-equivalent to beaks if, and only if, ${\ell }_i\left(0\right)=0$, $i=1,2$, ${\ell }_1^{\prime }\left(0\right){\ell }_2^{\prime }\left(0\right)>0$ and ${\left({\displaystyle \frac{\lambda }{1-\lambda }}\right)}^2{\displaystyle \frac{{\beta }_2^2\left(0\right)}{{\beta }_1^2\left(0\right)}}{\ell }_1^{\prime }\left(0\right)\ne {\ell }_2^{\prime }\left(0\right)$.
• M is $\mathcal{A}$-equivalent to swallowtail if, and only if, $\left({\displaystyle \frac{\lambda }{1-\lambda }}\right){\displaystyle \frac{{\beta }_2\left(0\right)}{{\beta }_1\left(0\right)}}{\ell }_1\left(0\right)=\mp {\ell }_2\left(0\right)$, ${\ell }_i\left(0\right)\ne 0$$(i=1,2)$, ${\beta }_2{\left({\displaystyle \frac{{\beta }_1}{{\ell }_1}}\right)}^{\prime }={\beta }_1{\left({\displaystyle \frac{{\beta }_2}{{\ell }_2}}\right)}^{\prime }$ at $(0,0)$ and
  $$\begin{gathered} -{\displaystyle \frac{{\beta }_1^{″}{\beta }_2{\ell }_2^3}{{\ell }_1^2}}-3{\displaystyle \frac{{\beta }_1^{\prime }{\beta }_2^{\prime }{\ell }_2^2}{{\ell }_1}}+{\beta }_1{\beta }_2^{″}{\ell }_2-3{\displaystyle \frac{{\beta }_1^{\prime }{\beta }_2{\ell }_2{\ell }_2^{\prime }}{{\ell }_1}}+{\displaystyle \frac{{\beta }_1{\beta }_2{\ell }_1^{″}{\ell }_2^3}{{\ell }_1^3}}+6{\displaystyle \frac{{\beta }_1{\beta }_2^{\prime }{\ell }_1^{\prime }{\ell }_2^2}{{\ell }_1^2}} \\ -3{\beta }_1{\beta }_2^{\prime }{\ell }_2^{\prime }-{\beta }_1{\beta }_2{\ell }_2^{″}+3{\displaystyle \frac{{\beta }_1{\left({\beta }_2^{\prime }\right)}^2{\ell }_2}{{\beta }_2}}\ne 0\mathrm{at}(0,0). \end{gathered}$$

Proof. 

Let $M(s_1,s_2)=(1-\lambda )\gamma \left(s_1\right)+\gamma \left(s_2\right)$ be the $\lambda$-point map between two Legendre plane curves. Suppose that ${\beta }_1\left(0\right)\ne 0$, ${\beta }_2\left(0\right)\ne 0$ and ${\omega }_1\left(0\right)=-{\omega }_2\left(0\right)$.

We choose vector field $\rho$, such that $dM{|}_{(0,0)}\left(\rho \right)=0$, thus we take $\rho ={\displaystyle \frac{\lambda {\beta }_2\left(0\right)}{(1-\lambda ){\beta }_1\left(0\right)}}{\displaystyle \frac{\partial }{\partial s_1}}+{\displaystyle \frac{\partial }{\partial s_2}}$. We can prove that $\Omega (s_1,s_2)=-{\beta }_1\left(s_1\right){\beta }_2\left(s_2\right){\omega }_1\left(s_1\right)·{\mu }_2\left(s_2\right)$.

For simplicity we omit $s_1$ and $s_2$, hence $\Omega =-{\beta }_1{\beta }_2{\omega }_1·{\mu }_2$. By a straightforward calculations at $(0,0)$, we have

$${\displaystyle \frac{\partial \Omega }{\partial s_1}}{|}_{(0,0)}={\beta }_1\left(0\right){\beta }_2\left(0\right){\ell }_1\left(0\right),{\displaystyle \frac{\partial \Omega }{\partial s_2}}{|}_{\left(0.0\right)}=-{\beta }_1\left(0\right){\beta }_2\left(0\right){\ell }_2\left(0\right),$$

$$\rho \Omega {|}_{(0,0)}={\beta }_2\left(0\right)\left({\displaystyle \frac{\lambda }{1-\lambda }}{\beta }_2\left(0\right){\ell }_1\left(0\right)-{\beta }_1\left(0\right){\ell }_2\left(0\right)\right),$$

$$\begin{array}{cc}\hfill {\rho }^2\Omega {|}_{(0,0)}& ={\displaystyle \frac{-2\lambda }{1-\lambda }}{\displaystyle \frac{{\beta }_1^{\prime }\left(0\right){\beta }_2^2\left(0\right){\ell }_2\left(0\right)}{{\beta }_1\left(0\right)}}+{\displaystyle \frac{{\lambda }^2}{{(1-\lambda )}^2}}{\displaystyle \frac{{\beta }_2^3\left(0\right){\ell }_1^{\prime }\left(0\right)}{{\beta }_1\left(0\right)}}+{\displaystyle \frac{3\lambda }{1-\lambda }}{\beta }_2\left(0\right){\beta }_2^{\prime }\left(0\right){\ell }_1\left(0\right)\hfill \\ & +{\displaystyle \frac{{\lambda }^2}{{(1-\lambda )}^2}}{\displaystyle \frac{{\beta }_1^{\prime }\left(0\right){\beta }_2^3\left(0\right){\ell }_1\left(0\right)}{{\beta }_1^2\left(0\right)}}-2{\beta }_1\left(0\right){\beta }_2^{\prime }\left(0\right){\ell }_2\left(0\right)-{\beta }_1\left(0\right){\beta }_2\left(0\right){\ell }_2^{\prime }\left(0\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\rho }^3\Omega {|}_{(0,0)}=& {\displaystyle \frac{2{\lambda }^3}{{(1-\lambda )}^3}}{\displaystyle \frac{{\beta }_1^{″}\left(0\right){\beta }_2^4\left(0\right){\ell }_1\left(0\right)}{{\beta }_1^3\left(0\right)}}+{\displaystyle \frac{-3{\lambda }^3}{{(1-\lambda )}^3}}{\displaystyle \frac{{\left({\beta }_1^{\prime }\right)}^2\left(0\right){\beta }_2^4\left(0\right){\ell }_1\left(0\right)}{{\beta }_1^4\left(0\right)}}+{\displaystyle \frac{6{\lambda }^2}{{(1-\lambda )}^2}}{\displaystyle \frac{{\beta }_1^{\prime }\left(0\right){\beta }_2^{\prime }\left(0\right){\beta }_2^2\left(0\right){\ell }_1\left(0\right)}{{\beta }_1^2\left(0\right)}}\hfill \\ & +{\displaystyle \frac{3{\lambda }^2}{{(1-\lambda )}^2}}{\displaystyle \frac{{\beta }_2^3\left(0\right){\ell }_1^2\left(0\right){\ell }_2\left(0\right)}{{\beta }_1\left(0\right)}}-{\displaystyle \frac{{\lambda }^3}{{(1-\lambda )}^3}}{\displaystyle \frac{{\beta }_2^4\left(0\right){\ell }_1^3\left(0\right)}{{\beta }_1^2\left(0\right)}}-{\displaystyle \frac{3\lambda }{1-\lambda }}{\beta }_2^2\left(0\right){\ell }_1\left(0\right){\ell }_2^2\left(0\right)\hfill \\ & +{\displaystyle \frac{4\lambda }{1-\lambda }}{\beta }_2\left(0\right){\beta }_2^{″}\left(0\right){\ell }_1\left(0\right)-{\displaystyle \frac{3{\lambda }^2}{{(1-\lambda )}^2}}{\displaystyle \frac{{\beta }_1^{″}\left(0\right){\beta }_2^3\left(0\right){\ell }_2\left(0\right)}{{\beta }_1^2\left(0\right)}}+{\displaystyle \frac{3{\lambda }^2}{{(1-\lambda )}^2}}{\displaystyle \frac{{\left({\beta }_1^{\prime }\right)}^2\left(0\right){\beta }_2^3\left(0\right){\ell }_2\left(0\right)}{{\beta }_1^3\left(0\right)}}\hfill \\ & -{\displaystyle \frac{3\lambda }{1-\lambda }}{\displaystyle \frac{{\beta }_1^{\prime }\left(0\right){\beta }_2^2\left(0\right){\ell }_2^{\prime }\left(0\right)}{{\beta }_1\left(0\right)}}-{\displaystyle \frac{9\lambda }{1-\lambda }}{\displaystyle \frac{{\beta }_2\left(0\right){\beta }_1^{\prime }\left(0\right){\beta }_2^{\prime }\left(0\right){\ell }_2\left(0\right)}{{\beta }_1\left(0\right)}}+{\displaystyle \frac{{\lambda }^3}{{(1-\lambda )}^3}}{\displaystyle \frac{{\beta }_2^4\left(0\right){\ell }_1^{″}\left(0\right)}{{\beta }_1^2\left(0\right)}}\hfill \\ & +{\displaystyle \frac{6{\lambda }^2}{{(1-\lambda )}^2}}{\displaystyle \frac{{\beta }_2^2\left(0\right){\beta }_2^{\prime }\left(0\right){\ell }_1^{\prime }\left(0\right)}{{\beta }_1\left(0\right)}}+{\displaystyle \frac{3\lambda }{1-\lambda }}{\left({\beta }_2^{\prime }\right)}^2\left(0\right){\ell }_1\left(0\right)+{\beta }_1\left(0\right){\beta }_2\left(0\right){\ell }_2^3\left(0\right)\hfill \\ & -3{\beta }_1\left(0\right){\beta }_2^{″}\left(0\right){\ell }_2\left(0\right)-3{\beta }_1\left(0\right){\beta }_2^{\prime }\left(0\right){\ell }_2^{\prime }\left(0\right)-{\beta }_1\left(0\right){\beta }_2\left(0\right){\ell }_2^{″}\left(0\right),\hfill \end{array}$$

and

$${\mathcal{H}}_{\Omega }(0,0)=\left(\begin{array}{ccc}2{\beta }_1^{\prime }\left(0\right){\beta }_2\left(0\right){\ell }_1\left(0\right)+{\beta }_1\left(0\right){\beta }_2\left(0\right){\ell }_1^{\prime }\left(0\right)& & {\beta }_1\left(0\right){\beta }_2^{\prime }\left(0\right){\ell }_1\left(0\right)-{\beta }_1^{\prime }\left(0\right){\beta }_2\left(0\right){\ell }_2\left(0\right)\\ \\ \\ {\beta }_1\left(0\right){\beta }_2^{\prime }\left(0\right){\ell }_1\left(0\right)-{\beta }_1^{\prime }\left(0\right){\beta }_2\left(0\right){\ell }_2\left(0\right)& & -(2{\beta }_1\left(0\right){\beta }_2^{\prime }\left(0\right){\ell }_2\left(0\right)+{\beta }_1\left(0\right){\beta }_2\left(0\right){\ell }_2^{\prime }\left(0\right))\end{array}\right).$$

Hence,

$$\begin{array}{cc}\hfill det\left({\mathcal{H}}_{\Omega }(0,0)\right)=& -(2{\beta }_1^{\prime }\left(0\right){\beta }_2\left(0\right){\ell }_1\left(0\right)+{\beta }_1\left(0\right){\beta }_2\left(0\right){\ell }_1^{\prime }\left(0\right))(2{\beta }_1\left(0\right){\beta }_2^{\prime }\left(0\right){\ell }_2\left(0\right)+{\beta }_1\left(0\right){\beta }_2\left(0\right){\ell }_2^{\prime }\left(0\right))\hfill \\ & -{({\beta }_1\left(0\right){\beta }_2^{\prime }\left(0\right){\ell }_1\left(0\right)-{\beta }_1^{\prime }\left(0\right){\beta }_2\left(0\right){\ell }_2\left(0\right))}^2.\hfill \end{array}$$

Therefore, applying Theorem 1 the results of this theorem hold. By a similar argument, we prove the case when ${\omega }_1\left(0\right)={\omega }_2\left(0\right)$ by choosing

$$\rho ={\displaystyle \frac{\lambda {\beta }_2\left(0\right)}{(1-\lambda ){\beta }_1\left(0\right)}}{\displaystyle \frac{\partial }{\partial s_1}}-{\displaystyle \frac{\partial }{\partial s_2}}.$$

□

Note that The results in [15] related to the midpoint map are special cases of Theorem 2.

3.2. The Case When ${\mathbf{\beta }}_{\mathbf{1}}\left(\mathbf{0}\right)=\mathbf{0}$ and ${\mathbf{\beta }}_{\mathbf{2}}\left(\mathbf{0}\right)\ne \mathbf{0}$

In this section, we study the case when one of the two curves is singular. Precisely, ${\beta }_1\left(0\right)=0$ and ${\beta }_2\left(0\right)\ne 0$. We give the conditions for the $\lambda$-point map between two Legendre plane curves to have fold and beaks singularities in the following theorem.

Theorem 3.

Let M be the λ-point map between two Legendre plane curves ${\gamma }_1$ and ${\gamma }_2$, such that ${\beta }_1\left(0\right)=0$ and ${\beta }_2\left(0\right)\ne 0$.

• If ${\omega }_1\left(0\right)\ne \pm {\omega }_2\left(0\right)$, then at $(0,0)$ M is $\mathcal{A}$-equivalent to fold if, and only if, ${\beta }_1^{\prime }\left(0\right)\ne 0$.
• If ${\omega }_1\left(0\right)=\pm {\omega }_2\left(0\right)$, then at $(0,0)$ M is $\mathcal{A}$-equivalent to beaks if, and only if, ${\beta }_1^{\prime }\left(0\right)\ne 0$ and ${\ell }_i\left(0\right)\ne 0$, $i=1,2$.

Proof. 

Let $M(s_1,s_2)=(1-\lambda ){\gamma }_1\left(s_1\right)+{\gamma }_2\left(s_2\right)$ be the $\lambda$-point map between two Legendre plane curves. Suppose that ${\beta }_1\left(0\right)=0$ and ${\beta }_2\left(0\right)\ne 0$. We prove this theorem by using Theorem 1. Now we choose vector field $\rho$, such that $dM{|}_{(0,0)}\left(\rho \right)=0$, so we take $\rho ={\displaystyle \frac{\partial }{\partial s_1}}$.

Then, by a straightforward calculations, we have

$${\displaystyle \frac{\partial \Omega }{\partial s_1}}=-{\beta }_1^{\prime }{\beta }_2{\omega }_1·{\mu }_2-{\beta }_1{\beta }_2{\ell }_1{\omega }_1·{\omega }_2,{\displaystyle \frac{\partial \Omega }{\partial s_2}}=-{\beta }_1{\beta }_2^{\prime }{\omega }_1·{\mu }_2+{\beta }_1{\beta }_2{\ell }_2{\omega }_1·{\omega }_2,$$

$$\rho \Omega =-{\beta }_1^{\prime }{\beta }_2{\omega }_1·{\mu }_2-{\beta }_1{\beta }_2{\ell }_1{\omega }_1·{\omega }_2,{\rho }^2\Omega =({\beta }_1{\beta }_2{\ell }_1^2-{\beta }_1^{″}{\beta }_2){\omega }_1·{\mu }_2-(2{\beta }_1^{\prime }{\beta }_2{\ell }_1+{\beta }_1{\beta }_2{\ell }_1^{\prime }){\omega }_1.{\omega }_2,$$

$${\rho }^3\Omega =(3{\beta }_1^{\prime }{\beta }_2{\ell }_1^2+3{\beta }_1{\beta }_2{\ell }_1{\ell }_1^{\prime }-{\beta }_1^{‴}{\beta }_2){\omega }_1·{\mu }_2+({\beta }_1{\beta }_2{\ell }_1^3-{\beta }_1{\beta }_2{\ell }_1^{″}-3{\beta }_1^{″}{\beta }_2{\ell }_1-3{\beta }_1^{\prime }{\beta }_2{\ell }_1^{\prime }){\omega }_1·{\omega }_2$$

and

$$\begin{array}{cc}\hfill det\left({\mathcal{H}}_{\Omega }\right)=& \left(({\beta }_1{\beta }_2{\ell }_1^2-{\beta }_1^{″}{\beta }_2){\omega }_1·{\mu }_2-(2{\beta }_1^{\prime }{\beta }_2{\ell }_1+{\beta }_1{\beta }_2{\ell }_1^{\prime }){\omega }_1·{\omega }_2\right)(({\beta }_1{\beta }_2{\ell }_2^2-{\beta }_1{\beta }_2^{″}){\omega }_1·{\mu }_2\hfill \\ & +(2{\beta }_1{\beta }_2^{\prime }{\ell }_2+{\beta }_1{\beta }_2{\ell }_2^{\prime }){\omega }_1·{\omega }_2)-{\left(-({\beta }_1^{\prime }{\beta }_2^{\prime }+{\beta }_1{\beta }_2{\ell }_1{\ell }_2){\omega }_1·{\mu }_2+({\beta }_1^{\prime }{\beta }_2{\ell }_2-{\beta }_1{\beta }_2^{\prime }{\ell }_1){\omega }_1·{\omega }_2\right)}^2.\hfill \end{array}$$

Thus, we have

$${\displaystyle \frac{\partial \Omega }{\partial s_1}}{|}_{(0,0)}=-{\beta }_1^{\prime }\left(0\right){\beta }_2\left(0\right){\omega }_1\left(0\right)·{\mu }_2\left(0\right),{\displaystyle \frac{\partial \Omega }{\partial s_2}}{|}_{(0,0)}=0,$$

$$\rho \Omega {|}_{(0,0)}=-{\beta }_1^{\prime }\left(0\right){\beta }_2\left(0\right){\omega }_1\left(0\right)·{\mu }_2\left(0\right),{\rho }^2\Omega {|}_{(0,0)}=-{\beta }_1^{″}\left(0\right){\beta }_2\left(0\right){\omega }_1\left(0\right)·{\mu }_2\left(0\right)-2{\beta }_1^{\prime }\left(0\right){\beta }_2\left(0\right){\ell }_1\left(0\right){\omega }_1\left(0\right).{\omega }_2\left(0\right),$$

$${\rho }^3\Omega {|}_{\left(0.0\right)}=\left(3{\beta }_1^{\prime }\left(0\right){\beta }_2\left(0\right){\ell }_1^2\left(0\right)-{\beta }_1^{‴}\left(0\right){\beta }_2\left(0\right)\right){\omega }_1\left(0\right)·{\mu }_2\left(0\right)-3\left({\beta }_1^{″}\left(0\right){\beta }_2\left(0\right){\ell }_1\left(0\right)+{\beta }_1^{\prime }\left(0\right){\beta }_2\left(0\right){\ell }_1^{\prime }\left(0\right)\right){\omega }_1\left(0\right)·{\omega }_2\left(0\right)$$

and

$$det\left({\mathcal{H}}_{\Omega }(0,0)\right)=-{(-{\beta }_1^{\prime }\left(0\right){\beta }_2^{\prime }\left(0\right){\omega }_1\left(0\right)·{\mu }_2\left(0\right)+{\beta }_1^{\prime }\left(0\right){\beta }_2\left(0\right){\ell }_2\left(0\right){\omega }_1\left(0\right)·{\omega }_2\left(0\right))}^2.$$

Therefore, applying Theorem 1 we obtain the result. □

Given proof of the above theorem, we have the following theorem.

Theorem 4.

Let M be the λ-point map between two Legendre plane curves ${\gamma }_1$ and ${\gamma }_2$ satisfying ${\beta }_1\left(0\right)=0$ and ${\beta }_2\left(0\right)\ne 0$. Then at $(0,0)$, M cannot be $\mathcal{A}$-equivalent to cusp, or lips or swallowtails singularity.

4. Classification of Corank Two Singularities of $\mathbf{\lambda }$-Point Map between Two Legendre Plane Curves

The criteria for sharksfin and deltoid singularities, which are generic singularities of corank two of maps from the plane into the plane (cf. [16]), have been obtain by kabata and Saji in [6].

Let $h:(U\subseteq {\mathbb{R}}^2,0)\to ({\mathbb{R}}^2,0)$ be a map germ with a corank two singularity at $(0,0)$. We call the function $\Omega :({\mathbb{R}}^2,0)\to (\mathbb{R},0)$ which is defined by $\Omega (s_1,s_2)=det\left({\displaystyle \frac{\partial h}{\partial s_1}},{\displaystyle \frac{\partial h}{\partial s_2}}\right)(s_1,s_2)$ a discriminant of singularities. The zeros of $\Omega$ are all the singular points of h. We define non-zero vector fields ${\rho }_1,{\rho }_2$ at a non-degenerate critical point of $\Omega$ which is a solution of the Hesse quadric of $\Omega$ at $(0,0)$. Recall that a vector field $({\rho }_{11},{\rho }_{12})$ is a solution of the Hesse quadric of $\Omega$

$$\left(\begin{array}{cc}{\rho }_{11}& {\rho }_{12}\end{array}\right){\mathcal{H}}_{\Omega }(0,0)\left(\begin{array}{c}{\rho }_{11}\\ {\rho }_{12}\end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right).$$

Our goal in this section is to give the geometric conditions for the $\lambda$-point map between two Legendre plan curves to have sharksfin singularity. The normal forms of sharksfin and deltoid singularities are $(xy,x^2+y^2+x^3)$ and $(xy,-x^2+y^2+x^3)$, respectively. We state the criteria for sharksfin and deltoid singularities.

Theorem 5

([6]). Let $h:({\mathbb{R}}^2,0)\to ({\mathbb{R}}^2,0)$ be a map germ with a corank two singularity at $(0,0)$ and suppose that Ω have a non-degenerate critical point at $(0,0)$.

Then h is a sharksfin (respectively, deltoid) at $(0,0)$ if, and only if, $det\left({\mathcal{H}}_{\Omega }(0,0)\right)<0$ (respectively, $det\left({\mathcal{H}}_{\Omega }(0,0)\right)>0$), $det({\rho }_1^{2}h,{\rho }_1^{3}h)(0,0)\ne 0$ and $det({\rho }_2^{2}h,{\rho }_2^{3}h)(0,0)\ne 0$. Here, $\rho \Omega$ means the directional derivative of Ω in the direction of the vector field ρ, and ${\rho }^{i}h=\rho \left({\rho }^{i-1}h\right)$.

Lemma 3.

Let M be the λ-point map between two Legendre plane curves ${\gamma }_1$ and ${\gamma }_2$. Then M is parametrized by a corank two singularity at $(0,0)$ if, and only if, ${\beta }_i\left(0\right)=0$, $i=1,2$.

Proof. 

The proof of this lemma is obvious. □

Lemma 4.

Let M be the λ-point map between two Legendre plane curves ${\gamma }_1$ and ${\gamma }_2$, such that ${\beta }_i\left(0\right)=0$, $i=1,2$. Then

• $(0,0)$ is a critical point of Ω.
• $(0,0)$ is a non-degenerate critical point of Ω if, and only if, ${\beta }_1^{\prime }\left(0\right)\ne 0$, ${\beta }_2^{\prime }\left(0\right)\ne 0$ and ${\omega }_1\left(0\right)\ne \pm {\omega }_2\left(0\right)$.

Proof. 

We define the discriminant of singularities $\Omega :{\mathbb{R}}^2\to \mathbb{R}$ of M by

$$\Omega (s_1,s_2)=det\left({\displaystyle \frac{\partial M}{\partial s_1}},{\displaystyle \frac{\partial M}{\partial s_2}}\right)=-{\beta }_1\left(s_1\right){\beta }_2\left(s_2\right){\omega }_1\left(s_1\right)·{\mu }_2\left(s_2\right).$$

It is easy to check that $(0,0)$ is a critical point of $\Omega$. A point $(0,0)$ is a non-degenerate if, and only if, $det\left({\mathcal{H}}_{\Omega }(0,0)\right)\ne 0$. Now

$${\mathcal{H}}_{\Omega }(0,0)=\left(\begin{array}{cc}{\displaystyle \frac{{\partial }^2\Omega }{{\partial }^2{s}_1}}{|}_{(0,0)}& {\displaystyle \frac{{\partial }^2\Omega }{\partial s_1\partial s_2}}{|}_{(0,0)}\\ \\ {\displaystyle \frac{{\partial }^2\Omega }{\partial s_1\partial s_2}}{|}_{(0,0)}& {\displaystyle \frac{{\partial }^2\Omega }{{\partial }^2{s}_2}}{|}_{(0,0)}\end{array}\right)=\left(\begin{array}{cc}0& -{\beta }_1^{\prime }\left(0\right){\beta }_2^{\prime }\left(0\right){\omega }_1\left(0\right)·{\mu }_2\left(0\right)\\ \\ -{\beta }_1^{\prime }\left(0\right){\beta }_2^{\prime }\left(0\right){\omega }_1\left(0\right)·{\mu }_2\left(0\right)& 0\end{array}\right)$$

Thus,

$$det\left({\mathcal{H}}_{\Omega }(0,0)\right)=-{({\beta }_1^{\prime }\left(0\right){\beta }_2^{\prime }\left(0\right){\omega }_1\left(0\right)·{\mu }_2\left(0\right))}^2\ne 0$$

if and only if ${\beta }_1^{\prime }\left(0\right){\beta }_2^{\prime }\left(0\right){\omega }_1\left(0\right)·{\mu }_2\left(0\right)\ne 0$. □

Now we introduce the main theorem of this section.

Theorem 6.

Let M be the λ-point map between two Legendre plane curves${\gamma }_1$and${\gamma }_2$with a corank two singularity at$(0,0)$and let$(0,0)$be non-degenerate critical point of Ω. Then M is$\mathcal{A}$-equivalent to a sharksfin if, and only if,${\ell }_i\left(0\right)\ne 0$, $i=1,2$.

Proof. 

Let $M(s_1,s_2)=(1-\lambda ){\gamma }_1\left(s_1\right)+\lambda {\gamma }_2\left(s_2\right)$ be the $\lambda$-point map between two Legendre plane curves ${\gamma }_1$ and ${\gamma }_2$, such that $rank\left(dM\right){|}_{(0,0)}=0$.

We will use Theorem 5 to prove this theorem. From Lemma 3 we have ${\beta }_1\left(0\right)={\beta }_2\left(0\right)=0$. Now we have

$${\mathcal{H}}_{\Omega }(0,0)=\left(\begin{array}{cc}0& -{\beta }_1^{\prime }\left(0\right){\beta }_2^{\prime }\left(0\right){\omega }_1\left(0\right)·{\mu }_2\left(0\right)\\ \\ -{\beta }_1^{\prime }\left(0\right){\beta }_2^{\prime }\left(0\right){\omega }_1\left(0\right)·{\mu }_2\left(0\right)& 0\end{array}\right).$$

Thus, $det\left({\mathcal{H}}_{\Omega }(0,0)\right)=-{({\beta }_1^{\prime }\left(0\right){\beta }_2^{\prime }\left(0\right){\omega }_1\left(0\right)·{\mu }_2\left(0\right))}^2<0$.

Now we choose vector fields ${\rho }_1={\displaystyle \frac{\partial }{\partial s_1}}$ and ${\rho }_2={\displaystyle \frac{\partial }{\partial s_2}}$ which satisfy the Hesse quadric of $\Omega$ at $(0,0)$. Calculations show that

$${\rho }_1^{2}M{|}_{(0,0)}=(1-\lambda ){\beta }_1^{\prime }\left(0\right){\mu }_1\left(0\right),$$

$${\rho }_1^{3}M{|}_{(0,0)}=(1-\lambda )({\beta }_1^{″}\left(0\right){\mu }_1\left(0\right)-2{\beta }_1^{\prime }\left(0\right){\ell }_1\left(0\right){\omega }_1\left(0\right)),$$

$${\rho }_2^{2}M{|}_{(0,0)}=\lambda {\beta }_2^{\prime }\left(0\right){\mu }_2\left(0\right)$$

and

$${\rho }_2^{3}M{|}_{(0,0)}=\lambda ({\beta }_2^{″}\left(0\right){\mu }_2\left(0\right)-2{\beta }_2^{\prime }\left(0\right){\ell }_2\left(0\right){\omega }_2\left(0\right)).$$

Now, $det({\rho }_1^{2}M,{\rho }_1^{3}M)\ne 0$ if, and only if, ${{\beta }^{\prime }}_1^2\left(0\right){\ell }_1\left(0\right)\ne 0.$ Additionally, $det({\rho }_2^{2}M,{\rho }_2^{3}M)$$\ne 0$ if, and only if, ${{\beta }^{\prime }}_2^2\left(0\right){\ell }_2\left(0\right)\ne 0.$ □

Given the proof of the above theorem, we have the following theorem.

Theorem 7.

The λ-point map between two Legendre plane curves ${\gamma }_1$ and ${\gamma }_2$ with a corank two singularity at $(0,0)$ cannot be $\mathcal{A}$-equivalent to a deltoid at $(0,0)$.

5. Examples

In this section, we present three examples for the $\lambda$-point map between two Legendre plane curves to have fold, beaks, and sharksfin singularities.

Example 1.

We give an example for part 1 of Theorem 2. Take ${\gamma }_1\left(s_1\right)=(2s_1,4s_1^2+s_1^4)$, ${\gamma }_2\left(s_2\right)=(-s_2,s_2^2-s_2^5)$, and $\lambda =\frac{1}{4}$. Then, the λ- point map between ${\gamma }_1$ and ${\gamma }_2$ is given by $M(s_1,s_2)=\left(\frac{3}{2}{s}_1-\frac{1}{4}{s}_2,\frac{3}{4}(4s_1^2+s_1^4)+\frac{1}{4}(s_2^2-s_2^5)\right)$. Clearly, M is singular at $(0,0)$, and direct calculation shows that ${\beta }_1\left(s_1\right)=2\sqrt{1+{(4s_1+2s_1^3)}^2}$, ${\ell }_1\left(s_1\right)={\displaystyle \frac{4+s_1^2}{1+{(4s_1+2s_1^3)}^2}}$, ${\omega }_1\left(s_1\right)=\left(\frac{4s_1+2s_1^3}{\sqrt{1+{(4s_1+2s_1^3)}^2}},\frac{-1}{\sqrt{1+{(4s_1+2s_1^3)}^2}}\right)$, ${\beta }_2\left(s_2\right)=\sqrt{1+{(2s_2-5s_2^4)}^2}$, ${\ell }_2\left(s_2\right)={\displaystyle \frac{20s_2^2-2}{1+{(2s_2-5s_2^4)}^2}}$, and ${\omega }_2\left(s_2\right)=\left(\frac{2s_2-5s_2^4}{\sqrt{1+{(2s_2-5s_2^4)}^2}},\frac{1}{\sqrt{1+{(2s_2-5s_2^4)}^2}}\right)$. Now at $s_1=0$ and $s_2=0$, we have ${\beta }_1\left(0\right)=2$, ${\beta }_2\left(0\right)=1$, ${\ell }_1\left(0\right)=4$, ${\ell }_2\left(0\right)=-2$, ${\omega }_1\left(0\right)=(0,-1)=-{\omega }_2\left(0\right)$, and $\left(\frac{\lambda }{1-\lambda }\right)\frac{{\beta }_2\left(0\right){\ell }_1\left(0\right)}{{\beta }_1\left(0\right)}=\frac{2}{3}\ne {\ell }_2\left(0\right)$. Thus, M is $\mathcal{A}$-equivalent to fold at $(0,0)$.

Example 2.

This example is dedicated to part 2 of Theorem 3. Let ${\gamma }_1\left(s_1\right)=(s_1^2,s_1^3)$, ${\gamma }_2\left(s_2\right)=(s_2,s_2^2)$, and $\lambda =\frac{1}{5}$. Then the λ-point map between ${\gamma }_1$ and ${\gamma }_2$ is given by $M(s_1,s_2)=(\frac{4}{5}{s}_1^2+\frac{1}{5}{s}_2,\frac{4}{5}{s}_1^3+\frac{1}{5}{s}_2^2)$. Direct calculation shows that ${\beta }_1\left(s_1\right)=s_1\sqrt{4+9s_1^2}$, ${\beta }_2\left(s_2\right)=\sqrt{1+4s_2^2}$, ${\ell }_1\left(s_1\right)=\frac{6}{4+9s_1^2}$, ${\ell }_2\left(s_2\right)=\frac{2}{1+4s_2^2}$, ${\omega }_1\left(s_1\right)=\left(\frac{3s_1}{\sqrt{4+9s_1^2}},\frac{-2}{\sqrt{4+9s_1^2}}\right)$, and ${\omega }_2\left(s_2\right)=\left(\frac{2s_2}{1+4s_2^2},\frac{-1}{1+4s_2^2}\right)$. At $(s_1,s_2)=(0,0)$, M as a corank one singularity and ${\beta }_1\left(0\right)=0$, ${\beta }_1^{\prime }\left(0\right)=2$, ${\beta }_2\left(0\right)=1$, ${\ell }_1\left(0\right)=\frac{3}{2}$, ${\ell }_2\left(0\right)=2$, and ${\omega }_1\left(0\right)=(0,1)={\omega }_2\left(0\right)$. Therefore, M is $\mathcal{A}$-equivalent to beaks at $(0,0)$.

Example 3.

This example illustrates the result in Theorem 6. Let ${\gamma }_1\left(s_1\right)=(\frac{1}{2}{s}_1^2,\frac{1}{3}{s}_1^3)$, ${\gamma }_2\left(s_2\right)=(2s_2^3,-s_2^2)$, and $\lambda =\frac{1}{3}$. The λ- point map between ${\gamma }_1$ and ${\gamma }_2$ is given by $M(s_1,s_2)=(\frac{1}{3}{s}_1^2+\frac{2}{3}{s}_2^3,\frac{2}{9}{s}_1^3-\frac{1}{3}{s}_2^2)$ and $\Omega (s_1,s_2)=-2s_1{s}_2(1+3s_1{s}_2)$. It is clear that $(0,0)$ is a non-degenerate critical point of Ω and M has a corank two singularity at $(0,0)$. Calculation shows that ${\beta }_1\left(s_1\right)=s_1\sqrt{1+s_1^2}$, ${\ell }_1\left(s_1\right)=\frac{1}{1+s_1^2}$, ${\omega }_1\left(s_1\right)=\left(\frac{s_1}{\sqrt{1+s_1^2}},\frac{-1}{\sqrt{1+s_1^2}}\right)$, ${\beta }_2\left(s_2\right)=-2s_2\sqrt{1+9s_2^2}$, ${\ell }_2\left(s_2\right)=\frac{3}{1+9s_2^2}$, and ${\omega }_2\left(s_2\right)=\left(\frac{1}{\sqrt{1+9s_2^2}},\frac{3s_2}{\sqrt{1+9s_2^2}}\right)$. At $(s_1,s_2)=(0,0)$, we have ${\ell }_1\left(0\right)=1$ and ${\ell }_2\left(0\right)=3$. Therefore, M is $\mathcal{A}$-equivalent to sharksfin at $(0,0)$.

6. Conclusions

Throughout this paper we introduce the $\lambda$-point map between two Legendre plane curves. The classifications of this map have been investigated for corank one and two singularities. All results obtained in this research are more general and many known plane map’s singularities are realized as special cases of these results. Moreover, three non-trivial examples are given throughout this research to illustrate some of the obtained results.