---
abstract: 'In the previous paper, the authors gave criteria for $A_{k+1}$-type singularities on wave fronts. Using them, we show in this paper that there is a duality between singular points and inflection points on wave fronts in the projective space. As an application, we show that the algebraic sum of $2$-inflection points (i.e. godron points) on an immersed surface in the real projective space is equal to the Euler number of $M_-$. Here $M^2$ is a compact orientable 2-manifold, and $M_-$ is the open subset of $M^2$ where the Hessian of $f$ takes negative values. This is a generalization of Bleecker and Wilson’s formula [@BW] for immersed surfaces in the affine $3$-space.'
address:
- ' Department of Mathematics, Faculty of Educaton, Gifu University, Yanagido 1-1, Gifu 151-1193, Japan'
- ' Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan '
- 'Department of Mathematics, Tokyo Institute of Technology, O-okayama, Meguro, Tokyo 152-8551, Japan'
author:
- Kentaro Saji
- Masaaki Umehara
- Kotaro Yamada
date: 'January 15, 2010'
title: |
     The duality between singular points and\
    inflection points on wave fronts
---

Introduction
============

We denote by ${\boldsymbol{K}}$ the real number field ${\boldsymbol{R}}$ or the complex number field ${\boldsymbol{C}}$. Let $n$ and $m$ be positive integers. A map $F\colon{}{\boldsymbol{K}}^n\to {\boldsymbol{K}}^m$ is called [*${\boldsymbol{K}}$-differentiable*]{} if it is a $C^\infty$-map when ${\boldsymbol{K}}={\boldsymbol{R}}$, and is a holomorphic map when ${\boldsymbol{K}}={\boldsymbol{C}}$. Throughout this paper, we denote by $P(V)$ the ${\boldsymbol{K}}$-projective space associated with a vector space $V$ over ${\boldsymbol{K}}$ and let $\pi:V\to P(V)$ be the canonical projection.

Let $M^{n}$ and $N^{n+1}$ be ${\boldsymbol{K}}$-differentiable manifolds of dimension $n$ and of dimension $n+1$, respectively. The projectified ${\boldsymbol{K}}$-cotangent bundle $$P(T^*N^{n+1}):=\bigcup_{p\in N^{n+1}} P(T^*_pN^{n+1})$$ has a canonical ${\boldsymbol{K}}$-contact structure. A ${\boldsymbol{K}}$-differentiable map $f : M^n
\to N^{n+1}$ is called a [*frontal*]{} if $f$ lifts to a ${\boldsymbol{K}}$-isotropic map $L_f$, i.e., a ${\boldsymbol{K}}$-differentiable map $L_f : M^n \to P(T^*N^{n+1})$ such that the image $dL_f(TM^n)$ of the ${\boldsymbol{K}}$-tangent bundle $TM^n$ lies in the contact hyperplane field on $P(T^*N^{n+1})$. Moreover, $f$ is called a [*wave front*]{} or a [*front*]{} if it lifts to a ${\boldsymbol{K}}$-isotropic immersion $L_f$. (In this case, $L_f$ is called a [*Legendrian immersion*]{}.) Frontals (and therefore fronts) generalize immersions, as they allow for singular points. A frontal $f$ is said to be [*co-orientable*]{} if its ${\boldsymbol{K}}$-isotropic lift $L_f$ can lift up to a ${\boldsymbol{K}}$-differentiable map into the ${\boldsymbol{K}}$-cotangent bundle $T^*N^{n+1}$, otherwise it is said to be [*non-co-orientable*]{}. It should be remarked that, when $N^{n+1}$ is a Riemannian manifold, a front $f$ is co-orientable if and only if there is a globally defined unit normal vector field $\nu$ along $f$.

Now we set $N^{n+1}={\boldsymbol{K}}^{n+1}$. Suppose that a ${\boldsymbol{K}}$-differentiable map $F:M^{n}\to {\boldsymbol{K}}^{n+1}$ is a frontal. Then, for each $p\in M^{n}$, there exist a neighborhood $U$ of $p$ and a map $$\nu:U\longrightarrow ({\boldsymbol{K}}^{n+1})^*\setminus \{{\boldsymbol{0}}\}$$ into the dual vector space $({\boldsymbol{K}}^{n+1})^*$ of ${\boldsymbol{K}}^{n+1}$ such that the canonical pairing $\nu\cdot dF(v)$ vanishes for any $v\in TU$. We call $\nu$ a [*local normal map*]{} of the frontal $F$. We set ${\mathcal{G}}:=\pi\circ\nu$, which is called a (local) [*Gauss map*]{} of $F$. In this setting, $F$ is a front if and only if $$L:=(F,{\mathcal{G}}):U\longrightarrow {\boldsymbol{K}}^{n+1}\times P\bigl(({\boldsymbol{K}}^{n+1})^*
\bigr)$$ is an immersion. When $F$ itself is an immersion, it is, of course, a front. If this is the case, for a fixed local ${\boldsymbol{K}}$-differentiable coordinate system $(x^1,\dots,x^{n})$ on $U$, we set $$\label{eq:nu}
 \nu_p:{\boldsymbol{K}}^{n+1}\ni v \longmapsto 
   \det(F_{x^1}(p),\dots,F_{x^{n}}(p),v)
   \in {\boldsymbol{K}}\qquad (p\in U),$$ where $F_{x^j}:=\partial F/\partial x^j$ ($j=1,\dots,n$) and $\det$ is the determinant function on ${\boldsymbol{K}}^{n+1}$. Then we get a ${\boldsymbol{K}}$-differentiable map $
    \nu:U\ni p \longmapsto \nu_p\in ({\boldsymbol{K}}^{n+1})^*,
$ which gives a local normal map of $F$.

Now, we return to the case that $F$ is a front. Then it is well-known that the local Gauss map $\mathcal G$ induces a global map $$\label{eq:G}
     {\mathcal{G}}:M^{n}\longrightarrow P\bigl(({\boldsymbol{K}}^{n+1})^*\bigr)$$ which is called the [*affine Gauss map*]{} of $F$. (In fact, the Gauss map ${\mathcal{G}}$ depends only on the affine structure of ${\boldsymbol{K}}^n$.)

We set $$\label{eq:h}
   h_{ij}:=\nu\cdot F_{x^ix^j}=-\nu_{x^i} \cdot F_{x^j}
      \qquad (i,j=1,\dots,n),$$ where $\cdot$ is the canonical pairing between ${\boldsymbol{K}}^{n+1}$ and $({\boldsymbol{K}}^{n+1})^*$, and $$F_{x^ix^j}=\frac{\partial^2 F}{\partial x^i\partial x^j},
\quad 
  F_{x^j}=\frac{\partial F}{\partial x^j},\quad
  \nu_{x^i}=\frac{\partial \nu}{\partial x^i}.$$ Then $$\label{eq:H}
    H:=\sum_{i,j=1}^{n}h_{ij}dx^i\, dx^j
     \qquad \left(
         dx^i\, dx^j:=\frac12(dx^i\otimes dx^j+dx^j\otimes dx^i)
        \right)$$ gives a ${\boldsymbol{K}}$-valued symmetric tensor on $U$, which is called the [*Hessian form*]{} of $F$ associated to $\nu$. Here, the ${\boldsymbol{K}}$-differentiable function $$\label{eq:hessian1}
   h:=\det(h_{ij}):U\longrightarrow {\boldsymbol{K}}$$ is called the [*Hessian*]{} of $F$. A point $p\in M^{n}$ is called an [*inflection point*]{} of $F$ if it belongs to the zeros of $h$. An inflection point $p$ is called [*nondegenerate*]{} if the derivative $dh$ does not vanish at $p$. In this case, the set of inflection points $I(F)$ consists of an embedded ${\boldsymbol{K}}$-differentiable hypersurface of $U$ near $p$ and there exists a non-vanishing ${\boldsymbol{K}}$-differentiable vector field $\xi$ along $I(F)$ such that $H(\xi,v)=0$ for all $v\in TU$. Such a vector field $\xi$ is called an [*asymptotic vector field*]{} along $I(F)$, and $[\xi]=\pi(\xi)\in P({\boldsymbol{K}}^{n+1})$ is called the [*asymptotic direction*]{}. It can be easily checked that the definition of inflection points and the nondegeneracy of inflection points are independent of choice of $\Omega$, $\nu$ and a local coordinate system.

In Section \[sec:prelim\], we shall define the terminology that

-   a ${\boldsymbol{K}}$-differentiable vector field $\eta$ along a ${\boldsymbol{K}}$-differentiable hypersurface $S$ of $M^n$ is [*$k$-nondegenerate at*]{} $p\in S$, and

-   [*$\eta$ meets $S$ at $p$ with multiplicity $k+1$.*]{}

Using this new terminology, $p(\in I(F))$ is called an [*$A_{k+1}$-inflection point*]{} or an [*inflection point of multiplicity $k+1$*]{}, if $\xi$ is $k$-nondegenerate at $p$ but does not meet $I(F)$ with multiplicity $k+1$. In Section \[sec:prelim\], we shall prove the following:

\[thm:A\] Let $F:M^{n}\to {\boldsymbol{K}}^{n+1}$ be an immersed ${\boldsymbol{K}}$-differentiable hypersurface. Then $p\in M^{n}$ is an $A_{k+1}$-inflection point $(k\leq n)$ if and only if the affine Gauss map $\mathcal G$ has an $A_k$-Morin singularity at $p$. $($See the appendix of [@SUY3] for the definition of $A_k$-Morin singularities.$)$

Though our definition of $A_k$-inflection points are given in terms of the Hessian, this assertion allows us to define $A_{k+1}$-inflection points by the singularities of their affine Gauss map, which might be more familiar to readers than our definition. However, the new notion “$k$-multiplicity” introduced in the present paper is very useful for recognizing the duality between singular points and inflection points. Moreover, as mentioned above, our definition of $A_k$-inflection points works even when $F$ is a front. We have the following dual assertion for the previous theorem:

\[prop:A\] Let $F:M^n\to {\boldsymbol{K}}^{n+1}$ be a front. Suppose that the affine Gauss map ${\mathcal{G}}:M^n\to P(({\boldsymbol{K}}^{n+1})^*)$ is a ${\boldsymbol{K}}$-immersion. Then $p\in M^{n}$ is an $A_{k+1}$-inflection point of ${\mathcal{G}}$ $(k\leq n)$ if and only if $F$ has an $A_k$-singularity at $p$. [(]{}See $(1.1)$ in [@SUY3] for the definition of $A_k$-singularities.[)]{}

In the case that ${\boldsymbol{K}}={\boldsymbol{R}}$, $n=3$ and $F$ is an immersion, an $A_{3}$-inflection point is known as a [*cusp of the Gauss map*]{} (cf. [@BGM]).

It can be easily seen that inflection points and the asymptotic directions are invariant under projective transformations. So we can define $A_{k+1}$-inflection points ($k\leq n$) of an immersion $f:M^{n}\to P({\boldsymbol{K}}^{n+2})$. For each $p\in M^{n}$, we take a local ${\boldsymbol{K}}$-differentiable coordinate system $(U;x^1,...,x^{n})(\subset M^n)$. Then there exists a ${\boldsymbol{K}}$-immersion $F:U\to {\boldsymbol{K}}^{n+2}$ such that $f=[F]$ is the projection of $F$. We set $$\label{eq:FG1}
 G:U\ni p \longmapsto 
  F_{x^1}(p)\wedge 
  F_{x^2}(p)\wedge 
  \cdots \wedge F_{x^n}(p)
  \wedge F(p)\in ({\boldsymbol{K}}^{n+2})^*.$$ Here, we identify $({\boldsymbol{K}}^{n+2})^*$ with $\bigwedge^{n+1}{\boldsymbol{K}}^{n+2}$ by $$\bigwedge\nolimits^{n+1} {\boldsymbol{K}}^{n+2}
     \ni v_1\wedge \cdots \wedge v_{n+1}
     \quad
     \longleftrightarrow 
     \quad
     \det(v_1, \dots, v_{n+1},*)\in ({\boldsymbol{K}}^{n+2})^*,$$ where $\det$ is the determinant function on ${\boldsymbol{K}}^{n+2}$. Then $G$ satisfies $$\label{eq:FG2}
     G \cdot F=0,\qquad G\cdot dF=dG\cdot F=0,$$ where $\cdot$ is the canonical pairing between ${\boldsymbol{K}}^{n+2}$ and $({\boldsymbol{K}}^{n+2})^*$. Since, $g:=\pi\circ G$ does not depend on the choice of a local coordinate system, the projection of $G$ induces a globally defined $K$-differentiable map $$g=[G]:M^n\to P\bigl(({\boldsymbol{K}}^{n+2})^*\bigr),$$ which is called the [*dual*]{} front of $f$. We set $$h:=\det(h_{ij}):U\longrightarrow {\boldsymbol{K}}\qquad \bigl(h_{ij}:=G\cdot F_{x^ix^j}
                =-G_{x^i}\cdot F_{x^j}\bigr),$$ which is called the [*Hessian*]{} of $F$. The inflection points of $f$ correspond to the zeros of $h$.

In Section \[sec:dual\], we prove the following

\[thm:B\] Let $f:M^{n}\to P({\boldsymbol{K}}^{n+2})$ be an immersed ${\boldsymbol{K}}$-differentiable hypersurface. Then $p\in M^{n}$ is an $A_k$-inflection point $(k\leq n)$ if and only if the dual front $g$ has an $A_k$-singularity at $p$.

Next, we consider the case of ${\boldsymbol{K}}={\boldsymbol{R}}$. In [@SUY1], we defined the [*tail part*]{} of a swallowtail. An $A_3$-inflection point $p$ of $f:M^{2}\to P({\boldsymbol{R}}^4)$ is called [*positive*]{} (resp. [*negative*]{}), if the Hessian takes negative (resp. positive) values on the tail part of the dual of $f$ at $p$. Let $p\in M^2$ be an $A_3$-inflection point. Then there exists a neighborhood $U$ such that $f(U)$ is contained in an affine space $A^3$ in $P({\boldsymbol{R}}^4)$. Then the affine Gauss map ${\mathcal{G}}:U\to P(A^3)$ has an elliptic cusp (resp. a hyperbolic cusp) if and only if it is positive (resp. negative) (see [@BGM p. 33]). In [@uribe], Uribe-Vargas introduced a projective invariant $\rho$ and studied the projective geometry of swallowtails. He proved that an $A_3$-inflection point is positive (resp. negative) if and only if $\rho>1$ (resp. $\rho<1$). The property that $h$ is negative is also independent of the choice of a local coordinate system. So we can define the set of negative points $$M_{-}:=\{p\in M^{2}\,;\, h(p)<0\}.$$ In Section \[sec:dual\], we shall prove the following assertion as an application.

\[thm:C\] Let $M^2$ be a compact orientable $C^\infty$-manifold without boundary, and $f:M^{2}\to P({\boldsymbol{R}}^4)$ an immersion. We denote by $i^+_2(f)$ [(]{}resp. $i^-_2(f)$[)]{} the number of positive $A_3$-inflection points [(]{}resp. negative $A_3$-inflection points[)]{} on $M^2$ [(]{}see Section \[sec:dual\] for the precise definition of $i^+_2(f)$ and $i^-_2(f)$[)]{}. Suppose that inflection points of $f$ consist only of $A_3$-inflection points. Then the following identity holds $$\label{eq:BW}
      i_2^+(f)-i_2^-(f)=2\chi(M_-).$$

The above formula is a generalization of that of Bleecker and Wilson [@BW] when $f(M^2)$ is contained in an affine plane.

\[cor:1\] Under the assumption of Theorem \[thm:C\], the total number $i_2^+(f)+i_2^-(f)$ of $A_3$-inflection points is even.

In [@uribe], this corollary is proved by counting the parity of a loop consisting of flecnodal curves which bound two $A_3$-inflection points.

\[cor:2\] The same formula holds for an immersed surface in the unit $3$-sphere $S^3$ or in the hyperbolic $3$-space $H^3$.

Let $\pi:S^3\to P({\boldsymbol{R}}^4)$ be the canonical projection. If $f:M^2\to S^3$ is an immersion, we get the assertion applying Theorem \[thm:C\] to $\pi\circ f$. On the other hand, if $f$ is an immersion into $H^3$, we consider the canonical projective embedding $\iota:H^3\to S^3_+$ where $S^3_+$ is the open hemisphere of $S^3$. Then we get the assertion applying Theorem \[thm:C\] to $\pi\circ \iota\circ f$.

Finally, in Section \[sec:cusp\], we shall introduce a new invariant for $3/2$-cusps using the duality, which is a measure for acuteness using the classical cycloid.

This work is inspired by the result of Izumiya, Pei and Sano [@ips] that characterizes $A_2$ and $A_3$-singular points on surfaces in $H^3$ via the singularity of certain height functions, and the result on the duality between space-like surfaces in hyperbolic $3$-space (resp. in light-cone), and those in de Sitter space (resp. in light-cone) given by Izumiya [@iz]. The authors would like to thank Shyuichi Izumiya for his impressive informal talk at Karatsu, 2005.

Preliminaries and a proof of Theorem \[thm:A\] {#sec:prelim}
==============================================

In this section, we shall introduce a new notion “multiplicity” for a contact of a given vector field along an immersed hypersurface. Then our previous criterion for $A_k$-singularities (given in [@SUY3]) can be generalized to the criteria for $k$-multiple contactness of a given vector field (see Theorem \[thm:actual\]).

Let $M^n$ be a ${\boldsymbol{K}}$-differentiable manifold and $S(\subset M^n)$ an embedded ${\boldsymbol{K}}$-differentiable hypersurface in $M^n$. We fix $p\in S$ and take a ${\boldsymbol{K}}$-differentiable vector field $$\eta:S\supset V\ni q \longmapsto \eta_q\in T_qM^n$$ along $S$ defined on a neighborhood $V\subset S$ of $p$. Then we can construct a ${\boldsymbol{K}}$-differential vector field $\tilde \eta$ defined on a neighborhood $U\subset M^n$ of $p$ such that the restriction $\tilde \eta|_S$ coincides with $\eta$. Such an $\tilde \eta$ is called [*an extension of $\eta$*]{}. (The local existence of $\tilde \eta$ is mentioned in [@SUY3 Remark 2.2].)

\[def:admissible\] Let $p$ be an arbitrary point on $S$, and $U$ a neighborhood of $p$ in $M^n$. A ${\boldsymbol{K}}$-differentiable function ${\varphi}:U\to {\boldsymbol{K}}$ is called [*admissible*]{} near $p$ if it satisfies the following properties

1.  $O:=U\cap S$ is the zero level set of ${\varphi}$, and

2.  $d{\varphi}$ never vanishes on $O$.

One can easily find an admissible function near $p$. We set ${\varphi}':=d{\varphi}(\tilde\eta):U\to {\boldsymbol{K}}$ and define a subset $S_2(\subset O\subset S)$ by $$S_2:=\{q\in O \,;\,{\varphi}'(q)=0\}=\{q\in O\,;\, \eta_q\in T_qS\}.$$ If $p\in S_2$, then $\eta$ is said to [*meet $S$ with multiplicity $2$ at $p$*]{} or equivalently, $\eta$ is said to [*contact $S$ with multiplicity $2$ at $p$*]{}. Otherwise, $\eta$ is said to [*meet $S$ with multiplicity $1$ at $p$*]{}. Moreover, if $d{\varphi}'(T_pO)\ne \{{\boldsymbol{0}}\}$, $\eta$ is said to [*be $2$-nondegenerate at $p$*]{}. The $(k+1)$-st multiple contactness and $k$-nondegeneracy are defined inductively. In fact, if the $j$-th multiple contactness and the submanifolds $S_j$ have been already defined for $j=1,\dots,k$, we set $${\varphi}^{(k)}:=d{\varphi}^{(k-1)}(\tilde\eta):
           U\longrightarrow {\boldsymbol{K}}\qquad ({\varphi}^{(1)}:={\varphi}')$$ and can define a subset of $S_{k}$ by $$S_{k+1}:=\{q\in S_k\,;\,{\varphi}^{(k)}(q)=0\}
          =\{q\in S_k\,;\, \eta_q\in T_qS_k\}.$$ We say that $\eta$ [*meets $S$ with multiplicity $k+1$ at $p$*]{} if $\eta$ is $k$-nondegenerate at $p$ and $p\in S_{k+1}$. Moreover, if $d{\varphi}^{(k)}(T_pS_{k})\ne \{{\boldsymbol{0}}\}$, $\eta$ is called [*$(k+1)$-nondegenerate*]{} at $p$. If $\eta$ is $(k+1)$-nondegenerate at $p$, then $S_{k+1}$ is a hypersurface of $S_k$ near $p$.

\[rmk:add\] Here we did not define $1$-nondegeneracyof $\eta$. However, from now on, [*any ${\boldsymbol{K}}$-differentiable vector field $\eta$ of $M^n$ along $S$ is always $1$-nondegenerate by convention*]{}. In the previous paper [@SUY3], $1$-nondegeneracy (i.e. nondegeneracy) is defined not for a vector field along the singular set but for a given singular point. If a singular point $p\in U$ of a front $f:U\to {\boldsymbol{K}}^{n+1}$ is nondegenerate in the sense of [@SUY3], then the function $\lambda:U\to {\boldsymbol{K}}$ defined in [@SUY3 (2.1)] is an admissible function, and the null vector field $\eta$ along $S(f)$ is given. When $k\ge 2$, by definition, $k$-nondegeneracy of the singular point $p$ is equivalent to the $k$-nondegeneracy of the null vector field $\eta$ at $p$ (cf. [@SUY3]).

\[prop:indep\] The $k$-th multiple contactness and $k$-nondegeneracy are both independent of the choice of an extension $\tilde \eta$ of $\eta$ and also of the choice of admissible functions as in Definition \[def:admissible\].

We can take a local coordinate system $(U;x^1,\dots,x^n)$ of $M^n$ such that $x^n={\varphi}$. Write $$\tilde \eta:=\sum_{j=1}^n c^j \frac{\partial}{\partial x^j},$$ where $c^j$ $(j=1,\ldots,n)$ are ${\boldsymbol{K}}$-differentiable functions. Then we have that ${\varphi}'=\sum_{j=1}^n c^j {\varphi}_{x^j}=c^n$.

Let $\psi$ be another admissible function defined on $U$. Then $$\psi'=
      \sum_{j=1}^n c^j \frac{\partial\psi}{\partial x^j}
       =c^n \frac{\partial\psi}{\partial x^n}
       ={\varphi}'\frac{\partial\psi}{\partial x^n}.$$ Thus $\psi'$ is proportional to ${\varphi}'$. Then the assertion follows inductively.

Corollary 2.5 in [@SUY3] is now generalized into the following assertion:

\[thm:actual\] Let $\tilde \eta$ be an extension of the vector field $\eta$. Let us assume $1\leq k\leq n$. Then the vector field $\eta$ is $k$-nondegenerate at $p$, but $\eta$ does not meet $S$ with multiplicity $k+1$ at $p$ if and only if $${\varphi}(p)={\varphi}'(p)=\dots ={\varphi}^{(k-1)}(p)=0,\quad
     {\varphi}^{(k)}(p)\ne 0$$ and the Jacobi matrix of ${\boldsymbol{K}}$-differentiable map $$\Lambda:=({\varphi},{\varphi}',\dots, {\varphi}^{(k-1)}):U\longrightarrow {\boldsymbol{K}}^k$$ is of rank $k$ at $p$, where ${\varphi}$ is an admissible ${\boldsymbol{K}}$-differentiable function and $${\varphi}^{(0)}:={\varphi},\quad 
    {\varphi}^{(1)}(={\varphi}'):=d{\varphi}(\tilde \eta), \quad\dots,\quad 
    {\varphi}^{(k)}:=d{\varphi}^{(k-1)}(\tilde \eta).$$

The proof of this theorem is completely parallel to that of Corollary 2.5 in [@SUY3].

To prove Theorem \[thm:A\] by applying Theorem \[thm:actual\], we shall review the criterion for $A_k$-singularities in [@SUY3]. Let $U^n$ be a domain in ${\boldsymbol{K}}^n$, and consider a map $\Phi:U^n\to {\boldsymbol{K}}^{m}$ where $m\ge n$. A point $p\in U^{n}$ is called a [*singular point*]{} if the rank of the differential map $d\Phi$ is less than $n$. Suppose that the singular set $S(\Phi)$ of $\Phi$ consists of a ${\boldsymbol{K}}$-differentiable hypersurface $U^n$. Then a vector field $\eta$ along $S$ is called a [*null vector field*]{} if $d\Phi(\eta)$ vanishes identically. In this paper, we consider the case $m=n$ or $m=n+1$. If $m=n$, we define a ${\boldsymbol{K}}$-differentiable function $\lambda:U^n\to {\boldsymbol{K}}$ by $$\label{eq:lambda-map}
     \lambda:=\det(\Phi_{x^1},\dots,\Phi_{x^n}).$$ On the other hand, if $\Phi:U^n\to{\boldsymbol{K}}^{n+1}$ ($m=n+1$) and $\nu$ is a non-vanishing ${\boldsymbol{K}}$-normal vector field (for a definition, see [@SUY3 Section 1]) we set $$\label{eq:lambda-fr}
     \lambda:=\det(\Phi_{x^1},\dots,\Phi_{x^n},\nu).$$ Then the singular set $S(\Phi)$ of the map $\Phi$ coincides with the zeros of $\lambda$. Recall that $p\in S(\Phi)$ is called [*nondegenerate*]{} if $d\lambda(p)\ne {\boldsymbol{0}}$ (see [@SUY3] and Remark \[rmk:add\]). Both of two cases and , the functions $\lambda$ are admissible near $p$, if $p$ is non-degenerate. When $S(\Phi)$ consists of nondegenerate singular points, then it is a hypersurface and there exists a non-vanishing null vector field $\eta$ on $S(\Phi)$. Such a vector field $\eta$ determined up to a multiplication of non-vanishing ${\boldsymbol{K}}$-differentiable functions. The following assertion holds as seen in [@SUY3].

\[eq:fact\_suy\] Suppose $m=n$ and $\Phi$ is a $C^\infty$-map [(]{}resp. $m=n+1$ and $\Phi$ is a front[)]{}. Then $\Phi$ has an $A_{k+1}$-Morin singularity [(]{}resp. $A_{k+1}$-singularity[)]{} at $p\in M^n$ if and only if $\eta$ is $k$-nondegenerate at $p$ but does not meet $S(\Phi)$ with multiplicity $k+1$ at $p$. [(]{}Here multiplicity $1$ means that $\eta$ meets $S(\Phi)$ at $p$ transversally, and $1$-nondegeneracy is an empty condition.[)]{}

As an application of the fact for $m=n$, we now give a proof of Theorem \[thm:A\]: Let $F:M^{n}\to {\boldsymbol{K}}^{n+1}$ be an immersed ${\boldsymbol{K}}$-differentiable hypersurface. Recall that a point $p\in M^{n}$ is called a [*nondegenerate inflection point*]{} if the derivative $dh$ of the local Hessian function $h$ (cf. ) with respect to $F$ does not vanish at $p$. Then the set $I(F)$ of inflection points consists of a hypersurface, called the [*inflectional hypersurface*]{}, and the function $h$ is an admissible function on a neighborhood of $p$ in $M^n$. A nondegenerate inflection point $p$ is called an [*$A_{k+1}$-inflection point*]{} of $F$ if the asymptotic vector field $\xi$ is $k$-nondegenerate at $p$ but does not meet $I(F)$ with multiplicity $k+1$ at $p$.

Let $\nu$ be a map given by , and ${\mathcal{G}}:M^n\to  P(({\boldsymbol{K}}^{n+1})^*)$ the affine Gauss map induced from $\nu$ by . We set $$\mu:=\det(\nu_{x^1},\nu_{x^2},\dots,\nu_{x^n},\nu),$$ where $\det$ is the determinant function of $({\boldsymbol{K}}^{n+1})^*$ under the canonical identification $({\boldsymbol{K}}^{n+1})^*\cong {\boldsymbol{K}}^{n+1}$, and $(x^1,\dots,x^n)$ is a local coordinate system of $M^n$. Then the singular set $S({\mathcal{G}})$ of ${\mathcal{G}}$ is just the zeros of $\mu$. By Theorem \[thm:actual\] and Fact \[eq:fact\_suy\], our criteria for $A_{k+1}$-inflection points (resp. $A_{k+1}$-singular points) are completely determined by the pair $(\xi, I(F))$ (resp. the pair $(\eta,S({\mathcal G}))$). Hence it is sufficient to show the following three assertions (1)–(3).

1.  \[ass:1\] $I(F)=S(\mathcal G)$,

2.  \[ass:2\] For each $p\in I(F)$, $p$ is a nondegenerate inflection point of $F$ if and only if it is a nondegenerate singular point of ${\mathcal{G}}$.

3.  \[ass:3\] The asymptotic direction of each nondegenerate inflection point $p$ of $F$ is equal to the null direction of $p$ as a singular point of ${\mathcal{G}}$.

Let $H=\sum_{i,j=1}^n h_{ij}dx^i\, dx^j$ be the Hessian form of $F$. Then we have that $$\label{eq:split}
  {{\begin{pmatrix}    h_{11}& \dots & h_{1n}& *\\
   \vdots & \ddots & \vdots & \vdots \\
   h_{n1}& \dots & h_{nn} & *\\
   0 & \dots & 0 & \nu\cdot {{\vphantom{\nu}}^t{\nu}}
    \end{pmatrix}}}
  =
  {{\begin{pmatrix}    \nu_{x^1}\\ \vdots\\ \nu_{x^n}\\ \nu
    \end{pmatrix}}} 
  (F_{x^1},\dots,F_{x^n},{{\vphantom{\nu}}^t{\nu}}),$$ where $\nu\cdot {{\vphantom{\nu}}^t{\nu}}=\sum_{j=1}^{n+1}(\nu^j)^2$ and $\nu=(\nu^1,...,\nu^n)$ as a row vector. Here, we consider a vector in ${\boldsymbol{K}}^n$ (resp. in $({\boldsymbol{K}}^n)^*$) as a column vector (resp. a row vector), and ${{\vphantom{(\cdot)}}^t{(\cdot)}}$ denotes the transposition. We may assume that $\nu(p)\cdot{{\vphantom{\nu}}^t{\nu}}(p)\ne0$ by a suitable affine transformation of ${\boldsymbol{K}}^{n+1}$, even when ${\boldsymbol{K}}={\boldsymbol{C}}$. Since the matrix $(F_{x^1},\dots,F_{x^n},{{\vphantom{\nu}}^t{\nu}})$ is regular, \[ass:1\] and \[ass:2\] follow by taking the determinant of . Also by , $\sum_{i=1}^n a_ih_{ij}=0$ for all $j=1,\dots,n$ holds if and only if $\sum_{i=1}^n a_i\nu_{x^i}={\boldsymbol{0}}$, which proves \[ass:3\].

Similar to the proof of Theorem \[thm:A\], it is sufficient to show the following properties, by virtue of Theorem \[thm:actual\].

1.  \[ass:1a\] $S(F)=I({\mathcal{G}})$, that is, the set of singular points of $F$ coincides with the set of inflection points of the affine Gauss map.

2.  \[ass:2a\] For each $p\in I({{\mathcal{G}}})$, $p$ is a nondegenerate inflection point if and only if it is a nondegenerate singular point of $F$.

3.  \[ass:3a\] The asymptotic direction of each nondegenerate inflection point coincides with the null direction of $p$ as a singular point of $F$.

Since ${\mathcal{G}}$ is an immersion, implies that $$\begin{aligned}
  I({\mathcal{G}}) &= \{p\,;\,(F_{x^1},\dots,F_{x^n},{{\vphantom{\nu}}^t{\nu}}) 
            \text{ are linearly dependent at $p$}\}\\
       &= \{p\,;\, \lambda(p)=0\}
        \qquad(\lambda:=\det(F_{x^1},\dots,F_{x^n},{{\vphantom{\nu}}^t{\nu}})).
 \end{aligned}$$ Hence we have \[ass:1a\]. Moreover, $h=\det(h_{ij})=\delta\lambda$ holds, where $\delta$ is a function on $U$ which never vanishes on a neighborhood of $p$. Thus \[ass:2a\] holds. Finally, by , $\sum_{j=1}^n b_j h_{ij}=0$ for $i=1,\dots,n$ if and only if $\sum_{j=1}^n b_j F_{x^j}={\boldsymbol{0}}$, which proves \[ass:3a\].

Let $\gamma(t):={{\vphantom{(x(t),y(t))}}^t{(x(t),y(t))}}$ be a ${\boldsymbol{K}}$-differentiable curve in ${\boldsymbol{K}}^2$. Then $\nu(t):=(-\dot y(t),\dot x(t))\in ({\boldsymbol{K}}^2)^*$ gives a normal vector, and $$h(t)=\nu(t)\cdot \ddot\gamma(t)=\det(\dot \gamma(t),
\ddot \gamma(t))$$ is the Hessian function. Thus $t=t_0$ is an $A_2$-inflection point if and only if $$\det(\dot \gamma(t_0),\ddot \gamma(t_0))=0,\qquad
   \det(\dot \gamma(t_0),\dddot \gamma(t_0))\ne 0.$$ Considering ${\boldsymbol{K}}^2\subset P({\boldsymbol{K}}^3)$ as an affine subspace, this criterion is available for curves in $P({\boldsymbol{K}}^3)$. When ${\boldsymbol{K}}={\boldsymbol{C}}$, it is well-known that non-singular cubic curves in $P({\boldsymbol{C}}^3)$ have exactly nine inflection points which are all of $A_2$-type. One special singular cubic curve is $2y^2-3x^3=0$ in $P({\boldsymbol{C}}^3)$ with homogeneous coordinates $[x,y,z]$, which can be parameterized as $\gamma(t)=[\sqrt[3]{2}t^2,\sqrt{3}t^3,1]$. The image of the dual curve of $\gamma$ in $P({\boldsymbol{C}}^3)$ is the image of $\gamma$ itself, and $\gamma$ has an $A_2$-type singular point $[0,0,1]$ and an $A_2$-inflection point $[0,1,0]$.

These two points are interchanged by the duality. (The duality on fronts are explained in Section \[sec:dual\].)

Let $F:{\boldsymbol{K}}^3\to{\boldsymbol{K}}^4$ be a map defined by $$F(u,v,w)
    =
      \raisebox{0.3cm}{${}^t$}\!\!\!
        \left(
          w,u,v,
          -u^2-\dfrac{3v^2}{2}+uw^2+vw^3
          -\dfrac{w^4}{4}+\dfrac{w^5}{5}-\dfrac{w^6}{6}
        \right)
            \qquad (u,v,w\in {\boldsymbol{K}}).$$ If we define ${\mathcal{G}}:{\boldsymbol{K}}^3\to P({\boldsymbol{K}}^4)\cong P(({\boldsymbol{K}}^4)^{*})$ by $${\mathcal{G}}(u,v,w) =
        [-2uw-3vw^2+w^3-w^4+w^5,2u-w^2,3v-w^3,1]$$ using the homogeneous coordinate system, ${\mathcal{G}}$ gives the affine Gauss map of $F$. Then the Hessian $h$ of $F$ is $$\det
   {{\begin{pmatrix} 
      -2&0&2w\\
       0&-3&3w^2\\
       2w&3w^2&2u+6vw-3w^2+4w^3-5w^4
     \end{pmatrix}}}
   =6(   2u+6vw-w^2+4w^3-2w^4).$$ The asymptotic vector field is $\xi=(w,w^2,1)$. Hence we have $$\begin{aligned}
  h&=6(2u+6vw-w^2+4w^3-2w^4),\\
  h'&=12(3v+6w^2-w^3),\qquad
  h''=144w,\qquad 
  h'''=144,
 \end{aligned}$$ where $h'=dh(\xi)$, $h''=dh'(\xi)$ and $h'''=dh''(\xi)$. The Jacobi matrix of $(h,h',h'')$ at ${\boldsymbol{0}}$ is $${{\begin{pmatrix}      2 & * & * \\ 
     0 & 36 & * \\ 
     0 & 0 & 144
       \end{pmatrix}}}.$$ This implies that $\xi$ is $3$-nondegenerate at ${\boldsymbol{0}}$ but does not meet $I(F)=h^{-1}(0)$ at $p$ with multiplicity $4$, that is, $F$ has an $A_{4}$-inflection point at ${\boldsymbol{0}}$. On the other hand, ${\mathcal{G}}$ has the ${A}_3$-Morin singularity at ${\boldsymbol{0}}$. In fact, by the coordinate change $$U=2u-w^2,\qquad V=3v-w^3,\qquad W=w,$$ it follows that ${\mathcal{G}}$ is represented by a map germ $$(U,V,W)\longmapsto
            -(UW+VW^2+W^4,U,V).$$ This coincides with the typical ${A}_3$-Morin singularity given in (A.3) in [@SUY3].

Duality on wave fronts {#sec:dual}
======================

Let $P({\boldsymbol{K}}^{n+2})$ be the $(n+1)$-projective space over ${\boldsymbol{K}}$. We denote by $[x]\in P({\boldsymbol{K}}^{n+2})$ the projection of a vector $x={{\vphantom{(}}^t{(}}x^0,\dots,x^{n+1})\in {\boldsymbol{K}}^{n+2}$. Consider a $(2n+3)$-submanifold of ${\boldsymbol{K}}^{n+2}\times ({\boldsymbol{K}}^{n+2})^*$ defined by $$\widetilde C:=
       \{(x,y)\in {\boldsymbol{K}}^{n+2}\times ({\boldsymbol{K}}^{n+2})^*\,;\, x\cdot y=0\},$$ and also a $(2n+1)$-submanifold of $P({\boldsymbol{K}}^{n+2})\times P(({\boldsymbol{K}}^{n+2})^*)$ $$C:=\{([x],[y])\in P({\boldsymbol{K}}^{n+2})\times P(({\boldsymbol{K}}^{n+2})^*)
\,;\, x\cdot y=0\}.$$ As $C$ can be canonically identified with the projective tangent bundle $PTP({\boldsymbol{K}}^{n+2})$, it has a canonical contact structure: Let $\pi:\widetilde C\to C$ be the canonical projection, and define a $1$-from $$\omega:=\sum_{j=0}^{n+1} \left(x^jdy^j-y^j d{x^j}\right)$$ which is considered as a $1$-form of $\widetilde C$. The tangent vectors of the curves $t\mapsto (tx,y)$ and $t\mapsto (x,ty)$ at $(x,y)\in \widetilde C$ generate the kernel of $d\pi$. Since these two vectors also belong to the kernel of $\omega$ and $\dim(\ker\omega)=2n+2$, $$\Pi:=d\pi(\ker \omega)$$ is a $2n$-dimensional vector subspace of $T_{\pi(x,y)}C$. We shall see that $\Pi$ is the contact structure on $C$, which is defined to be [*the canonical contact structure*]{}: Let $U$ be an open subset of $C$ and $s:U\to {\boldsymbol{K}}^{n+2}
\times ({\boldsymbol{K}}^{n+2})^*$ a section of the fibration $\pi$. Since $d\pi\circ ds$ is the identity map, it can be easily checked that $\Pi$ is contained in the kernel of the $1$-form $s^*\omega$. Since $\Pi$ and the kernel of the $1$-form $s^*\omega$ are the same dimension, they coincide. Moreover, suppose that $p=\pi(x,y)\in C$ satisfies $x^i\ne 0$ and $y^j\ne 0$. We then consider a map of ${\boldsymbol{K}}^{n+1}\times ({\boldsymbol{K}}^{n+1})^*
\cong {\boldsymbol{K}}^{n+1}\times {\boldsymbol{K}}^{n+1}$ into ${\boldsymbol{K}}^{n+2}\times({\boldsymbol{K}}^{n+2})^*\cong {\boldsymbol{K}}^{n+2}\times {\boldsymbol{K}}^{n+2}$ defined by $$(a^0,\dots,a^{n},b^0,\dots,b^n)
  \mapsto 
  (a^0,\dots,a^{i-1},1,a^{i+1},\dots,a^n,b^0,\dots,b^{j-1},1,b^{j+1},\dots,b^n),$$ and denote by $s_{i,j}$ the restriction of the map to the neighborhood of $p$ in $C$. Then one can easily check that $$s_{i,j}^*
  \left[
       \omega\wedge \left(\bigwedge\nolimits^{n}d\omega\right)
  \right]$$ does not vanish at $p$. Thus $s_{i,j}^*\omega$ is a contact form, and the hyperplane field $\Pi$ defines a canonical contact structure on $C$. Moreover, the two projections from $C$ into $P({\boldsymbol{K}}^{n+2})$ are both Legendrian fibrations, namely we get a double Legendrian fibration. Let $f=[F]:M^{n}\to P({\boldsymbol{K}}^{n+2})$ be a front. Then there is a Legendrian immersion of the form $L=([F],[G]):M^{n}\to C$. Then $g=[G]:M^{n}\to P(({\boldsymbol{K}}^{n+2})^*)$ satisfies and . In particular, $L:=\pi(F,G):M^{n}\to C$ gives a Legendrian immersion, and $f$ and $g$ can be regarded as mutually dual wave fronts as projections of $L$.

Since our contact structure on $C$ can be identified with the contact structure on the projective tangent bundle on $P({\boldsymbol{K}}^{n+2})$, we can apply the criterion of $A_k$-singularities as in Fact \[eq:fact\_suy\]. Thus a nondegenerate singular point $p$ is an $A_k$-singular point of $f$ if and only if the null vector field $\eta$ of $f$ (as a wave front) is $(k-1)$-nondegenerate at $p$, but does not meet the hypersurface $S(f)$ with multiplicity $k$ at $p$. Like as in the proof of Theorem \[thm:A\], we may assume that ${{\vphantom{F}}^t{F}}(p)\cdot F(p)\ne0$ and $G(p)\cdot {{\vphantom{G}}^t{G}}(p)\ne0$ simultaneously by a suitable affine transformation of ${\boldsymbol{K}}^{n+2}$, even when ${\boldsymbol{K}}={\boldsymbol{C}}$. Since $(F_{x^1},\dots,F_{x^n}, F, {{\vphantom{G}}^t{G}})$ is a regular $(n+2)\times (n+2)$-matrix if and only if $f=[F]$ is an immersion, the assertion immediately follows from the identity $$\label{eq:split2}
  {{\begin{pmatrix} 
    h_{11}& \dots & h_{1n}& 0 & *\\
    \vdots & \ddots & \vdots & \vdots & \vdots  \\
    h_{n1}& \dots & h_{nn} & 0 & *\\
    0 & \dots & 0 & 0 & G\cdot {{\vphantom{G}}^t{G}} \\
    * & \dots & * & {{\vphantom{F}}^t{F}}\cdot F & 0
     \end{pmatrix}}}=
   {{\begin{pmatrix} 
     G_{x^1}\\ \vdots\\ G_{x^n}\\ G \\ {{\vphantom{F}}^t{F}}
     \end{pmatrix}}} 
   (F_{x^1},\dots,F_{x^n}, F, {{\vphantom{G}}^t{G}}).$$

Let $g:M^2\to P(({\boldsymbol{R}}^4)^*)$ be the dual of $f$. We fix $p\in M^2$ and take a simply connected and connected neighborhood $U$ of $p$.

Then there are lifts $\hat f,\hat g:U\to S^3$ into the unit sphere $S^3$ such that $$\hat f\cdot \hat g=0,\qquad 
    d\hat f(v)\cdot \hat g=d\hat g(v)\cdot \hat f=0 \quad (v\in TU),$$ where $\cdot$ is the canonical inner product on ${\boldsymbol{R}}^4\supset S^3$. Since $\hat f\cdot \hat f=1$, we have $$d\hat f(v)\cdot \hat f(p)
          =0\qquad (v\in T_pM^2).$$ Thus $$d\hat f(T_pM^2)=\{\zeta \in S^3\,;\,
       \zeta\cdot \hat f(p)=\zeta \cdot \hat g(p)=0\},$$ which implies that $df(TM^2)$ is equal to the limiting tangent bundle of the front $g$. So we apply (2.5) in [@SUY2] for $g$: Since the singular set $S(g)$ of $g$ consists only of cuspidal edges and swallowtails, the Euler number of $S(g)$ vanishes. Then it holds that $$\chi(M_+)+\chi(M_-)=\chi(M^2)=\chi(M_+)-\chi(M_-)
           +i_2^+(f)-i_2^-(f),$$ which proves the formula.

When $n=2$, the duality of two fronts in the unit $2$-sphere $S^2$ (as the double cover of $P({\boldsymbol{R}}^3)$) plays a crucial role for obtaining the classification theorem in [@MU] for flat fronts with embedded ends in ${\boldsymbol{R}}^3$. Also, a relationship between the number of inflection points and the number of double tangents on certain class of simple closed regular curves in $P({\boldsymbol{R}}^3)$ is given in [@tu3]. (For the geometry and a duality of fronts in $S^2$, see [@A].) In [@porteous], Porteous investigated the duality between $A_k$-singular points and $A_k$-inflection points when $k=2,3$ on a surface in $S^3$.

Cuspidal curvature on $3/2$-cusps {#sec:cusp}
=================================

Relating to the duality between singular points and inflection points, we introduce a curvature on $3/2$-cusps of planar curves:

Suppose that $(M^2,g)$ is an oriented Riemannian manifold, $\gamma:I\to M^2$ is a front, $\nu(t)$ is a unit normal vector field, and $I$ an open interval. Then $t=t_0\in I$ is a $3/2$-cusp if and only if $\dot\gamma(t_0)={\boldsymbol{0}}$ and $\Omega(\ddot\gamma(t_0),\dddot\gamma(t_0))\ne 0$, where $\Omega$ is the unit $2$-form on $M^2$, that is, the Riemannian area element, and the dot means the covariant derivative. When $t=t_0$ is a $3/2$-cusp, $\dot\nu(t)$ does not vanish (if $M^2={\boldsymbol{R}}^2$, it follows from Proposition $\mathrm A'$). Then we take the (arclength) parameter $s$ near $\gamma(t_0)$ so that $|\nu'(s)|=\sqrt{g(\nu'(s),\nu'(s))}=1$ ($s\in I$), where $\nu'=d\nu/ds$. Now we define the [*cuspidal curvature*]{} $\mu$ by $$\mu:=
     2\operatorname{sgn}(\rho)\left. \sqrt{\left |{ds}/{d\rho}\right |}
     \right|_{s=s_0} \qquad (\rho=1/\kappa_g),$$ where we choose the unit normal $\nu(s)$ so that it is smooth around $s=s_0$ ($s_0=s(t_0)$). If $\mu>0$ (resp. $\mu<0$), the cusp is called [*positive*]{} (resp. [*negative*]{}). It is interesting phenomenon that the left-turning cusps have negative cuspidal curvature, although the left-turning regular curves have positive geodesic curvature (see Figure \[cusps\]).

![a positive cusp and a negative cusp[]{data-label="cusps"}](cusp23.eps){width="4cm"}

$(\mu>0)\qquad \qquad (\mu<0)$

Then it holds that $$\label{eq:general3}
 \mu=    \left.\frac{\Omega(\ddot \gamma(t),\dddot \gamma(t))}
     {|\ddot \gamma(t)|^{5/2}}\right|_{t=t_0}
    = 
    \left. 2\frac{\Omega(\nu(t),\dot \nu(t))}{         \sqrt{|\Omega(\ddot\gamma(t),\nu(t))|}}
\right|_{t=t_0}.$$ We now examine the case that $(M^2,g)$ is the Euclidean plane ${\boldsymbol{R}}^2$, where $\Omega(v,w)$ ($v,w\in {\boldsymbol{R}}^2$) coincides with the determinant $\det(v,w)$ of the $2\times 2$-matrix $(v,w)$. A [*cycloid*]{} is a rigid motion of the curve given by $c(t):=a(t-\sin t,1-\cos t)$ ($a>0$), and here $a$ is called the [*radius*]{} of the cycloid. The cuspidal curvature of $c(t)$ at $t\in 2\pi \mathbf Z$ is equal to $-1/\sqrt{a}$. In [@ume], the second author proposed to consider the curvature as the inverse of radius of the cycloid which gives the best approximation of the given $3/2$-cusp. As shown in the next proposition, $\mu^2$ attains this property:

Suppose that $\gamma(t)$ has $3/2$-cusp at $t=t_0$. Then by a suitable choice of the parameter $t$, there exists a unique cycloid $c(t)$ such that $$\gamma(t)-c(t)=o((t-t_0)^3),$$ where $o((t-t_0)^3)$ denotes a higher order term than $(t-t_0)^3$. Moreover, the square of the absolute value of cuspidal curvature of $\gamma(t)$ at $t=t_0$ is equal to the inverse of the radius of the cycloid $c$.

Without loss of generality, we may set $t_0=0$ and $\gamma(0)={\boldsymbol{0}}$. Since $t=0$ is a singular point, there exist smooth functions $a(t)$ and $b(t)$ such that $\gamma(t)=t^2(a(t),b(t))$. Since $t=0$ is a $3/2$-cusp, $(a(0),b(0))\ne {\boldsymbol{0}}$. By a suitable rotation of $\gamma$, we may assume that $b(0)\ne 0$ and $a(0)=0$. Without loss of generality, we may assume that $b(0)>0$. By setting $s=t\sqrt{b(t)}$, $\gamma(s)=\gamma(t(s))$ has the expansion $$\gamma(s)=(\alpha s^3, s^2)+o(s^3) \qquad (\alpha\ne 0).$$ Since the cuspidal curvature changes sign by reflections on ${\boldsymbol{R}}^2$, it is sufficient to consider the case $\alpha>0$. Then, the cycloid $$c(t):=\frac{2}{9\alpha^2}(t-\sin t,1-\cos t)$$ is the desired one by setting $s=t/(3\alpha)$.

It is well-known that the cycloids are the solutions of the brachistochrone problem. We shall propose to call the number $1/|\mu|^2$ the [*cuspidal curvature radius*]{} which corresponds the radius of the best approximating cycloid $c$.

During the second author’s stay at Saitama University, Toshizumi Fukui pointed out the followings: Let $\gamma(t)$ be a regular curve in ${\boldsymbol{R}}^2$ with non-vanishing curvature function $\kappa(t)$. Suppose that $t$ is the arclength parameter of $\gamma$. For each $t=t_0$, there exists a unique cycloid $c$ such that a point on $c$ gives the best approximation of $\gamma(t)$ at $t=t_0$ (namely $c$ approximates $\gamma$ up to the third jet at $t_0$). The angle $\theta(t_0)$ between the axis (i.e. the normal line of $c$ at the singular points) of the cycloid and the normal line of $\gamma$ at $t_0$ is given by $$\label{c:theta}
\sin \theta=\frac{\kappa^2}{\sqrt{\kappa^4+\dot\kappa^2}},$$ and the radius $a$ of the cycloid is given by $$\label{c:k}
a:=\frac{\sqrt{\kappa^4+\dot\kappa^2}}{|\kappa|^3}.$$ One can prove and by straightforward calculations. The cuspidal curvature radius can be considered as the limit.

[99]{} V. I. Arnold, [*The geometry of spherical curves and the algebra of quaternions*]{}, Uspekhi Mat. Nauk [**50**]{}(1995) 3-68, English transl. in Russian Math. Surveys [**50**]{} (1995).

T. Banchoff, T. Gaffney and C. McCrory, [Cusps of Gauss Mappings]{}, Pitman publ. (1982).

D. Bleecker and L. Wilson, [*Stability of Gauss maps*]{}, Illinois J. of Math. [**22**]{} (1978) 279–289.

S. Izumiya, D. Pei and T. Sano, [*Singularities of hyperbolic Gauss maps*]{}, Proc. London Math. Soc. [**86**]{} (2003), 485–512

S. Izumiya, [*Legendrian dualities and spacelike hypersurfaces in the lightcone*]{}, Moscow Math. J. [**9**]{} (2009), 325–357.

S. Murata and M. Umehara, [ *Flat surfaces with singularities in Euclidean $3$-space*]{}, J. of Differential Geometry, [**82**]{} (2009), 279–316.

I. R. Porteous, [ *Some remarks on duality in $S^3$*]{}, Geometry and topology of caustics, Banach Center Publ. [**50**]{}, Polish Acad. Sci., Warsaw, (2004), 217–226.

K. Saji, M. Umehara and K. Yamada, [*The geometry of fronts*]{}, Ann. of Math. [**169**]{} (2009), 491–529.

, [*Behavior of corank one singular points on wave fronts*]{}, Kyushu J. of Mathematics [**62**]{} (2008), 259–280.

, [*$A_k$-singularities of wave fronts*]{}, Math. Proc. Cambridge Philosophical Soc., [**146**]{} (2009), 731-746.

G. Thorbergsson and M. Umehara, [*Inflection points and double tangents on anti-convex curves in the real projective plane*]{}, Tôhoku Math. J. [**60**]{} (2008), 149–181. M. Umehara, , (in Japanese), The world of Singularities (ed. H. Arai, T. Sunada and K. Ueno) Nippon-Hyoron-sha Co., Ltd. (2005), 50–64. R. Uribe-Vargas, [*A projective invariant for swallowtails and godrons, and global theorems on the flecnodal curve*]{}, Moscow Math. J. [**6**]{}, (2006), 731–768.
