# Hyperplane families creating envelopes

Takashi Nishimura  Research Institute of Environment and Information Sciences,
Yokohama National University, 240-8501 Yokohama, JAPAN  nishimura-takashi-
<EMAIL_ADDRESS>
###### Abstract.

A simple geometric mechanism: “the locus of intersections of perpendicular
bisectors and normal lines”, often arises in many guises in Nonlinear
Sciences. In this paper, a new application of this simple geometric mechanism
is given. Namely, we show that this mechanism gives answers to all four basic
problems on envelopes created by hyperplane families (existence problem,
representation problem, equivalence problem of definitions, uniqueness
problem) at once.

###### Key words and phrases:

Hyperplane family, Frontal, Envelope, Creative, Creator, Mirror-image mapping,
Anti-orthotomic, Orthotomic, Cahn-Hoffman vector formula.

###### 2010 Mathematics Subject Classification:

57R45, 58C25

##  1\. Introduction

Throughout this paper, let $n$ be a positive integer. Moreover, all manifolds,
functions and mappings are of class $C^{\infty}$ unless otherwise stated.

A simple geometric mechanism: “the locus of intersections of perpendicular
bisectors and normal lines”, often arises in many guises in Physical Sciences.
For example, as Richard Feynman elegantly explained in [9], the orbit of a
planet around the sun can be understood as a consequence of this mechanism
under the assumption of the inverse-square law (see Figure 2 where the circle
is the hodograph of the velocity vectors of a planet , that is to say, the
circle is a curve drawn by the end points of the vectors that are parallel to
the velocity vectors and start at a fixed point $P$. The orbit of the planet
is similar to the locus of intersections $B_{t}$ of the perpendicular
bisectors of velocity vectors $\overrightarrow{PA_{t}}$ and the normal lines
to the circle at $A_{t}$). This is an example in Celestial Mechanics. In the
same book [9], one can find that even the historical discovery of atomic
nucleus due to Ernest Rutherford can be explained as a consequence of this
simple geometric mechanism (see Figure 2 where the center of circle $O$ is an
atomic nucleus. The orbit of an $\alpha$ particle is the locus of
intersections $B_{t}$ of the perpendicular bisectors of the segment
$\overline{PA_{t}}$ and the normal lines to the circle at $A_{t}$ ). This is
an example in Nuclear Physics.

Figure 1.  Locus similar to the orbit of a planet.

Figure 2.  Locus of an $\alpha$ particle.

In Crystallography, one can find such the mechanism in the so-called Wulff
construction for the equilibrium shape of a crystal. A brief explanation of
the Wulff construction is as follows. Given an equilibrium crystal, take an
arbitrary point $P$ inside the crystal and fix it. Georg Wulff discovered in
[20] the so-called Gibbs-Wulff theorem which asserts that the length from the
fixed point $P$ to the foot of the perpendicular to the tangent space to the
face of the crystal is proportional to its surface energy density of the face.
Let $\gamma:S^{2}\to\mathbb{R}$ be the surface energy density function of the
equilibrium crystal. The graph of $\gamma$ with respect to the polar
coordinates about the point $P$ defines the mapping
$g:S^{2}\to\mathbb{R}^{3}$. The mapping $g$ is often called the polar plot of
$\gamma$ or the $\gamma$-plot or the Wulff plot. Set $f=2g$ and suppose that
the image $f(S^{2})$ has the well-defined normal vectors at any point $f(x)$.
Then, by the Gibbs-Wulff theorem, the accurate shape of the crystal surface is
proportional to the shape obtained by our simple geometric mechanism: “the
locus of the intersection of the perpendicular bisector of the vector
$\overrightarrow{Pf(x)}$ and the normal line to $f(S^{2})$ at $f(x)$”. This is
the Wulff construction and the constructed shape is called the Wulff shape.
Notice that in general $f$ is a continuous mapping and thus from the viewpoint
of rigorous mathematics, the Wulff construction is not a well-defined
construction method in general. Nevertheless, Hoffman and Cahn showed in [11]
that if $\gamma:S^{2}\to\mathbb{R}$ is differentiable, then the image
$f(S^{2})$ has a well-defined normal vector at each point $f(x)$ and the set
$\\{\nabla\gamma(x)+\gamma(x)x\;|\;x\in S^{2}\\}$ is exactly the shape
obtained by our simple geometric mechanism for the point $P$ and the surface
$f(S^{2})$. The Wulff construction and the Cahn-Hoffman formula in the plane
is depicted in Figure 3. For details on the Wulff construction and Wulff
shapes, see for instance [8, 10].

Figure 3. The Wulff construction and the Cahn-Hoffman vector formula in the
plane.

Moreover, it is a surprising fact that our simple geometric mechanism: “the
locus of intersections of perpendicular bisectors and normal lines”can be
applied even to Seismic Survey (see 7.14 (9) of [6]).

In Mathematics, our simple geometric mechanism: “the locus of intersections of
perpendicular bisectors and normal lines”is called the anti-orthotomic of a
mapping $f$ having a well-defined normal vector to its image at each point
(for details on anti-orthotomics, see 7.14 of [6]. See also [15] where anti-
orthotomics are generalized to frontals and [16] where more elementary
explanation on anti-orthotomics can be found). In Mathematics as well, there
are examples where anti-orthotomics are effectively applied (see [6]).

In order to understand better the powerfulness of the simple geometric
mechanism, we would like to have more striking examples in Mathematics where
anti-orthotomics are effectively applied. Namely, we want to seek mathematical
problems which can be geometrically solved by our simple geometric mechanism
though it seems difficult to solve them by other methods. This is the
primitive motivation of this paper. In this paper, we show that the existence
and uniqueness problem of envelopes for a given hyperplane family is one of
such problems. Namely, we give a necessary and sufficient condition (see
Definition 2) for a given hyperplane family to create an envelope. And then,
we give a necessary and sufficient condition for the uniqueness of created
envelopes if the given hyperplane family creates an envelope. It seems
difficult to prove that the condition given in Definition 2 is actually a
sufficient condition to create envelopes by other methods. In order to apply
our simple geometric mechanism, we need some geometric objects to which the
normal line can be reasonably well-defined at each point. Hyperplane families
themselves are far from the reasonable geometric objects for our purpose. The
reasonable geometric objects are frontals (the definition of frontal is given
in the next paragraph). In order to obtain a frontal from a given hyperplane
family, the mirror-image mapping will be locally introduced. Then, it turns
out that if the given hyperplane family is creative (see Definition 2 below),
then the mirror-image mapping is actually a frontal such that the normal line
at each point intersects the corresponding hyperplane. Thus, we can apply the
anti-orthotomic method developed in [15] to obtain Theorem 1 and Theorem 2.
The existence and uniqueness problem of envelopes for a given hyperplane
family can be easily interpreted as the existence and uniqueness problem of
solutions for a certain type of system of first order differential equations
with one constraint condition. In the author’s opinion, one of the most
attractive features of our simple geometric mechanism is that it can make all
solutions and their precise expressions clear in one shot by geometry without
the need to solve the corresponding system of differential equations with one
constraint condition.

Let $S^{n}$ be the $n$-dimensional unit sphere in the $(n+1)$-dimensional
vector space $\mathbb{R}^{n+1}$. Given a point $P$ of $\mathbb{R}^{n+1}$ and
an $(n+1)$-dimensional unit vector ${\bf n}\in S^{n}\subset\mathbb{R}^{n+1}$,
the hyperplane $H_{\left(P,{\bf n}\right)}$ relative to $P$ and ${\bf n}$ is
naturally defined as follows, where the dot in the center stands for the
standard scalar product of two vectors $(X-P)$ and ${\bf n}$ in the vector
space $\mathbb{R}^{n+1}$.

$H_{\left(P,{\bf n}\right)}=\\{X\in\mathbb{R}^{n+1}\;|\;(X-P)\cdot{\bf
n}=0\\}.$

Let $N$ be an $n$-dimensional manifold without boundary. Given two mappings
$\widetilde{\varphi}:N\to\mathbb{R}^{n+1}$ and $\widetilde{\nu}:N\to S^{n}$,
the _hyperplane family_
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ relative to
$\widetilde{\varphi}$ and $\widetilde{\nu}$ is naturally defined as follows.

$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left\\{H_{\left(\widetilde{\varphi}(x),\widetilde{\nu}(x)\right)}\right\\}_{x\in
N}}.$

A mapping $\widetilde{f}:N\to\mathbb{R}^{n+1}$ is called a frontal if there
exists a mapping $\widetilde{\nu}:N\to S^{n}$ such that
$d\widetilde{f}_{x}({\bf v})\cdot\widetilde{\nu}(x)=0$ for any $x\in N$ and
any ${\bf v}\in T_{x}N$, where two vector spaces
$T_{\widetilde{f}(x)}\mathbb{R}^{n+1}$ and $\mathbb{R}^{n+1}$ are identified.
By definition, it is natural to call $\widetilde{\nu}:N\to S^{n}$ a Gauss
mapping of the frontal $\widetilde{f}$. The notion of frontal has been
recently investigated (for instance, see [13]). In this paper, as the
definition of envelope created by a hyperplane family, the following is
adopted.

######  Definition 1.

Let $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ be a
hyperplane family. A mapping $\widetilde{f}:N\to\mathbb{R}^{n+1}$ is called an
envelope created by
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ if the
following two conditions are satisfied.

  1. (a)

$\widetilde{f}(x)\in
H_{\left(\widetilde{\varphi}(x),\widetilde{\nu}(x)\right)}$ for any $x\in N$.

  2. (b)

$d\widetilde{f}_{x}({\bf v})\cdot\widetilde{\nu}(x)=0$ for any $x\in N$ and
any ${\bf v}\in T_{x}N$.

In other words, an envelope created by
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is a mapping
$\widetilde{f}:N\to\mathbb{R}^{n+1}$ giving a solution of the following system
of first order differential equations with one constraint condition, where
$\left(U,\left(x_{1},\ldots,x_{n}\right)\right)$ is an arbitrary coordinate
neighborhood of $N$ such that $x\in U$.

$\left\\{\begin{array}[]{ccc}\frac{\partial\widetilde{f}}{\partial
x_{1}}(x)\cdot\widetilde{\nu}(x)&=&0,\\\ \vdots&{}\hfil&{}\hfil\\\
\frac{\partial\widetilde{f}}{\partial x_{n}}(x)\cdot\widetilde{\nu}(x)&=&0,\\\
\left(\widetilde{f}(x)-\widetilde{\varphi}(x)\right)\cdot\widetilde{\nu}(x)&=&0.\end{array}\right.$

By definition, any envelope $\widetilde{f}:N\to\mathbb{R}^{n+1}$ created by a
hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ must be a
frontal with Gauss mapping $\widetilde{\nu}:N\to S^{n}$. For details on
envelopes created by families of plane regular curves, refer to [6]. In
Chapter 5 of [6], several definitions for envelope are given. For a hyperplane
family $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$,
Definition 1 is a generalization of their definition $E_{2}$ from a viewpoint
of parametrization ($E_{2}$ envelope is a variety tangent to all lines of the
given line family. Thus, in the case of plane, an envelope defined by
Definition 1 is the same notion of $E_{2}$ envelope. For details on the
definition $E_{2}$, see 5.12 of [6]). The following definition, which may be
regarded as a higher dimensional generalization of $E_{1}$ from a viewpoint of
parametrization ($E_{1}$ envelope is the set of the limits of intersections
with nearby members of the given line family.  For details on the definition
$E_{1}$, see 5.8 of [6] and for the relation between Definition 2 in the plane
case and $E_{1}$, see Subsection 2.3), is the key notion for this paper.

######  Definition 2.

Let $N$ be an $n$-dimensional manifold without boundary and let
$\widetilde{\varphi}:N\to\mathbb{R}^{n+1}$, $\widetilde{\nu}:N\to S^{n}$ be
mappings. Let $\widetilde{\gamma}:N\to\mathbb{R}$ be the function defined by
$\widetilde{\gamma}(x)=\widetilde{\varphi}(x)\cdot\widetilde{\nu}(x)$. Let
$T^{*}S^{n}$ be the cotangent bundle of $S^{n}$. A hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is said to be
creative if there exists a mapping $\widetilde{\Omega}:N\to T^{*}S^{n}$ with
the form
$\widetilde{\Omega}(x)=\left(\widetilde{\nu}(x),\widetilde{\omega}(x)\right)$
such that for any $x_{0}\in N$ the equality
$d\widetilde{\gamma}=\widetilde{\omega}$ holds as germs of $1$-form at
$x_{0}$.

${T^{*}S^{n}}$${N}$${S^{n}}$$\scriptstyle{\widetilde{\Omega}}$$\scriptstyle{\widetilde{\nu}}$

Namely, $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is
creative if there exists a $1$-form $\widetilde{\Omega}$ along
$\widetilde{\nu}$ such that for any $x_{0}\in N$ by using of a coordinate
neighborhood $\left(U,\left(x_{1},\ldots,x_{n}\right)\right)$ of $N$ at
$x_{0}$ and a normal coordinate neighborhood
$\left(V,\left(\Theta_{1},\ldots,\Theta_{n}\right)\right)$ of $S^{n}$ at
$\widetilde{\nu}\left(x_{0}\right)$, the $1$-form germ $d\widetilde{\gamma}$
at $x_{0}$ is expressed as follows.

$d\widetilde{\gamma}=\sum_{i=1}^{n}\left(\widetilde{\omega}(x)\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Pi}_{\left(\widetilde{\nu}(x),\widetilde{\nu}(x_{0})\right)}\left(\frac{\partial}{\partial\Theta_{i}}\right)\right)\right)d\left(\Theta_{i}\circ\widetilde{\nu}\right),$

where  a normal coordinate neighborhood
$\left(V,\left(\Theta_{1},\ldots,\Theta_{n}\right)\right)$ is a local
coordinate neighborhood at $\widetilde{\nu}\left(x_{0}\right)$ obtained by the
inverse mappping of the exponential mapping at
$\widetilde{\nu}\left(x_{0}\right)$, $S^{n}$ inherits its metric from the
ambient space $\mathbb{R}^{n+1}$ and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Pi}_{\left(\widetilde{\nu}(x),\widetilde{\nu}(x_{0})\right)}:T_{\widetilde{\nu}\left(x_{0}\right)}S^{n}\to
T_{\widetilde{\nu}\left(x\right)}S^{n}$ is the Levi-Civita translation. Notice
that our objective manifold is the unit sphere $S^{n}$ with metric inherited
from $\mathbb{R}^{n+1}$. Therefore, the Levi-Civita translation
$\Pi_{\left(\widetilde{\nu}(x),\widetilde{\nu}(x_{0})\right)}$ is the
restriction of the rotation $R:\mathbb{R}^{n+1}\to\mathbb{R}^{n+1}$ satisfying
$R(\widetilde{\nu}(x_{0}))=\widetilde{\nu}(x)$ to the tangent space
$T_{\widetilde{\nu}\left(x_{0}\right)}S^{n}$. In particular, in the case
$n=1$, a normal coordinate $\Theta$ at $\widetilde{\nu}\left(x\right)$ is
nothing but the radian (or, its negative) between two unit vectors
$\widetilde{\nu}\left(x_{0}\right)$ and $\widetilde{\nu}\left(x\right)$ and
the Levi-Civita translation
$\Pi_{\left(\widetilde{\nu}(x),\widetilde{\nu}(x_{0})\right)}$ is just the
restriction of the plane rotation through $\Theta$ to the tangent space
$T_{\widetilde{\nu}\left(x_{0}\right)}S^{1}$.

######  Remark 1.1.

  1. (1)

It is reasonable to say that $\widetilde{\gamma}$ is totally differentiable
with respect to $\widetilde{\nu}$ if
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is creative.

  2. (2)

For a creative hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$, the map-germ
$\left(\widetilde{\nu},\widetilde{\gamma}\right):(N,x_{0})\to
S^{n}\times\mathbb{R}$ at any $x_{0}\in N$ is called an opening of
$\widetilde{\nu}:(N,x_{0})\to S^{n}$ (for opening germs, see for example
[12]). Thus, Definition 2 may be regarded as a globalization of the notion of
opening.

  3. (3)

Definition 2 may be interpreted as follows. Let $\theta$ be a canonical
contact $1$-form on $J^{1}(S^{n},\mathbb{R})$, namely at any
$(X_{0},Y_{0},P_{0})\in J^{1}\left(S^{n},\mathbb{R}\right)$ the $1$-form germ
$\theta$ is expressed as $\theta=dY-\sum_{i=1}^{n}C_{i}d\Theta_{i}$, where
$\left(\Theta_{1},\ldots,\Theta_{n}\right)$ is a normal coordinate system at
$X_{0}$ and $\left(\Theta_{1},\ldots,\Theta_{n},Y,C_{1},\ldots,C_{n}\right)$
is a canonical coordinate system of $J^{1}\left(S^{n},\mathbb{R}\right)$ at
$(X_{0},Y_{0},P_{0})$. Then, a hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is creative
if there exists a mapping ${\Omega}:N\to J^{1}\left(S^{n},\mathbb{R}\right)$
with the form
${\Omega}(x)=\left(\widetilde{\nu}(x),\widetilde{\gamma}(x),\widetilde{c}_{1}(x),\ldots,\widetilde{c}_{n}(x)\right)$
such that ${\Omega}^{*}\theta=0$, where
$\widetilde{c}_{1},\ldots,\widetilde{c}_{n}:N\to\mathbb{R}$ are some
functions.

${J^{1}\left(S^{n},\mathbb{R}\right)}$${N}$${S^{n}}$$\scriptstyle{\Omega}$$\scriptstyle{\widetilde{\nu}}$

Notice that in Legendrian Singularity Theory, at any point $x_{0}\in N$, the
map-germ ${\Omega}:\left(N,x_{0}\right)\to J^{1}\left(S^{n},\mathbb{R}\right)$
is assumed to be immersive and it is called a Legendrian immersion; and for
Legendrian immersion ${\Omega}$, the mapping $N\ni
x\mapsto\left(\widetilde{\nu}(x),\widetilde{\gamma}(x)\right)$ is called a
wavefront or front (for details on Legendrian Singularity Theory and fronts,
see for instance [1, 2, 17]). On the other hand, in Definition 2, ${\Omega}$
is not assumed to be immersive in general and the mapping ${\Omega}$ is called
a Legendrian mapping (for details on Legendrian mappings, see for instance
[12, 13, 18]). Thus, in Definition 2, in general, the set-germ
$\left({\Omega}(N),{\Omega}\left(x_{0}\right)\right)$ may be singular at some
point $x_{0}\in N$ (for example, see Example 4.1(4)).

  4. (4)

Notice that the $1$-form $\widetilde{\Omega}$ along $\widetilde{\nu}$ in
Definition 2 is not necessarily the pullback of a $1$-form over $S^{n}$ by
$\widetilde{\nu}$ (for example, see Example 4.1(3), (4)) and the
“creativeness” does not depend on the particular choice of
$\widetilde{\varphi},\widetilde{\nu}$ and depends only on the hyperplane
family $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$. In
the case that $N=S^{n}$ and $\widetilde{\nu}:S^{n}\to S^{n}$ is the identity
mapping, for any $\widetilde{\varphi}:S^{n}\to\mathbb{R}^{n+1}$ the hyperplane
family $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is
always creative by the following equality.

$d\widetilde{\gamma}=\sum_{i=1}^{n}\frac{\partial\widetilde{\gamma}}{\partial\Theta_{i}}d\Theta_{i}.$

More generally, if $\widetilde{\gamma}:U\to\mathbb{R}$ may be expressed as the
composition of $\widetilde{\nu}:U\to S^{n}$ and a certain function
$\xi:S^{n}\to\mathbb{R}$ over an open set $U\subset N$, then the hyperplane
family
$\mathcal{H}_{\left(\widetilde{\varphi}|_{U},\widetilde{\nu}|_{U}\right)}$ is
creative. However, there are examples showing that there does not exist a
function $\widetilde{\alpha}:S^{n}\to\mathbb{R}$ such that
$\widetilde{\gamma}=\widetilde{\alpha}\circ\widetilde{\nu}$ although
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is creative
(for example, see Example 4.1(3), (4)). Moreover, there are many examples such
that
$\mathcal{H}_{\left(\widetilde{\varphi}|_{U},\widetilde{\nu}|_{U}\right)}$ is
not creative. For instance, for any constant mapping
$\widetilde{\nu}:\mathbb{R}\to S^{1}$, the line family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is not
creative where $\widetilde{\varphi}:\mathbb{R}\to\mathbb{R}^{2}$ is defined by
$\widetilde{\varphi}(t)=t^{2}\widetilde{\nu}(t)$. And, it is clear in this
case that $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$
does not create an envelope in the sense of Definition 1. However, it is
easily seen that

$\displaystyle\qquad\qquad\quad\mathcal{D}$ $\displaystyle=$
$\displaystyle\left\\{\left(X_{1},X_{2}\right)\in\mathbb{R}^{2}\,|\,\exists
t\mbox{ s.t. }F\left(X_{1},X_{2},t\right)=\frac{\partial F}{\partial
t}\left(X_{1},X_{2},t\right)=0\right\\}$ $\displaystyle=$
$\displaystyle\left\\{(X_{1},X_{2})\in\mathbb{R}^{2}\,|\,\left(X_{1},X_{2}\right)\cdot\widetilde{\nu}(0)=0\right\\}\neq\emptyset,$

where
$F\left(X_{1},X_{2},t\right)=\left(\left(X_{1},X_{2}\right)-\widetilde{\varphi}(t)\right)\cdot\widetilde{\nu}(t)$.
Thus, for this example, the envelope defined by Definition 1 is different from
the envelope in the sense of classical definition (see 5.3 of [6]), For more
examples on creative/non-creative hyperplane families and on comparison of
Definition 2 with the classical envelope $\mathcal{D}$, see Section 4.
Therefore, it seems that the current situation on both the definitions of
envelope and the relation of the creative condition (Defnition 2) with an
envelope seems to be wrapped in mystery.

By definition, any frontal $\widetilde{f}:N\to\mathbb{R}^{n+1}$ with Gauss
mapping $\widetilde{\nu}:N\to S^{n}$ is an envelope created by
$\mathcal{H}_{(\widetilde{f},\widetilde{\nu})}$. Therefore, the notion of
envelope created by a hyperplane family is the same as the notion of frontal.
Moreover, it is clear that for any mapping $\widetilde{\nu}:N\to S^{n}$, a
constant mapping $\widetilde{f}:N\to\mathbb{R}^{n+1}$ is an envelope created
by $\mathcal{H}_{\left(\widetilde{f},\widetilde{\nu}\right)}$. On the other
hand, for a constant mapping $\widetilde{\nu}:\mathbb{R}\to S^{1}$, if the
line family $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$
does not create an envelope then
$\widetilde{\varphi}:\mathbb{R}\to\mathbb{R}^{2}$ must be not constant. From
these elementary observations, it is natural to ask to obtain a necessary and
sufficient condition for a given hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ to create an
envelope $\widetilde{f}:N\to\mathbb{R}^{n+1}$ in terms of
$\widetilde{\gamma}:N\to\mathbb{R}$ and $\widetilde{\nu}:N\to S^{n}$.
Moreover, it is also desirable to solve the following two incidentally.
“Suppose that a given hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ creates an
envelope $\widetilde{f}:N\to\mathbb{R}^{n+1}$. Then, obtain a representation
formula of $\widetilde{f}$.” “Suppose that $n=1$. Then, find the precise
relation between $E_{1}$ envelope and $E_{2}$ envelope.” In this paper, as an
application of our simple geometric mechanism, all of these problems are
solved as follows.

######  Theorem 1.

Let $N$ be an $n$-dimensional manifold without boundary and let
$\widetilde{\varphi}:N\to\mathbb{R}^{n+1}$, $\widetilde{\nu}:N\to S^{n}$ be
mappings. Then, the following three hold.

  1. (1)

The hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ creates an
envelope if and only if it is creative.

  2. (2)

Suppose that the hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ creates an
envelope $\widetilde{f}:N\to\mathbb{R}^{n+1}$. Then, for any $x\in N$, under
the canonical identifications $T^{*}_{\widetilde{\nu}(x)}S^{n}\cong
T_{\widetilde{\nu}(x)}S^{n}\subset
T_{\widetilde{\nu}(x)}\mathbb{R}^{n+1}\cong\mathbb{R}^{n+1}$, the
$(n+1)$-dimensional vector $\widetilde{f}(x)$ is represented as follows.

$\widetilde{f}(x)=\widetilde{\omega}(x)+\widetilde{\gamma}(x)\widetilde{\nu}(x),$

where the $(n+1)$-dimensional vector $\widetilde{\omega}(x)$ is identified
with the corresponding $n$-dimensional cotangent vector
$\widetilde{\omega}(x)$ under these identifications.

  3. (3)

Suppose that $n=1$. Then, the line family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ creates an
envelope ($E_{2}$-envelope) if and only if it creates an $E_{1}$ envelope.
Moreover, these two envelopes are exactly the same.

By Theorem 1, it is natural to call $\widetilde{\omega}$ the creator for an
envelope $\widetilde{f}$ created by
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$. Recall that
$E_{1}$ envelope (resp., $E_{2}$ envelope) is the set of the limit of
intersections with nearby lines (resp., a parametrization tangent to all
members of the given family). Thus, even in the case of plane, $E_{2}$
envelope is exactly the same as the envelope in Definition 1.

The key idea for the proof of Theorem 1 is to regard the given hyperplane
family as a moving mirror parametrized by $x\in N$. Then, for any parameter
$x_{0}\in N$, by taking a point $P\in\mathbb{R}^{n+1}$ outside the mirror
$H_{\left(\widetilde{\varphi}(x_{0}),\widetilde{\nu}\left(x_{0}\right)\right)}$,
the mirror-image

$f_{{}_{P}}(x)=2\left(\left(\widetilde{\varphi}(x)-P\right)\cdot\widetilde{\nu}(x)\right)\widetilde{\nu}(x)+P$

of $P$ by the mirror
$H_{\left(\widetilde{\varphi}(x),\widetilde{\nu}(x)\right)}$ must have the
same information as the mirror since the mirror is reconstructed as the
perpendicular bisectors of the segment $\overline{Pf_{{}_{P}}(x)}$, where $x$
is a point in a sufficiently small neighborhood $U_{{}_{P}}$ of $x_{0}$.

Figure 4.  The mirror-image mapping
$f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$.

Hence, investigation of the given hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi}|_{{}U_{{}P}},\widetilde{\nu}|_{{}U_{{}P}}\right)}$
may be replaced with analyzing the associated mirror-image mapping
$f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ (see Figure 4). This suggests
applicability of results in [15] to the problem of this paper.

A sketch of the proof of Theorem 1 (1) may be given as follows. Suppose that
the hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is creative.
Then, by definition, there exists a mapping $\widetilde{\Omega}:N\to
T^{*}S^{n}$ having the form
$\widetilde{\Omega}(x)=\left(\widetilde{\nu}(x),\widetilde{\omega}(x)\right)$
such that the equality $d\widetilde{\gamma}=\widetilde{\omega}$ holds as germs
of $1$-form at $x_{0}$. Then, by investigating the Jacobian matrix of the
mirror-image mapping $f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ at $x\in
U_{{}_{P}}$ directly, it turns out that for any $x\in U_{{}_{P}}$ the non-zero
vector

${\bf
v}_{{}_{P}}(x)=\sum_{i=1}^{n}\left(\left(\widetilde{\omega}\left(x\right)-P\right)\left(\frac{\partial}{\partial\Theta_{\left(i,\widetilde{\nu}(x)\right)}}\right)\right)\frac{\partial}{\partial\Theta_{\left(i,\widetilde{\nu}(x)\right)}}-\left(\left(\widetilde{\varphi}(x)-P\right)\cdot\widetilde{\nu}\left(x\right)\right)\widetilde{\nu}\left(x\right)$

is perpendicular to the vector $d\left(f_{{}_{P}}\right)_{x}\left({\bf
v}\right)$ for any ${\bf v}\in T_{x}N$, where $\mathbb{R}^{n+1}$,
$T_{\widetilde{\nu}\left(x\right)}\mathbb{R}^{n+1}$ and
$T^{*}_{\widetilde{\nu}\left(x\right)}\mathbb{R}^{n+1}$ are identified and
$\frac{\partial}{\partial\Theta_{\left(i,\widetilde{\nu}(x)\right)}}=P_{\left(\widetilde{\nu}(x),\widetilde{\nu}\left(x_{0}\right)\right)}\left(\frac{\partial}{\partial\Theta_{i}}\right)$.
Thus, $f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ is a frontal. From the
construction, the mapping $\widetilde{f}_{{}_{P}}={\bf
v}_{{}_{P}}+f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ must be exactly the same
as the mapping $\widetilde{f}_{{}_{P}}$ given in Theorem 1 of [15]. Therefore,
by Theorem 1 of [15] asserting that $\widetilde{f}_{{}_{P}}$ satisfies both
conditions (a), (b) of Definition 1, $\widetilde{f}_{{}_{P}}$ is an envelope
created by the hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi}|_{U_{{}_{P}}},\widetilde{\nu}|_{U_{{}_{P}}}\right)}$.
The mapping $\widetilde{f}_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ is called
the anti-orthotomic of $f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ relative to
$P$. Calculation shows

$None$
$\widetilde{f}_{{}_{P}}(x_{0})=\widetilde{\omega}\left(x_{0}\right)+\widetilde{\gamma}\left(x_{0}\right)\widetilde{\nu}\left(x_{0}\right).$

Thus, unlike $f_{{}_{P}}(x_{0})$, the location $\widetilde{f}_{{}_{P}}(x_{0})$
does not depend on the particular choice of $P$. In other words, in order to
discover the formula $(*)$, the role of $P$ is merely an auxiliary point just
like an auxiliary line in elementary geometry (see Figure 5).

Figure 5. The location $\widetilde{f}_{{}_{P}}(x_{0})$ does not depend on the
particular choice of $P$.

Since $x_{0}$ is an arbitrary point of $N$, the hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ creates an
envelope $\widetilde{f}:N\to\mathbb{R}^{n+1}$.

Conversely, suppose that the given hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ creates an
envelope $\widetilde{f}:N\to\mathbb{R}^{n+1}$. Then, the mirror-image mapping
$f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ (resp., the mapping
$g_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ defined by
$g_{{}_{P}}(x)=\left(\widetilde{f}(x)-P\right)\cdot\widetilde{\nu}(x)+P$) is
called the orthotomic (resp., pedal) of $\widetilde{f}|_{U_{{}_{P}}}$ relative
to the point $P$. It is known that both the orthotomic $f_{{}_{P}}$ and the
pedal $g_{{}_{P}}$ are frontals (see Proposition 1 and Corollary 1 of [15]).
We prefer to investigate the orthotomic $f_{{}_{P}}$ rather than the pedal
$g_{{}_{P}}$ because its Gauss mapping $\nu_{{}_{P}}:U_{{}_{P}}\to S^{n}$ has
characteristic properties:
$\nu_{{}_{P}}(x)=\frac{\widetilde{f}(x)-f_{{}_{P}}(x)}{||\widetilde{f}(x)-f_{{}_{P}}(x)||}$
and $\widetilde{\nu}(x)\cdot\nu_{{}_{P}}(x)\neq 0$ for any $x\in U_{{}_{P}}$,
and thus we can take a bird’s eye view of $\widetilde{f}(x)$. Set
$\widetilde{\omega}(x)=\widetilde{f}(x)-\widetilde{\gamma}(x)\widetilde{\nu}(x)$
and
$\widetilde{\Omega}(x)=\left(\widetilde{\nu}(x),\widetilde{\omega}(x)\right)$
for any $x\in U_{{}_{P}}$. Then, under the identification of
$\mathbb{R}^{n+1}$ and $T^{*}_{\widetilde{\nu}(x)}\mathbb{R}^{n+1}$,
$\widetilde{\Omega}$ having the form
$\widetilde{\Omega}(x)=\left(\widetilde{\nu}(x),\widetilde{\omega}(x)\right)$
is a well-defined mapping $U_{{}_{P}}\to T^{*}S^{n}$. By investigating the
Jacobian matrix of the mirror image mapping $f_{{}_{P}}$ at $x\in U_{{}_{P}}$
directly again, it turns out that $\widetilde{\omega}$ is actually the creator
for the envelope $\widetilde{f}|_{U_{{}_{P}}}$. Since the vector
$\widetilde{\omega}(x_{0})$ does not depend on the particular choice of $P$
and the point $x_{0}$ is an arbitrary point of $N$,
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is creative.

Theorem 1 (2) is a direct by-product of the proof of Theorem 1 (1) (see Figure
5). Theorem 1 (3) seems to be not a direct by-product of the proof of Theorem
1 (1) although it can be proved relatively easily by using the above argument
(see Subsection 2.3).

When $N=S^{n}$ and $\widetilde{\nu}:S^{n}\to S^{n}$ is the identity mapping,
it is easily seen $\widetilde{\omega}(x)=\nabla\widetilde{\gamma}(x)$.
Therefore, in the case that $N=S^{n}$ and $\widetilde{\nu}:S^{n}\to S^{n}$ is
the identity mapping,  Theorem 1 (2) has been known as the Cahn-Hoffman vector
formula ([11]). Theorem 1 (2) is a comprehensive generalization of their
formula.  Any Wulff shape is clearly a convex body and conversely it is known
that any convex body can be constructed by the Wulff construction (for
instance, see [19]). There are many Wulff shapes such that the surface energy
density functions $\gamma:S^{n}\to\mathbb{R}$ are not differentiable (convex
polytopes are typical examples). Thus, for studing Wulff shapes having non-
smooth surface energy functions, it is very significant to answer the
following two problems: “(a) Generalize Cahn-Hoffman vector formula to the
corresponding formula for any $\widetilde{\nu}:N\to S^{n}$”ȧnd “(b) Resolution
of singularities of the boundary of a convex body having non-smooth boundary
by a frontal $\widetilde{f}:S^{n}\to\mathbb{R}^{n+1}$”. By Theorem 1 (2), the
problem (a) is completely solved. As for the problem (b), to the best of
author’s knowledge, only the boundary of a square has been realized as a
frontal $\widetilde{f}:S^{1}\to\mathbb{R}^{2}$ so far (see [15]). Although
there are apparently no published proofs at present, it is a comparatively
straightforward generalization of this result to show that the boundary of a
convex polygon is realized as a frontal
$\widetilde{f}:S^{1}\to\mathbb{R}^{2}$. However, even in the plane case, the
problem (b) for the boundary of a convex body in general seems to be wrapped
in mystery.

Moreover, Theorem 1 (2) might be useful even for the study of force problems
in higher dimensional vector spaces. In [4], Petr Blaschke discovered that
pedal coordinates are more suitable settings to study force problems in
$\mathbb{R}^{2}$. Readers who want to confirm their usefulness are recommeded
to refer to [4] (see also 7.24 (6) of [6] though this is not a force problem
but a very suitable problem for understanding how useful pedal coordinates
are). Theorem 1 (2) may be regarded as a higher dimensional generalization of
pedal coordinates. Hence, it is expected that Theorem 1 (2) is a very suitable
expression to study force problems etc. in all finite-dimensional vector
spaces over $\mathbb{R}$. Example 4.2 (2) might be regarded as examples in
which higher dimensional version of pedal coordinates are effectively used.

As an application of Theorem 1, a characterization for a hyperplane family to
create a unique envelope is given as follows.

######  Theorem 2.

Let $\widetilde{\varphi}:N\to\mathbb{R}^{n+1}$, $\widetilde{\nu}:N\to S^{n}$
be mappings. Then, the hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ creates a
unique envelope if and only if it is creative and the set consisting of
regular points of $\widetilde{\nu}$ is dense in $N$.

Under the assumption that $\Omega$ in Remark 1.1 (2) is immersive and some
conditions are satisfied, a unique existence result of envelopes for
hyperplane families has been obtained in [7]. Since their assumptions clearly
imply that the creative condition defined in Definition 2 is satisfied and the
set consisting of regular points of $\widetilde{\nu}$ is dense, their result
follows from Theorem 1 and Theorem 2.

Notice that non-unique existence cases, too, are intriguing cases since
Theorem 1 may be effectively applied even in such cases (see Example 4.2 (1),
(2)).

This paper is organized as follows. Theorem 1 and Theorem 2 are proved in
Section 2 and Section 3 respectively. In Section 4, examples are given.
Section 5 is an appendix where an alternative proof of Theorem 1 except for
Theorem 1 (3) is given. The alternative proof is a proof by a gauge theoretic
approach. In order to avoid unnecessary complication, the alternative proof is
given only in the case $n=1$. The author has no idea on how to prove Theorem 1
(3) by using the alternative proof.

##  2\. Proof of Theorem 1

###  2.1. Proof of Theorem 1 (1)

####  2.1.1. Proof of “if” part

Let $x_{0}$ be an arbitrary point of $N$. Take one point $P$ of
$\mathbb{R}^{n+1}-H_{\left(\widetilde{\varphi}\left(x_{0}\right),\widetilde{\nu}\left(x_{0}\right)\right)}$
and fix it. It follows
$\left(\widetilde{\varphi}\left(x_{0}\right)-P\right)\cdot\widetilde{\nu}\left(x_{0}\right)\neq
0$. Let $\widetilde{U}_{{}_{P}}$ be the set of points $x\in N$ satisfying

$None$ $\left(\widetilde{\varphi}(x)-P\right)\cdot\widetilde{\nu}(x)\neq 0.$

Then, it is clear that $\widetilde{U}_{{}_{P}}$ is an open neighborhood of
$x_{0}$ and the mirror image of the fixed point $P$ by the mirror
$H_{\left(\widetilde{\varphi}(x),\widetilde{\nu}(x)\right)}$ is given by

$2\left(\left(\widetilde{\varphi}(x)-P\right)\cdot\widetilde{\nu}(x)\right)\widetilde{\nu}(x)+P$

for any $x\in\widetilde{U}_{{}_{P}}$.

Figure 6.  The mirror-image mapping $f_{{}_{P}}:U_{P}\to\mathbb{R}^{n+1}$.

Since the hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is assumed to
be creative, there exists a mapping $\widetilde{\Omega}:N\to T^{*}S^{n}$ with
the form
$\widetilde{\Omega}(x)=\left(\widetilde{\nu}(x),\widetilde{\omega}(x)\right)$
such that for any $x\in N$ the following equality holds as $1$-form germs at
$x$.

$d\widetilde{\gamma}=\widetilde{\omega}.$

Let $\left(V,\left(\Theta_{1},\ldots,\Theta_{n}\right)\right)$ be a normal
coordinate neighborhood of $S^{n}$ at $\widetilde{\nu}\left(x_{0}\right)$. Set
$U_{{}_{P}}=\widetilde{U}_{{}_{P}}\cap\widetilde{\nu}^{-1}(V)$. Consider the
mirror-image mapping $f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ defined by

$f_{{}_{P}}(x)=2\left(\left(\widetilde{\varphi}(x)-P\right)\cdot\widetilde{\nu}(x)\right)\widetilde{\nu}(x)+P$

for any $x\in U_{{}_{P}}$. In order to show that $f_{{}_{P}}$ is a frontal, it
is sufficient to construct a Gauss mapping with respect to $f_{{}_{P}}$. By
using the mapping $\widetilde{\Omega}|_{U_{{}_{P}}}$, a Gauss mapping for
$f_{{}_{P}}$ is constructed as follows. For any $x\in U_{{}_{P}}$ set
$X=\widetilde{\nu}(x)$. Let
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Pi}_{\left(X,X_{0}\right)}:T_{X_{0}}S^{n}\to
T_{X}S^{n}$ be the Levi-Civita translation. For any $i$ $(1\leq i\leq n)$, set
$\frac{\partial}{\partial\Theta_{\left(i,X\right)}}={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\Pi}_{\left(X,X_{0}\right)}\left(\frac{\partial}{\partial\Theta_{i}}\right)$.
Then notice that for any $x\in U_{{}_{P}}$, under the identification of
$\mathbb{R}^{n+1}$ and $T_{f_{{}_{P}}(x)}\mathbb{R}^{n+1}$,

$\left\langle\frac{\partial}{\partial\Theta_{\left(1,X\right)}},\ldots,\frac{\partial}{\partial\Theta_{\left(n,X\right)}},\widetilde{\nu}(x)\right\rangle$

is an orthonormal basis of the tangent vector space
$T_{f_{{}_{P}}(x)}\mathbb{R}^{n+1}$.

Figure 7.  Figure for Proof of “if” part.

######  Lemma 2.1.

For any $x\in U_{{}_{P}}$, the following equality holds.

$d\left(P\cdot\widetilde{\nu}\right)=\sum_{i=1}^{n}\left(P\cdot\frac{\partial}{\partial\Theta_{\left(i,X\right)}}\right)d\left(\Theta_{i}\circ\widetilde{\nu}\right).$

Proof of Lemma 2.1.

$\displaystyle d\left(P\cdot\widetilde{\nu}\right)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\frac{\partial\left(P\cdot\widetilde{\nu}\right)}{\partial
x_{j}}(x)dx_{j}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\left(P\cdot\left(\sum_{i=1}^{n}\frac{\partial\left(\Theta_{i}\circ\widetilde{\nu}\right)}{\partial
x_{j}}(x)\frac{\partial}{\partial\Theta_{(i,X)}}\right)\right)dx_{j}$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(P\cdot\frac{\partial}{\partial\Theta_{(i,X)}}\right)\left(\sum_{j=1}^{n}\frac{\partial\left(\Theta_{i}\circ\widetilde{\nu}\right)}{\partial
x_{j}}(x)dx_{j}\right)$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(P\cdot\frac{\partial}{\partial\Theta_{\left(i,X\right)}}\right)d\left(\Theta_{i}\circ\widetilde{\nu}\right).$

$\Box$

By Lemma 2.1, under the identification of $T_{\widetilde{\nu}(x)}S^{n}$ and
$T^{*}_{\widetilde{\nu}(x)}S^{n}$, it follows

$\displaystyle
d\left(\left(\widetilde{\varphi}-P\right)\cdot\widetilde{\nu}\right)$
$\displaystyle=$ $\displaystyle
d\left(\widetilde{\varphi}\cdot\widetilde{\nu}\right)-d\left(P\cdot\widetilde{\nu}\right)$
$\displaystyle=$ $\displaystyle
d\widetilde{\gamma}-d\left(P\cdot\widetilde{\nu}\right)$ $\displaystyle=$
$\displaystyle\widetilde{\omega}-d\left(P\cdot\widetilde{\nu}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(\widetilde{\omega}(x)\cdot\frac{\partial}{\partial\Theta_{\left(i,X\right)}}\right)d\left(\Theta_{i}\circ\widetilde{\nu}\right)-\sum_{i=1}^{n}\left(P\cdot\frac{\partial}{\partial\Theta_{\left(i,X\right)}}\right)d\left(\Theta_{i}\circ\widetilde{\nu}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(\left(\widetilde{\omega}(x)-P\right)\cdot\frac{\partial}{\partial\Theta_{\left(i,X\right)}}\right)d\left(\Theta_{i}\circ\widetilde{\nu}\right)$

for any $x\in U_{{}_{P}}$. Set

${\bf
v}_{{}_{P}}(x)=\sum_{i=1}^{n}\left(\left(\widetilde{\omega}(x)-P\right)\cdot\frac{\partial}{\partial\Theta_{\left(i,X\right)}}\right)\frac{\partial}{\partial\Theta_{(i,X)}}-\left(\left(\widetilde{\varphi}(x)-P\right)\cdot\widetilde{\nu}(x)\right)\widetilde{\nu}(x)$

for any $x\in U_{{}_{P}}$ where $\mathbb{R}^{n+1}$ and
$T_{f_{{}_{P}}\left(x\right)}\mathbb{R}^{n+1}$ are identified and
$T_{f_{{}_{P}}\left(x\right)}S^{n}$ and
$T_{f_{{}_{P}}\left(x\right)}^{*}S^{n}$ are identified. By ($None$), ${\bf
v}_{{}_{P}}(x)$ is not the zero vector. Moreover, the following holds.

######  Lemma 2.2.

For any ${\bf v}\in T_{x_{0}}N$, ${\bf v}_{{}_{P}}(x_{0})$ is perpendicular to
$d\left(f_{{}_{P}}\right)_{{}x_{0}}({\bf v})$.

Proof of Lemma 2.2. Calculation of the product of the vector ${\bf
v}_{{}_{P}}\left(x_{0}\right)$ and the Jacobian matrix of $f_{{}_{P}}$ at
$x_{0}$ (denoted by $J\left(f_{{}_{P}}\right)_{x_{0}}$) is carried out as
follows, where $\mathbb{R}^{n+1}$ and
$T_{f_{{}_{P}}\left(x_{0}\right)}\mathbb{R}^{n+1}$ are identified and
$T_{f_{{}_{P}}\left(x_{0}\right)}S^{n}$ and
$T_{f_{{}_{P}}\left(x_{0}\right)}^{*}S^{n}$ are identified.

$\displaystyle{\bf
v}_{{}_{P}}\left(x_{0}\right)J\left(f_{{}_{P}}\right)_{x_{0}}$
$\displaystyle=$ $\displaystyle
2\sum_{i=1}^{n}\left(\left(\widetilde{\omega}\left(x_{0}\right)-P\right)\cdot\frac{\partial}{\partial\Theta_{i}}\right)\left(\left(\widetilde{\varphi}\left(x_{0}\right)-P\right)\cdot\widetilde{\nu}\left(x_{0}\right)\right)d\left(\Theta_{i}\circ\widetilde{\nu}\right)$
$\displaystyle\qquad-2\left(\left(\widetilde{\varphi}\left(x_{0}\right)-P\right)\cdot\widetilde{\nu}\left(x_{0}\right)\right)d\left(\left(\widetilde{\varphi}-P\right)\cdot\widetilde{\nu}\right)_{\mbox{at
}x_{0}}$ $\displaystyle=$ $\displaystyle
2\left(\left(\widetilde{\varphi}\left(x_{0}\right)-P\right)\cdot\widetilde{\nu}\left(x_{0}\right)\right)\sum_{i=1}^{n}\left(\left(\widetilde{\omega}\left(x_{0}\right)-P\right)\cdot\frac{\partial}{\partial\Theta_{i}}\right)d\left(\Theta_{i}\circ\widetilde{\nu}\right)$
$\displaystyle\qquad-2\left(\left(\widetilde{\varphi}\left(x_{0}\right)-P\right)\cdot\widetilde{\nu}\left(x_{0}\right)\right)\sum_{i=1}^{n}\left(\left(\widetilde{\omega}\left(x_{0}\right)-P\right)\cdot\frac{\partial}{\partial\Theta_{i}}\right)d\left(\Theta_{i}\circ\widetilde{\nu}\right)$
$\displaystyle=$ $\displaystyle 0.$

$\Box$

We may consider that the point $x_{0}$ is an arbitrary point of $U_{{}_{P}}$.
Thus we have the following.

######  Lemma 2.3.

The mapping $f_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ is a frontal with Gauss
mapping $\nu_{{}_{P}}:U_{{}_{P}}\to S^{n}$ such that
$\nu_{{}_{P}}(x)\cdot\widetilde{\nu}(x)\neq 0$, where
$\nu_{{}_{P}}(x)=\frac{{\bf v}_{{}_{P}}(x)}{\|{\bf v}_{{}_{P}}(x)\|}$.

By Lemma 2.3, the hyperplane
$H_{\left(\widetilde{\varphi}(x),\widetilde{\nu}(x)\right)}$ and the line
$\ell_{x}=\left\\{f_{{}_{P}}(x)+t\nu_{{}_{P}}(x)\left|\right.t\in\mathbb{R}\right\\}$
must intersect only at one point for any $x\in U_{{}_{P}}$. Define the mapping
$\widetilde{f}_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ by

$\left\\{\widetilde{f}_{{}_{P}}(x)\right\\}=H_{\left(\widetilde{\varphi}(x),\widetilde{\nu}(x)\right)}\cap\ell_{x}.$

Then, from the construction, $\widetilde{f}_{{}_{P}}$ must have the following
form (see p.7 of [15]).

$\widetilde{f}_{{}_{P}}(x)=f_{{}_{P}}(x)-\frac{||f_{{}_{P}}(x)-P||^{2}}{2\left(f_{{}_{P}}(x)-P\right)\cdot\nu_{{}_{P}}(x)}\nu_{{}_{P}}(x).$

By Theorem 1 of [15] (more precisely, by 3.1 in p.9 of [15]) and Lemma 2.3, we
have the following.

######  Lemma 2.4.

The mapping $\widetilde{f}_{{}_{P}}$ is a frontal with Gauss mapping
$\widetilde{\nu}|_{U_{{}_{P}}}:U_{{}_{P}}\to S^{n}$. In other words,
$\widetilde{f}_{{}_{P}}:U_{{}_{P}}\to\mathbb{R}^{n+1}$ is an envelope created
by the hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi}|_{U_{{}_{P}}},\widetilde{\nu}|_{U_{{}_{P}}}\right)}$.

On the other hand, it is easily seen that
$\left(f_{{}_{P}}\left(x_{0}\right)+{\bf
v}_{{}_{P}}\left(x_{0}\right)-\widetilde{\varphi}\left(x_{0}\right)\right)\cdot\widetilde{\nu}\left(x_{0}\right)=0$
(see Figure 7). Thus, the vector $f_{{}_{P}}\left(x_{0}\right)+{\bf
v}_{{}_{P}}\left(x_{0}\right)$ must belong to
$H_{\left(\widetilde{\varphi}\left(x_{0}\right),\widetilde{\nu}\left(x_{0}\right)\right)}$.
From the construction and by using the equality
$P=\sum_{i=1}^{n}\left(P\cdot\frac{\partial}{\partial\Theta_{i}}\right)\frac{\partial}{\partial\Theta_{i}}+\left(P\cdot\widetilde{\nu}\left(x_{0}\right)\right)\widetilde{\nu}\left(x_{0}\right),$
we have the following.

$\displaystyle\widetilde{f}_{{}_{P}}\left(x_{0}\right)$ $\displaystyle=$
$\displaystyle f_{{}_{P}}(x)+{\bf v}_{{}_{P}}\left(x_{0}\right)$
$\displaystyle=$ $\displaystyle
2\left(\left(\widetilde{\varphi}\left(x_{0}\right)-P\right)\cdot\widetilde{\nu}\left(x_{0}\right)\right)\widetilde{\nu}\left(x_{0}\right)+P$
$\displaystyle\quad\quad+\sum_{i=1}^{n}\left(\left(\widetilde{\omega}\left(x_{0}\right)-P\right)\cdot\frac{\partial}{\partial\Theta_{i}}\right)\frac{\partial}{\partial\Theta_{i}}-\left(\left(\widetilde{\varphi}\left(x_{0}\right)-P\right)\cdot\widetilde{\nu}\left(x_{0}\right)\right)\widetilde{\nu}\left(x_{0}\right)$
$\displaystyle=$
$\displaystyle\left(\left(\widetilde{\varphi}\left(x_{0}\right)-P\right)\cdot\widetilde{\nu}\left(x_{0}\right)\right)\widetilde{\nu}\left(x_{0}\right)+P+\sum_{i=1}^{n}\left(\left(\widetilde{\omega}\left(x_{0}\right)-P\right)\cdot\frac{\partial}{\partial\Theta_{i}}\right)\frac{\partial}{\partial\Theta_{i}}$
$\displaystyle=$
$\displaystyle\left(\widetilde{\varphi}\left(x_{0}\right)\cdot\widetilde{\nu}\left(x_{0}\right)\right)\widetilde{\nu}\left(x_{0}\right)+\sum_{i=1}^{n}\left(\widetilde{\omega}\left(x_{0}\right)\cdot\frac{\partial}{\partial\Theta_{i}}\right)\frac{\partial}{\partial\Theta_{i}}$
$\displaystyle=$
$\displaystyle\widetilde{\gamma}\left(x_{0}\right)\widetilde{\nu}\left(x_{0}\right)+\widetilde{\omega}\left(x_{0}\right).$

This proves the following lemma.

######  Lemma 2.5.

The following equality holds.

$\widetilde{f}_{{}_{P}}\left(x_{0}\right)=\widetilde{\gamma}\left(x_{0}\right)\widetilde{\nu}\left(x_{0}\right)+\widetilde{\omega}\left(x_{0}\right).$

Lemma 2.5 shows that $\widetilde{f}_{{}_{P}}\left(x_{0}\right)$ does not
depend on the particular choice of
$P\in\mathbb{R}^{n+1}-H_{\left(\widetilde{\varphi}\left(x_{0}\right),\widetilde{\nu}\left(x_{0}\right)\right)}$.
Define the mapping $\widetilde{f}:N\to\mathbb{R}^{n+1}$ by
$\widetilde{f}(x)=\widetilde{\gamma}(x)\widetilde{\nu}(x)+\widetilde{\omega}(x)$.
Since $x_{0}$ is an arbitrary point of $N$, by Lemma 2.4 and Lemma 2.5, it
follows that the mapping $\widetilde{f}:N\to\mathbb{R}^{n+1}$ is an envelope
created by $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$.
This completes the proof of “if” part. $\Box$

####  2.1.2. Proof of “only if” part

Suppose that the hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ creates an
envelope $\widetilde{f}:N\to\mathbb{R}^{n+1}$. Then, by definition,
$\widetilde{f}$ is a frontal such that the inclusion
$\widetilde{f}(x)+d\widetilde{f}_{x}(T_{x}N)\subset
H_{\left(\widetilde{\varphi}(x),\widetilde{\nu}(x)\right)}$ holds for any
$x\in N$. Let $\widetilde{\omega}:N\to\mathbb{R}^{n+1}$ be the mapping defined
by
$\widetilde{\omega}(x)=\widetilde{f}(x)-\widetilde{\gamma}(x)\widetilde{\nu}(x)$
(see Figure 8).

Figure 8.  Figure for Proof of “only if” part.

It is sufficient to show that under some identifications, $\widetilde{\omega}$
is actually a creator for the envelope $\widetilde{f}$.

It is easily seen that $\widetilde{\omega}(x)\cdot\widetilde{\nu}(x)=0$ for
any $x\in N$. Thus, under the identification of $\mathbb{R}^{n+1}$ and
$T^{*}_{\widetilde{\nu}(x)}\mathbb{R}^{n+1}$, we have

######  Lemma 2.6.

For any $x\in N$, $\widetilde{\omega}(x)\in T^{*}_{\widetilde{\nu}(x)}S^{n}$
holds.

Let $\widetilde{\Omega}:N\to T^{*}S^{n}$ be the mapping defined by
$\widetilde{\Omega}(x)=\left(\widetilde{\nu}(x),\widetilde{\omega}(x)\right)$.
Let $x_{0}$ be an arbitrary point of $N$ and let $P$ be a point of
$\mathbb{R}^{n+1}-H_{\left(\widetilde{\varphi}(x_{0}),\widetilde{\nu}(x_{0})\right)}$.
Again, we consider the mirror-image mapping
$f_{{}_{P}}:\widetilde{U}_{{}_{P}}\to\mathbb{R}^{n+1}$ defined by

$f_{{}_{P}}(x)=2\left(\left(\widetilde{\varphi}(x)-P\right)\cdot\widetilde{\nu}(x)\right)\widetilde{\nu}(x)+P,$

where $\widetilde{U}_{{}_{P}}=\left\\{x\in
N\,\left|\,\left(\widetilde{\varphi}(x)-P\right)\cdot\widetilde{\nu}(x)\neq
0\right\\}\right.$. The mapping $f_{{}_{P}}$ is exactly the orthotomic of
$\widetilde{f}|_{\widetilde{U}_{{}_{P}}}$ relative to the point $P$. Thus, by
Proposition 1 of [15] (more precisely, by 2.1 in pp. 7–8 of [15]) ,
$f_{{}_{P}}$ is a frontal and the mapping
$\nu_{{}_{P}}:\widetilde{U}_{{}_{P}}\to S^{n}$ define by

$\nu_{{}_{P}}(x)=\frac{\widetilde{f}(x)-f_{{}_{P}}(x)}{\|\widetilde{f}(x)-f_{{}_{P}}(x)\|}$

is its Gauss mapping. In particular, we have the following.

######  Lemma 2.7.

For any $x\in\widetilde{U}_{{}_{P}}$ and any ${\bf v}\in T_{x}N$, the
following holds.

$\left(\widetilde{f}(x)-f_{{}_{P}}(x)\right)\cdot
d\left(f_{{}_{P}}\right)_{x}({\bf v})=0.$

For any $x\in\widetilde{U}_{{}_{P}}$, set

$g_{{}_{P}}(x)=\frac{1}{2}\left(f_{{}_{P}}(x)-P\right)+P=\left(\left(\widetilde{\varphi}(x)-P\right)\cdot\widetilde{\nu}(x)\right)\widetilde{\nu}(x)+P.$

Then, since $f_{{}_{P}}(x)$ is the mirror-image of $P$ with respect to the
mirror $H_{\left(\widetilde{\varphi}(x),\widetilde{\nu}(x)\right)}$, the
following clearly holds.

######  Lemma 2.8.

The vector $\widetilde{f}(x)-g_{{}_{P}}(x)$ is perpendicular to the vector
$g_{{}_{P}}(x)-f_{{}_{P}}(x)=-\left(\left(\widetilde{\varphi}(x)-P\right)\cdot\widetilde{\nu}(x)\right)\widetilde{\nu}(x)$
for any $x\in\widetilde{U}_{{}_{P}}$.

Thus,

$\widetilde{f}(x)-f_{{}_{P}}(x)=\left(\widetilde{f}(x)-g_{{}_{P}}(x)\right)+\left(g_{{}_{P}}(x)-f_{{}_{P}}(x)\right)$

is an orthogonal decomposition of $\widetilde{f}(x)-f_{{}_{P}}(x)$ for any
$x\in\widetilde{U}_{{}_{P}}$ (see Figure 8).

In order to decompose the vector $\widetilde{f}(x)-g_{{}_{P}}(x)$ reasonably,
the open neighborhood $\widetilde{U}_{{}_{P}}$ of $x_{0}$ is reduced as
follows. Let $\left(V,\left(\Theta_{1},\ldots,\Theta_{n}\right)\right)$ be a
normal coordinate neighborhood of $S^{n}$ at
$\widetilde{\nu}\left(x_{0}\right)$. Set again
$U_{{}_{P}}=\widetilde{U}_{{}_{P}}\cap\widetilde{\nu}^{-1}(V)$. Notice that
$\left\langle d\Theta_{1},\ldots,d\Theta_{n}\right\rangle$ is an orthonormal
basis of the cotangent space $T^{*}_{\widetilde{\nu}\left(x_{0}\right)}S^{n}$.

######  Lemma 2.9.

The equality

$\widetilde{f}\left(x_{0}\right)-g_{{}_{P}}\left(x_{0}\right)=\widetilde{\omega}\left(x_{0}\right)-\sum_{i=1}^{n}\left(P\cdot\frac{\partial}{\partial\Theta_{i}}\right)\frac{\partial}{\partial\Theta_{i}}$

holds where three vector spaces $\mathbb{R}^{n+1}$,
$T_{\widetilde{\nu}\left(x_{0}\right)}\mathbb{R}^{n+1}$ and
$T^{*}_{\widetilde{\nu}\left(x_{0}\right)}\mathbb{R}^{n+1}$ are identified.

Proof of Lemma 2.9.

$\displaystyle\widetilde{f}\left(x_{0}\right)-g_{{}_{P}}\left(x_{0}\right)$
$\displaystyle=$
$\displaystyle\widetilde{f}\left(x_{0}\right)-\left(\left(\left(\widetilde{\varphi}\left(x_{0}\right)-P\right)\cdot\widetilde{\nu}\left(x_{0}\right)\right)\widetilde{\nu}\left(x_{0}\right)+P\right)$
$\displaystyle=$
$\displaystyle\left(\widetilde{f}\left(x_{0}\right)-\left(\widetilde{\varphi}\left(x_{0}\right)\cdot\widetilde{\nu}\left(x_{0}\right)\right)\widetilde{\nu}\left(x_{0}\right)\right)+\left(\left(P\cdot\widetilde{\nu}\left(x_{0}\right)\right)\widetilde{\nu}\left(x_{0}\right)-P\right)$
$\displaystyle=$
$\displaystyle\left(\widetilde{f}\left(x_{0}\right)-\widetilde{\gamma}\left(x_{0}\right)\widetilde{\nu}\left(x_{0}\right)\right)+\left(\left(P\cdot\widetilde{\nu}\left(x_{0}\right)\right)\widetilde{\nu}\left(x_{0}\right)-P\right)$
$\displaystyle=$
$\displaystyle\widetilde{\omega}\left(x_{0}\right)-\sum_{i=1}^{n}\left(P\cdot\frac{\partial}{\partial\Theta_{i}}\right)\frac{\partial}{\partial\Theta_{i}}.$

$\Box$

By Lemma 2.9, the following holds.

$\displaystyle\widetilde{f}\left(x_{0}\right)-f_{{}_{P}}\left(x_{0}\right)$
$\displaystyle=$
$\displaystyle\left(\widetilde{f}\left(x_{0}\right)-g_{{}_{P}}\left(x_{0}\right)\right)+\left(g_{{}_{P}}\left(x_{0}\right)-f_{{}_{P}}\left(x_{0}\right)\right)$
$\displaystyle=$
$\displaystyle\widetilde{\omega}\left(x_{0}\right)-\sum_{i=1}^{n}\left(P\cdot\frac{\partial}{\partial\Theta_{i}}\right)\frac{\partial}{\partial\Theta_{i}}-\left(\left(\widetilde{\varphi}\left(x_{0}\right)-P\right)\cdot\widetilde{\nu}\left(x_{0}\right)\right)\widetilde{\nu}\left(x_{0}\right).$

Hence, by Lemma 2.1 and Lemma 2.7, the germ of $1$-form $d\widetilde{\gamma}$
at $x_{0}$ is calculated as follows, where $X=\widetilde{\nu}(x)$,
$\frac{\partial}{\partial\Theta_{\left(i,X\right)}}=P_{\left(X,X_{0}\right)}\left(\frac{\partial}{\partial\Theta_{i}}\right).$
and $P_{\left(X,X_{0}\right)}:T_{X_{0}}S^{n}\to T_{X}S^{n}$ is the Levi-Civita
translation.

$\displaystyle d\widetilde{\gamma}$ $\displaystyle=$ $\displaystyle
d\widetilde{\gamma}-d\left(P\cdot\widetilde{\nu}\right)+d\left(P\cdot\widetilde{\nu}\right)$
$\displaystyle=$ $\displaystyle
d\left(\left(\widetilde{\varphi}-P\right)\cdot\widetilde{\nu}\right)+d\left(P\cdot\widetilde{\nu}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(\left(\widetilde{\omega}-P\right)\cdot\frac{\partial}{\partial\Theta_{\left(i,X\right)}}\right)d\left(\Theta_{i}\circ\widetilde{\nu}\right)+\sum_{i=1}^{n}\left(P\cdot\frac{\partial}{\partial\Theta_{\left(i,X\right)}}\right)d\left(\Theta_{i}\circ\widetilde{\nu}\right)$
$\displaystyle=$
$\displaystyle\left(\widetilde{\omega}-\sum_{i=1}^{n}\left(P\cdot\frac{\partial}{\partial\Theta_{\left(i,X\right)}}\right)d\left(\Theta_{i}\circ\widetilde{\nu}\right)\right)+\sum_{i=1}^{n}\left(P\cdot\frac{\partial}{\partial\Theta_{\left(i,X\right)}}\right)d\left(\Theta_{i}\circ\widetilde{\nu}\right)$
$\displaystyle=$ $\displaystyle\widetilde{\omega}.$

This calculation proves the following lemma.

######  Lemma 2.10.

The equality

$d\widetilde{\gamma}=\widetilde{\omega}$

holds as germs of $1$-form at $x_{0}$.

Since $x_{0}$ is an arbitrary point of $N$, by Lemma 2.10,
$\widetilde{\omega}$ is actually the creator for the given envelope
$\widetilde{f}:N\to\mathbb{R}^{n+1}$. This completes the proof of “only if”
part. $\Box$

###  2.2. Proof of Theorem 1 (2)

Theorem 1 (2) is a direct by-product of the proof of Theorem 1 (1). ∎

###  2.3. Proof of Theorem 1 (3)

Recall that the line family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is said to
create an $E_{1}$ envelope (denoted by $(E_{1})$ in this subsection) if for
any fixed $t_{0}\in N$ and any $t\in N$ near $t_{0}$ the limit $\lim_{t\to
t_{0}}H_{\left(\widetilde{\varphi}(t),\widetilde{\nu}(t)\right)}\cap
H_{\left(\widetilde{\varphi}(t_{0}),\widetilde{\nu}(t_{0})\right)}$ exists. On
the other hand, the line family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is said to
create an $E_{2}$ envelope (denoted by $(E_{2})$ in this subsection) if it
creates an envelope in the sense of Definition 1.

$(E_{1})\Rightarrow(E_{2})$ Let $t_{0}$ be a point of $N$ and let $t_{i}\in N$
$(i=1,2,\ldots)$ be a sequence conversing to $t_{0}$. Since $(E_{1})$ is
assumed, we can assume that a point $X_{t_{i}}$ can be taken from the
intersection $H_{\left(\widetilde{\varphi}(t),\widetilde{\nu}(t)\right)}\cap
H_{\left(\widetilde{\varphi}(t_{0}),\widetilde{\nu}(t_{0})\right)}$ such that
$\lim_{t_{i}\to t_{0}}X_{t_{i}}$ exists. Denote the limit by $X_{t_{0}}$.
Then, we have the following.

$\displaystyle\left(X_{t_{i}}-\widetilde{\varphi}({t_{i}})\right)\cdot\widetilde{\nu}(t_{i})$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\left(X_{t_{i}}-\widetilde{\varphi}(t_{0})\right)\cdot\widetilde{\nu}(t_{0})$
$\displaystyle=$ $\displaystyle 0.$

This implies

$X_{t_{i}}\cdot\left(\widetilde{\nu}(t_{i})-\widetilde{\nu}(t_{0})\right)=\widetilde{\gamma}(t_{i})-\widetilde{\gamma}(t_{0}).$

Thus we have

$X_{t_{0}}\cdot\frac{\partial\widetilde{\nu}}{\partial
t}(t_{0})=\frac{\partial\widetilde{\gamma}}{\partial t}(t_{0}).$

This implies that there exists a real number $\alpha(t_{0})$ such that the
following identity holds where $d\left(\Theta\circ\widetilde{\nu}\right)$ and
$d\widetilde{\gamma}$ stand for the $1$-dimensional cotangent vectors in
$T_{t_{0}}^{*}N$, namely the following identity is nothing but the identity of
two real numbers.

$\alpha(t_{0})d\left(\Theta\circ\widetilde{\nu}\right)=d\widetilde{\gamma}.$

It is not difficult to see that the function $\alpha:N\to\mathbb{R}$ is of
class $C^{\infty}$. This means that the line family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is creative.
Therefore, by Theorem 1 (1), the line family creates an $E_{2}$ envelope. ∎

$(E_{2})\Rightarrow(E_{1})$ For the proof of this implication, it is used the
notions and notations introduced in the proof of Theorem 1 (1). The assumption
$(E_{2})$ implies that $\widetilde{\gamma}$ is totally differentiable with
respect to $\widetilde{\nu}$. Take an arbitrary point $t_{0}\in N$ and fixed
it. Since $\widetilde{\gamma}$ is totally differentiable with respect to
$\widetilde{\nu}$ at $t_{0}$, for any $t$ near $t_{0}$ if the length of the
vector $\overrightarrow{f_{P}(t_{0})f_{P}(t)}$ is positive, then the
horizontal vector of $\overrightarrow{f_{P}(t_{0})f_{P}(t)}$ must be non-zero,
where $P$ is a point taken outside the line
$H_{\left(\widetilde{\varphi}(t_{0}),\widetilde{\nu}(t_{0})\right)}$ and
$f_{P}$ is a mirror-image mapping introduced in the proof of Theorem 1 (1).
Denote the intersection of the perpendicular bisector of
$\overrightarrow{f_{P}(t_{0})f_{P}(t)}$ and the line
$H_{\left(\widetilde{\varphi}(t_{0}),\widetilde{\nu}(t_{0})\right)}$ by
$J_{t}$. Then, from the construction, it follows that the triangre $\triangle
J_{t}f_{P}(t_{0})f_{P}(t)$ is an isosceles triangle with legs
$J_{t}f_{P}(t_{0})$ and $J_{t}f_{P}(t)$. This implies the following (see
Figure 9).

$J_{t}\in H_{\left(\widetilde{\varphi}(t),\widetilde{\nu}(t)\right)}\cap
H_{\left(\widetilde{\varphi}(t_{0}),\widetilde{\nu}(t_{0})\right)}.$

Figure 9.  Figure for $(E_{2})\Rightarrow(E_{1})$.

Notice that $\lim_{t\to t_{0}}||J_{t}f_{P}(t_{0})||$ is positive. Thus, we
have

$\lim_{t\to t_{0}}\angle J_{t}f_{P}(t_{0})f_{P}(t)=\lim_{t\to t_{0}}\angle
J_{t}f_{P}(t)f_{P}(t_{0})=\frac{\pi}{2}.$

By Proposition 1 of [15] asserting that $f_{P}$ is a frontal with its Gauss
mapping
$\frac{f_{P}(t_{0})-\widetilde{f}_{P}(t_{0})}{||f_{P}(t_{0})-\widetilde{f}_{P}(t_{0})||}$,
it follows

$\lim_{t\to t_{0}}J_{t}=\widetilde{f}_{P}(t_{0}),$

where $\widetilde{f}_{P}$ is the anti-orthotomic of $f_{P}$ relative to the
point $P$ introduced in the proof of Theorem 1 (1). Since $t_{0}$ is an
arbitrary point of $N$, the given $E_{2}$ envelope must be an $E_{1}$ envelope
by Theorem 1 (1). ∎

##  3\. Proof of Theorem 2

Proof of “if” part. Since the hyperplane
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is creative,
by Theorem 1, it creates an envelope. Let
$\widetilde{f}_{1},\widetilde{f}_{2}:N\to\mathbb{R}^{n+1}$ be envelopes
created by $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$.

Let $x_{0}\in N$ be a regular point of $\widetilde{\nu}$. Then, there exists
an open coordinate neighborhood
$\left(U,\left(x_{1},\ldots,x_{n}\right)\right)$ such that $x_{0}\in U$ and
$\widetilde{\nu}|_{U}:U\to\widetilde{\nu}(U)$ is a diffeomorphism. Then, the
germ of $1$-form $d\left(\widetilde{\varphi}\cdot\widetilde{\nu}\right)$ at
$x_{0}\in U$ is

$\displaystyle d\left(\widetilde{\varphi}\cdot\widetilde{\nu}\right)$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\frac{\partial\left(\widetilde{\varphi}\cdot\widetilde{\nu}\right)}{\partial
x_{j}}(x)dx_{j}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{n}\frac{\partial\left(\widetilde{\varphi}\cdot\widetilde{\nu}\right)}{\partial
x_{j}}(x)\left(\sum_{i=1}^{n}\frac{\partial\left(x_{j}\circ\widetilde{\nu}^{-1}\right)}{\partial\Theta_{\left(i,\widetilde{\nu}(x)\right)}}\left(\widetilde{\nu}(x)\right)d\Theta_{i}\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\left(\sum_{j=1}^{n}\frac{\partial\left(\widetilde{\varphi}\cdot\widetilde{\nu}\right)}{\partial
x_{j}}(x)\frac{\partial\left(x_{j}\circ\widetilde{\nu}^{-1}\right)}{\partial\Theta_{\left(i,\widetilde{\nu}(x)\right)}}\left(\widetilde{\nu}(x)\right)\right)d\Theta_{i}.$

Let $\widetilde{\Omega}:N\to T^{*}S^{n}$ be the mapping with the form
$\widetilde{\Omega}(x)=\left(\widetilde{\nu}(x),\widetilde{\omega}(x)\right)$
such that $\widetilde{\omega}$ is the creator for $\widetilde{f}$. Then, by
the above calculation, $\widetilde{\omega}|_{U}$ must have the following form.

$\widetilde{\omega}|_{U}(x)=\sum_{i=1}^{n}\left(\sum_{j=1}^{n}\frac{\partial\left(\widetilde{\varphi}\cdot\widetilde{\nu}\right)}{\partial
x_{j}}(x)\frac{\partial\left(x_{j}\circ\widetilde{\nu}^{-1}\right)}{\partial\Theta_{\left(i,\widetilde{\nu}(x)\right)}}\left(\widetilde{\nu}(x)\right)\right)d\Theta_{i}.$

Hence, by Theorem 1 (2), we have the following.

######  Lemma 3.1.

At a regular point $x_{0}\in N$ of $\widetilde{\nu}$, the equality
$\widetilde{f}_{1}(x_{0})=\widetilde{f}_{2}(x_{0})$ holds.

Let $x_{0}\in N$ be a singular point of $\widetilde{\nu}$. Then, since we have
assumed that the set of regular points of $\widetilde{\nu}$ is dense, there
exists a point-sequence $\left\\{y_{i}\right\\}_{i=1,2,\ldots}\subset N$ such
that $y_{i}$ is a regular point of $\widetilde{\nu}$ for any $i\in\mathbb{N}$
and $\lim_{i\to\infty}y_{i}=x_{0}$. Then, by Lemma 3.1, we have

$\widetilde{f}_{1}(x_{0})=\widetilde{f}_{1}\left(\lim_{i\to\infty}y_{i}\right)=\lim_{i\to\infty}\widetilde{f}_{1}(y_{i})=\lim_{i\to\infty}\widetilde{f}_{2}(y_{i})=\widetilde{f}_{2}\left(\lim_{i\to\infty}y_{i}\right)=\widetilde{f}_{2}(x_{0}).$

Thus, we have the following.

######  Lemma 3.2.

Even at a singular point $x_{0}\in N$ of $\widetilde{\nu}$, the equality
$\widetilde{f}_{1}(x_{0})=\widetilde{f}_{2}(x_{0})$ holds.

$\Box$

Proof of “only if” part. Suppose that the hyperplane
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is creative
and the set of regular points of $\widetilde{\nu}$ is not dense in $N$. Then,
there exists an open set $U$ of $N$ such that any point $x\in U$ is a singular
point of $\widetilde{\nu}$. Then, there exist an integer $k$ $(0\leq k<n)$ and
an open set $U_{k}$ such that $U_{k}\subset U$ and the rank of
$\widetilde{\nu}$ at $x$ is $k$ for any $x\in U_{k}$. Let $x_{0}$ be a point
of $U_{k}$. We may assume that $U_{k}$ is sufficiently small open neighborhood
of $x_{0}$. Then, by the rank theorem (for the rank theorem, see for example
[5]), we have the following.

######  Lemma 3.3.

There exist functions $\eta_{1},\ldots,\eta_{k}:N\to\mathbb{R}$ such that the
following three hold.

  1. (1)

For any $i$ $(1\leq i\leq n)$, $\eta_{i}(x)=0$ if $x\not\in U_{k}$.

  2. (2)

There exists an $i$ $(1\leq i\leq n)$ such that
$\eta_{i}\left(x_{0}\right)\neq 0$.

  3. (3)

The following equality holds for any $x\in N$.

$\sum_{i=1}^{n}\eta_{i}(x)d\left(\Theta_{i}\circ\widetilde{\nu}\right)=0.$

Since we have assumed that
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is creative,
there exists a mapping $\widetilde{\Omega}:N\to T^{*}S^{n}$ with he form
$\widetilde{\Omega}(x)=\left(\widetilde{\nu}(x),\widetilde{\omega}(x)\right)$
such that
$d\left(\widetilde{\varphi}\cdot\widetilde{\nu}\right)=\widetilde{\omega}$. By
Lemma 3.3, the following holds.

######  Lemma 3.4.

For any function $\alpha:N\to\mathbb{R}$ and any $x\in N$, the following
equality holds as germs of $1$-form at $x$.

$d\left(\widetilde{\varphi}\cdot\widetilde{\nu}\right)=\widetilde{\omega}(x)+\alpha(x)\sum_{i=1}^{n}\eta_{i}(x)d\left(\Theta_{i}\circ\widetilde{\nu}\right).$

Therefore, by Theorem 1 (2),  uncountably many distinct envelopes
$\widetilde{f}$ are created by the same hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$. $\Box$

##  4\. Examples

######  Example 4.1 (Uniform spin of affine tangent lines).

  1. (1)

Let $\alpha:\mathbb{R}\to\mathbb{R}$ be a non-constant function. Notice that
$\alpha$ is of class $C^{\infty}$ as stated at the top of Section 1. Let
$\widetilde{\varphi}:\mathbb{R}\to\mathbb{R}^{2}$ be the mapping defined by
$\widetilde{\varphi}(t)=(\alpha(t),0)$. Let $\widetilde{\nu}:\mathbb{R}\to
S^{1}$ be the constant mapping $\widetilde{\nu}(t)=(0,1)$. For any fixed
$\theta_{0}\in\mathbb{R}$, let
$R_{\theta_{0}}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ be the linear mapping
representing the rotation through angle $\theta_{0}$. Set
$\widetilde{\nu}_{\theta_{0}}(t)=R_{\theta}\circ\widetilde{\nu}(t)=\left(-\sin\theta_{0},\cos\theta_{0}\right)$
and
$\widetilde{\gamma}_{\theta_{0}}(t)=\widetilde{\varphi}(t)\cdot\widetilde{\nu}_{\theta_{0}}(t)=-\alpha(t)\sin\theta_{0}$.
Figure is depicted in Figure 10.

Figure 10.  Figure for Example 4.1 (1).

It follows $d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)\equiv 0$ and
$d\gamma_{\theta_{0}}=-\sin\theta_{0}d\alpha$. Since $\alpha$ is non-constant,
there exists a regular point of $\alpha$, that is to say, there exists a
$t\in\mathbb{R}$ such that $\alpha^{\prime}(t)\neq 0$. Therefore, by Theorem
1, the line family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}_{\theta_{0}}\right)}$
creates an envelope if and only if $\theta_{0}\in\pi\mathbb{Z}$. Suppose that
$\theta_{0}\in\pi\mathbb{Z}$. In this case, by Theorem 2, uncountably many
distinct envelope $\widetilde{f}:\mathbb{R}\to\mathbb{R}^{2}$ can be created
by the given line family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}_{\theta_{0}}\right)}$.
Let $\beta:\mathbb{R}\to\mathbb{R}$ be a function. Since
$d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)\equiv 0$ and
$d\gamma_{\theta_{0}}\equiv 0$ in this case, the $1$-form
$t\mapsto\beta(t)d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)$ along
$\widetilde{\nu}_{\theta_{0}}$ may be a creator $\widetilde{\omega}$ for the
line family. By Theorem 1 (2),  the envelope $\widetilde{f}$ has the following
form.

$\widetilde{f}(t)=\widetilde{\omega}(t)+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\widetilde{\gamma}_{\theta_{0}}(t)\cdot\widetilde{\nu}_{\theta_{0}}(t)}}\right)\widetilde{\nu}_{\theta_{0}}(t)}}=({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pm}}}\beta(t),0)+(0,0)=({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pm}}}\beta(t),0),$

where double sign should be read in the same order and
$\beta(t)d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)$,
$\beta(t)R_{\frac{\pi}{2}}\circ\widetilde{\nu}_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}{\theta_{0}}}(t)$
are identified (both are denoted by the same symbol $\widetilde{\omega}(t)$).

Set
$F_{\theta_{0}}\left(X_{1},X_{2},t\right)=\left(X_{1}-\alpha(t),X_{2}\right)\cdot\widetilde{\nu}_{\theta_{0}}(t)$.
Suppose that $\theta_{0}\not\in\pi\mathbb{Z}$. In this case, the classical
common definition of envelope $\mathcal{D}$ relative to $F_{\theta_{0}}$ is as
follows.

$\mathcal{D}=\left\\{\left(X_{1},X_{2}\right)\,|\,\exists t\mbox{ s.t.
}\alpha^{\prime}(t)=0,X_{1}=\cot\theta_{0}X_{2}+\alpha(t)\right\\}.$

Therefore, in this case, $\mathcal{D}=E_{1}=E_{2}=\emptyset$ if and only if
$\alpha$ is non-singular. Suppose that $\theta_{0}\in\pi\mathbb{Z}$. Then,

$\mathcal{D}=\left\\{\left(X_{1},X_{2}\right)\,|\,X_{2}=0\right\\}.$

Therefore, in this case, $E_{1}=E_{2}=\mathcal{D}$ if and only if $\beta$ is
surjective.

  2. (2)

Let $\widetilde{\nu}:\mathbb{R}\to S^{1}$ be the mapping given by
$\widetilde{\nu}(t)=(\cos t,\sin t)$. Set
$\widetilde{\nu}_{\theta_{0}}=R_{\theta_{0}}\circ\widetilde{\nu}$, where
$R_{\theta_{0}}$ is the rotation defined in the above example. Then, since
$\frac{d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)}{dt}(t)=1$, it
follows $d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)=dt$. Thus, by
Theorem 1(1) and Theorem 2, for any
$\widetilde{\varphi}:\mathbb{R}\to\mathbb{R}^{2}$ the line family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}_{\theta_{0}}\right)}$
creates a unique envelope $\widetilde{f}_{\theta_{0}}$. For any
$\widetilde{\varphi}:\mathbb{R}\to\mathbb{R}^{2}$, set
$\widetilde{\gamma}_{\theta_{0}}(t)=\widetilde{\varphi}(t)\cdot\widetilde{\nu}_{\theta_{0}}(t)$.
Since
$d\widetilde{\gamma}_{\theta_{0}}=\frac{d\widetilde{\gamma}_{\theta_{0}}}{dt}(t)d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)$,
by Theorem 1 (2),  it follows

$\displaystyle\widetilde{f}(t)$ $\displaystyle=$
$\displaystyle\frac{d\widetilde{\gamma}_{\theta_{0}}}{dt}(t)R_{\pi/2}\circ\widetilde{\nu}_{\theta_{0}}\left(t\right)+\widetilde{\gamma}_{\theta_{0}}(t)\widetilde{\nu}_{\theta_{0}}(t)$
$\displaystyle=$
$\displaystyle\frac{d\widetilde{\gamma}_{\theta_{0}}}{dt}(t)R_{\pi/2}\circ\widetilde{\nu}_{\theta_{0}}\left(t\right)+\widetilde{\gamma}_{\theta_{0}}(t)\left(\cos\left(t+\theta_{0}\right),\sin\left(t+\theta_{0}\right)\right),$

where the $1$-form $d\left(\Theta\circ\widetilde{\nu}\right)$ and the vector
field $R_{\pi/2}\circ\widetilde{\nu}_{\theta_{0}}\left(t\right)$ are
identified. Let $\alpha:\mathbb{R}\to\mathbb{R}$ be a function and set
$\widetilde{\varphi}(t)=\widetilde{\nu}(t)+\alpha(t)R_{\pi/2}\circ\widetilde{\nu}_{\theta_{0}}(t)$.

Figure 11.  Figure for Example 4.1 (2).

Then, it follows $\frac{d\widetilde{\gamma}_{\theta_{0}}}{dt}(t)\equiv 0$.
Thus, as expected, the envelope created by the line family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}_{\theta_{0}}\right)}$
in this case is actually the circle with radius $|c|$ centered at the origin,
where $c=\widetilde{\gamma}_{\theta_{0}}(t)=\cos\theta_{0}$ (see Figure 11).

  3. (3)

Let $\widetilde{\nu}:\mathbb{R}\to S^{1}$ be the mapping defined by
$\widetilde{\nu}(t)=\frac{1}{\sqrt{1+9t^{4}}}\left(-3t^{2},1\right)$. Set
$\widetilde{\nu}_{\theta_{0}}=R_{\theta_{0}}\circ\widetilde{\nu}$ where
$R_{\theta_{0}}$ is as above. Let $\alpha:\mathbb{R}\to\mathbb{R}$ be a
function and set
$\widetilde{\varphi}_{\theta_{0}}(t)=(t,t^{3})+\alpha(t)R_{\pi/2}\circ\widetilde{\nu}_{\theta_{0}}(t)$.
Set
$\widetilde{\gamma}_{\theta_{0}}(t)=\widetilde{\varphi}_{\theta_{0}}(t)\cdot\widetilde{\nu}_{\theta_{0}}(t)$.
It is easily seen that $0$ is a singular point of
$\widetilde{\gamma}_{\theta_{0}}$ if and only if $\theta_{0}\in\pi\mathbb{Z}$.
On the other hand, by calculation, we have
$\frac{d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)}{dt}(t)=\frac{6t}{1+9t^{4}}$
and thus $0$ is a unique singular point of $\widetilde{\nu}_{\theta_{0}}$ for
any $\theta_{0}$. Therefore, by Theorem 1, the hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}_{{}_{\theta}}\right)}$
does not create an envelope if
$\theta_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}0}\not\in\pi\mathbb{Z}$.

Next, suppose that $\theta_{0}\in\pi\mathbb{Z}$. Then, calculations show

$\quad\quad
d\left(\widetilde{\gamma}_{\theta_{0}}\right)=\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(6t^{2}+18t^{6})}{(1+9t^{4})^{\frac{3}{2}}}dt=\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(t+3t^{5})}{\sqrt{1+9t^{4}}}\frac{d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)}{dt}(t)dt=\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(t+3t^{5})}{\sqrt{1+9t^{4}}}d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},}$

where double sign should be read in the same order. Set
$\widetilde{\omega}(t)=\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(t+3t^{5})}{\sqrt{1+9t^{4}}}d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)$.
By Theorem 1 and Theorem 2, the hyperplane family
$\mathcal{H}_{(\widetilde{\varphi},\widetilde{\nu}_{\theta_{0}})}$ creates a
unique envelope with the desired form

$\displaystyle\widetilde{f}(t)$ $\displaystyle=$
$\displaystyle\widetilde{\omega}(t)+\widetilde{\gamma}_{\theta_{0}}(t)\widetilde{\nu}_{\theta_{0}}(t)$
$\displaystyle=$
$\displaystyle\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(t+3t^{5})}{1+9t^{4}}\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp}1,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp}3t^{2}\right){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp}\frac{2t^{3}}{1+9t^{4}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp}3t^{2},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pm}1)$
$\displaystyle=$
$\displaystyle\frac{1}{1+9t^{4}}\left(t+3t^{5}+6t^{5},3t^{3}+9t^{7}-2t^{3}\right)$
$\displaystyle=$ $\displaystyle\left(t,t^{3}\right),$

where for each $t\in\mathbb{R}$ the cotangent vector
$\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(t+3t^{5})}{\sqrt{1+9t^{4}}}d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)$
and the vector
$\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(t+3t^{5})}{\sqrt{1+9t^{4}}}R_{\pi/2}\circ\widetilde{\nu}_{\theta_{0}}(t)$
in the vector space $\mathbb{R}^{2}$ are identified (see Figure 12).

Figure 12.  Figure for Example 4.1 (3) in the case $\theta_{0}\in
2\pi\mathbb{Z}$.

Set $U=\mathbb{R}-\\{0\\}$. It is easily seen that
$\widetilde{\nu}_{\theta_{0}}|_{U}$ is non-singular even in the case
$\theta_{0}\not\in\pi\mathbb{Z}$. Hence, by Theorem 1 and Theorem 2, the
hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi}|_{U},\widetilde{\nu}_{\theta_{0}}|_{U}\right)}$
creates a unique envelope $\widetilde{f}_{\theta_{0}}:U\to\mathbb{R}^{2}$ even
when $\theta_{0}\not\in\pi\mathbb{Z}$ and $\lim_{t\to
0}\|\widetilde{f}_{\theta_{0}}(t)\|=\infty$ when
$\theta_{0}\not\in\pi\mathbb{Z}$.

  4. (4)

Let $\widetilde{\nu}:\mathbb{R}\to S^{1}$ be the mapping defined by
$\widetilde{\nu}(t)=\frac{1}{\sqrt{4+25t^{6}}}\left(-5t^{3},2\right)$. Set
$\widetilde{\nu}_{\theta_{0}}=R_{\theta_{0}}\circ\widetilde{\nu}$ where
$R_{\theta_{0}}$ is as above. Let $\alpha:\mathbb{R}\to\mathbb{R}$ be a
function and set
$\widetilde{\varphi}_{\theta_{0}}(t)=(t^{2},t^{5})+\alpha(t)R_{\pi/2}\circ\widetilde{\nu}_{\theta_{0}}(t)$.
Set
$\widetilde{\gamma}_{\theta_{0}}(t)=\widetilde{\varphi}_{\theta_{0}}(t)\cdot\widetilde{\nu}_{\theta_{0}}(t)=\frac{-3t^{5}\cos\theta_{0}-2t^{2}\sin\theta_{0}-5t^{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}8}\sin\theta_{0}}{\sqrt{4+25t^{6}}}$.
By calculation, we have
$\frac{d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)}{dt}(t)=\frac{30t^{2}}{4+25t^{6}}$.
Therefore, the hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}_{{}_{\theta}}\right)}$
is not creative if $\theta\not\in\pi\mathbb{Z}$ and it creates no envelope in
this case by Theorem 1.

Next, suppose that $\theta_{0}\in\pi\mathbb{Z}$. Then, calculation shows

$\displaystyle d\left(\widetilde{\gamma}_{\theta_{0}}\right)$ $\displaystyle=$
$\displaystyle\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp
30t^{2}\left(2t^{2}+5t^{8}\right)}{(4+25t^{6})\sqrt{4+25t^{6}}}dt$
$\displaystyle=$
$\displaystyle\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(2t^{2}+5t^{8})}{\sqrt{4+25t^{6}}}\frac{d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)}{dt}(t)dt=\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(2t^{2}+5t^{8})}{\sqrt{4+25t^{6}}}d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},}$

where double sign should be read in the same order. Therefore, the hyperplane
family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}_{{}_{\theta}}\right)}$
is creative. Set
$\widetilde{\omega}(t)=\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(2t^{2}+5t^{8})}{\sqrt{4+25t^{6}}}.d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)$.
By Theorem 1 and Theorem 2,
$\mathcal{H}_{(\widetilde{\varphi},\widetilde{\nu}_{\theta_{0}})}$ creates a
unique envelope with the desired form

$\displaystyle\widetilde{f}(t)$ $\displaystyle=$
$\displaystyle\widetilde{\omega}(t)+\widetilde{\gamma}_{\theta_{0}}(t)\widetilde{\nu}_{\theta_{0}}(t)$
$\displaystyle=$
$\displaystyle\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(2t^{2}+5t^{8})}{4+25t^{6}}\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp}2,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp}5t^{3}\right)+\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp}3t^{5}}{4+25t^{6}}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp}5t^{3},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pm}2)$
$\displaystyle=$
$\displaystyle\frac{1}{4+25t^{6}}\left(4t^{2}+10t^{8}+15t^{8},10t^{5}+25t^{11}-6t^{5}\right)$
$\displaystyle=$ $\displaystyle\left(t^{2},t^{5}\right),$

where for each $t\in\mathbb{R}$ the cotangent vector
$\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(2t^{2}+5t^{8})}{\sqrt{4+25t^{6}}}d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)$
and the vector
$\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(2t^{2}+5t^{8})}{\sqrt{4+25t^{6}}}R_{\pi/2}\circ\widetilde{\nu}_{\theta_{0}}(t)$
in the vector space $\mathbb{R}^{2}$ are identified (see Figure 13).

Figure 13.  Figure for Example 4.1 (4) in the case $\theta_{0}\in
2\pi\mathbb{Z}$.

In the case $\theta_{0}=0$, consider the mapping
$\widetilde{\Omega}:\mathbb{R}\to T^{*}S^{1}$ given in Definition 2 and
$\Omega:\mathbb{R}\to J^{1}\left(S^{1},\mathbb{R}\right)$ given in Remark
1.1(1). Namely, consider the following two mappings.

$\displaystyle\widetilde{\Omega}(t)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{\sqrt{4+25t^{6}}}\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp}5t^{3},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pm}2\right),\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp
30t^{2}(2t^{2}+5t^{8})}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(4+25t^{6}\right)^{\frac{3}{2}}}\right),$
$\displaystyle\Omega(t)$ $\displaystyle=$
$\displaystyle\left(\frac{1}{\sqrt{4+25t^{6}}}\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp}5t^{3},{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\pm}2\right),\frac{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp}3t^{5}}{\sqrt{4+25t^{6}}},\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp
30t^{2}(2t^{2}+5t^{8})}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\left(4+25t^{6}\right)^{\frac{3}{2}}}\right).$

Since
$d\left(\widetilde{\gamma}_{\theta_{0}}\right)=\frac{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\mp(2t^{2}+5t^{8})}{\sqrt{4+25t^{6}}}d\left(\Theta\circ\widetilde{\nu}_{\theta_{0}}\right)$,
the map-germ of $\Omega$ at any $t$ is nothing but an opening of the map-germ
$\widetilde{\Omega}:(\mathbb{R},t)\to T^{*}S^{1}$. At $t=0$, the map-germ of
each of them is not immersive and has singular images.

Set $U=\mathbb{R}-\\{0\\}$. It is easily seen that
$\widetilde{\nu}_{\theta_{0}}|_{U}$ is non-singular even in the case
$\theta_{0}\not\in\pi\mathbb{Z}$. Hence, by Theorem 1 and Theorem 2, the
hyperplane family
$\mathcal{H}_{\left(\widetilde{\varphi}|_{U},\widetilde{\nu}_{\theta_{0}}|_{U}\right)}$
creates a unique envelope $\widetilde{f}_{\theta_{0}}:U\to\mathbb{R}^{2}$ even
when $\theta_{0}\not\in\pi\mathbb{Z}$ and $\lim_{t\to
0}\|\widetilde{f}_{\theta_{0}}(t)\|=\infty$ when
$\theta_{0}\not\in\pi\mathbb{Z}$.

######  Example 4.2 (Unit speed curves).

  1. (1)

Let ${\bf r}:\mathbb{R}\to\mathbb{R}^{2}$ be a unit speed curve. As usual, set
${\bf t}(s)={\bf r}^{\prime}(s)$ and ${\bf n}(s)$ is defined from t(s) by
rotating anticlockwise through $\frac{\pi}{2}$. The Serret-Frenet formulas for
the plane curve ${\bf r}$ is as follows.

$\left\\{\begin{array}[]{cccc}{\bf t}^{\prime}(s)&=&{}\hfil&\kappa(s){\bf
n}(s)\\\ {\bf n}^{\prime}(s)&=&-\kappa(s){\bf
t}(s).&{}\hfil\end{array}\right.$

Set $\widetilde{\varphi}={\bf r}$ and $\widetilde{\nu}={\bf n}$. Then, the
line family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}=\mathcal{H}_{\left({\bf
r},{\bf n}\right)}$ is the affine tangent line family of the curve ${\bf r}$.
In this case, the correspondence ${\bf r}\mapsto\mathcal{H}_{\left({\bf
r},{\bf n}\right)}$ may be regarded as the Legendre transformation of the
given curve ${\bf r}$. Set
$\widetilde{\gamma}(s)=\widetilde{\varphi}(s)\cdot\widetilde{\nu}(s)$. Then,

$\widetilde{\gamma}^{\prime}(s)={\bf r}(s)\cdot\left(-\kappa(s){\bf
t}(s)\right)={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}-}\left({\bf
r}(s)\cdot{\bf t}(s)\right)\left(\Theta_{{\bf
t}}\circ\widetilde{\nu}\right)^{\prime}(s){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0},}$

where $\widetilde{\nu}(s)=\left(\cos\Theta_{{\bf
t}}\circ\widetilde{\nu}(s),\sin\Theta_{{\bf
t}}\circ\widetilde{\nu}(s)\right)$. Therefore, by Theorem 1, the line family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ creates an
envelope.

Suppose that the set of regular points of $\widetilde{\nu}$ is dense, that is
to say, the set $\\{s\in\mathbb{R}\;|\;\kappa(s)\neq 0\\}$ is dense. Then, by
Theorem 2, the created envelopes are unique. By Theorem 1, the unique envelope
is as follows (see Figure 14).

$\displaystyle\widetilde{f}(s)$ $\displaystyle=$
$\displaystyle\widetilde{\omega}(s)+\widetilde{\gamma}(s)\cdot\widetilde{\nu}(s)$
$\displaystyle=$ $\displaystyle\left({\bf r}(s)\cdot{\bf t}(s)\right){\bf
t}(s)+\left({\bf r}(s)\cdot{\bf n}(s)\right){\bf n}(s)$ $\displaystyle=$
$\displaystyle{\bf r}(s).$

Notice that if there is a point $s\in\mathbb{R}$ such that $\kappa(s)=0$, then
the full discriminant of the line family is different from the unique desired
envelope since the full discriminant includes the affine tangent line at $s$.
This is one of advantages of our method. The correspondence

$\mathcal{H}_{\left({\bf r},{\bf n}\right)}\mapsto{\bf r}$

may be regarded as the inverse Legendre transformation for plane curves.

Next, suppose that the set of regular points of $\widetilde{\nu}$ is not
dense. Then, there exists an open interval $(a,b)$ such that $\kappa(s)=0$ for
any $s\in(a,b)$. Then, for any $s\in(a,b)$ and any function
$\alpha:\mathbb{R}\to\mathbb{R}$ such that $\alpha(\mathbb{R}-(a,b))=\\{0\\}$,
it follows

$\widetilde{\gamma}^{\prime}(s)=\alpha(s)\left(\Theta_{{\bf
t}}\circ\widetilde{\nu}\right)^{\prime}(s).$

By Theorem 1,

$\displaystyle\widetilde{f}(s)$ $\displaystyle=$
$\displaystyle\widetilde{\omega}(s)+\widetilde{\gamma}(s)\cdot\widetilde{\nu}(s)$
$\displaystyle=$ $\displaystyle\alpha(s){\bf t}(s)+\left({\bf r}(s)\cdot{\bf
n}(s)\right){\bf n}(s)$ $\displaystyle=$
$\displaystyle\left(\left(\alpha(s)-\left({\bf r}(s)\cdot{\bf
t}(s)\right)\right)+\left({\bf r}(s)\cdot{\bf t}(s)\right)\right){\bf
t}(s)+\left({\bf r}(s)\cdot{\bf n}(s)\right){\bf n}(s)$ $\displaystyle=$
$\displaystyle{\bf r}(s)+\beta(s){\bf t}(s),$

where $\beta(s)=\alpha(s)-\left({\bf r}(s)\cdot{\bf t}(s)\right)$. Hence, in
this case, the inverse Legendre transformation does not work well.

Figure 14.  Example 4.2 (1).

  1. (2)

Let ${\bf r}:\mathbb{R}\to\mathbb{R}^{3}$ be a unit speed space curve. As
usual, set ${\bf t}(s)={\bf r}^{\prime}(s)$ and assume $||{\bf
t}^{\prime}(s)||>0$ for any $s\in\mathbb{R}$ so that the principal normal
vector ${\bf n}(s)$ can be defined by ${\bf t}^{\prime}(s)=||{\bf
t}^{\prime}(s)||{\bf n}(s)$. As usual, the binormal vector ${\bf b}(s)$ is
defined by $\det\left({\bf t}(s),{\bf n}(s),{\bf b}(s)\right)=1$. The Serret-
Frenet formulas for the space curve ${\bf r}$ is as follows.

$\left\\{\begin{array}[]{ccccc}{\bf t}^{\prime}(s)&=&{}\hfil&\kappa(s){\bf
n}(s)&{}\hfil\\\ {\bf n}^{\prime}(s)&=&-\kappa(s){\bf
t}(s)&{}\hfil&+\tau(s){\bf b}(s)\\\ {\bf b}^{\prime}(s)&=&{}\hfil&-\tau(s){\bf
n}(s)&{}.\end{array}\right.$

Define $\widetilde{\varphi}:\mathbb{R}^{2}\to\mathbb{R}^{3}$ and
$\widetilde{\nu}:\mathbb{R}^{2}\to S^{2}$ by $\widetilde{\varphi}(s,u)={\bf
r}(s)$ and $\widetilde{\nu}(s,u)={\bf b}(s)$ respectively. Then, the plane
family $\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is the
family of osculating planes of the space curve ${\bf r}$. Set
$\widetilde{\gamma}(s,u)={\bf r}(s)\cdot{\bf b}(s)$. Then, all of the
following six identities are clear.

$\frac{\partial\widetilde{\gamma}}{\partial s}(s,u)={\bf
r}(s)\cdot\left(-\tau(s){\bf
n}(s)\right),\quad\frac{\partial\widetilde{\gamma}}{\partial
u}(s,u)=0,\quad\frac{\partial\left(\Theta_{{\bf
t}}\circ\widetilde{\nu}\right)}{\partial s}(s,u)=0,\quad$
$\frac{\partial\left(\Theta_{{\bf t}}\circ\widetilde{\nu}\right)}{\partial
u}(s,u)=0,\quad\frac{\partial\left(\Theta_{{\bf
n}}\circ\widetilde{\nu}\right)}{\partial
s}(s,u)=-\tau(s),\quad\frac{\partial\left(\Theta_{{\bf
n}}\circ\widetilde{\nu}\right)}{\partial u}(s,u)=0.$

Therefore, we have the following.

$\displaystyle\frac{\partial\widetilde{\gamma}}{\partial s}(s,u)$
$\displaystyle=$ $\displaystyle\alpha_{1}(s,u)\frac{\partial\left(\Theta_{{\bf
t}}\circ\widetilde{\nu}\right)}{\partial s}(s,u)+\left({\bf r}(s)\cdot{\bf
n}(s)\right)\frac{\partial\left(\Theta_{{\bf
n}}\circ\widetilde{\nu}\right)}{\partial s}(s,u),$
$\displaystyle\frac{\partial\widetilde{\gamma}}{\partial u}(s,u)$
$\displaystyle=$ $\displaystyle\alpha_{2}(s,u)\frac{\partial\left(\Theta_{{\bf
t}}\circ\widetilde{\nu}\right)}{\partial
u}(s,u)+\alpha_{3}(s,u)\frac{\partial\left(\Theta_{{\bf
n}}\circ\widetilde{\nu}\right)}{\partial u}(s,u),$

where $\alpha_{1},\alpha_{2},\alpha_{3}:\mathbb{R}^{2}\to\mathbb{R}$ are
arbitrary functions. Thus, by Theorem 1, the plane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ creates an
envelope if and only if $\left({\bf r}(s)\cdot{\bf
n}(s)\right)=\alpha_{3}(s,u)$ and $\alpha_{1}(s,u)=\alpha_{2}(s,u)$.
Therefore, again by Theorem 1, we have the following concrete expression of
the created envelopes.

$\displaystyle\widetilde{f}(s,u)$ $\displaystyle=$
$\displaystyle\widetilde{\omega}(s,u)+\widetilde{\gamma}(s)\widetilde{\nu}(s)$
$\displaystyle=$ $\displaystyle\left({\bf r}(s)\cdot{\bf n}(s)\right){\bf
n}(s)+\alpha(s,u){\bf t}(s)+\left({\bf r}(s)\cdot{\bf b}(s)\right){\bf b}(s)$
$\displaystyle=$ $\displaystyle\left({\bf r}(s)\cdot{\bf n}(s)\right){\bf
n}(s)+\left({\bf r}(s)\cdot{\bf t}(s)\right){\bf
t}(s)+\left(\alpha(s,u)-\left({\bf r}(s)\cdot{\bf t}(s)\right)\right){\bf
t}(s)+\left({\bf r}(s)\cdot{\bf b}(s)\right){\bf b}(s)$ $\displaystyle=$
$\displaystyle{\bf r}(s)+\beta(s,u){\bf t}(s),$

where $\alpha(s,u)=\alpha_{1}(s,u)=\alpha_{2}(s,u)$ and
$\beta(s,u)=\alpha(s,u)-\left({\bf r}(s)\cdot{\bf t}(s)\right)$. All envelopes
created by the osculating family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ can be
exactly expressed as above. Hence, for example, both the tangent developable
of ${\bf r}$ (in the case $\beta(s,u)=u$) and the space curve ${\bf r}$ (in
the case $\beta(s,u)=0$) are envelopes of
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$. Not only
these two, there are uncountably many envelopes created by
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$. All
envelopes for the osculating plane family are created only by the given curve
${\bf r}$ and its unit tangent curve ${\bf t}$.

Next, we consider envelopes created by $\mathcal{H}_{\left({\bf r},{\bf
b}\right)}$ and $\mathcal{H}_{\left(\widetilde{f},{\bf n}\right)}$. Namely, we
obtain all solutions
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{g}}(s,u)$
for the following system of PDEs with one constraint condition.

$\left\\{\begin{array}[]{ccc}\frac{\partial{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{g}}}{\partial
s}(s,u)\cdot{\bf b}(s)&=&0,\\\
\frac{\partial{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{g}}}{\partial
u}(s,u)\cdot{\bf b}(s)&=&0,\\\
\frac{\partial{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{g}}}{\partial
s}(s,u)\cdot{\bf n}(s)&=&0,\\\
\frac{\partial{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{g}}}{\partial
u}(s,u)\cdot{\bf n}(s)&=&0,\\\
\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{g}}(s,u)-{\bf
r}(s)\right)\cdot{\bf b}(s)&=&0.\end{array}\right.$

Since $\kappa(s)>0$ for any $s\in\mathbb{R}$ and

$\displaystyle\frac{\partial\widetilde{f}}{\partial s}(s,u)$ $\displaystyle=$
$\displaystyle{\bf t}(s)+\frac{\partial\beta}{\partial s}(s,u){\bf
t}(s)+\beta(s,u)\left(\kappa(s){\bf n}(s)\right),$
$\displaystyle\frac{\partial\widetilde{f}}{\partial u}(s,u)$ $\displaystyle=$
$\displaystyle\frac{\partial\beta}{\partial u}(s,u){\bf t}(s),$

if $\widetilde{f}$ itself is a solution of the above system of PDEs, then
$\beta(s,u)$ must be constant $0$. Conversely, it is clear that ${\bf r}$
itself is a solution of the above system of PDEs with one constraint
condition. Therefore, for the above system of PDEs with one constraint
condition, there are no solutions except for the trivial solution ${\bf r}$.
This implies that even for a space curve ${\bf
r}:\mathbb{R}\to\mathbb{R}^{3}$, the inverse Legendre transformation

$\mathcal{H}_{\left({\bf r},\\{{\bf b},{\bf n}\\}\right)}\mapsto{\bf r}$

works well.

Finally, we consider envelopes created by $\mathcal{H}_{\left({\bf r},{\bf
b}\right)}$ and $\mathcal{H}_{\left(\widetilde{f},{\bf t}\right)}$. Namely, we
obtain all solutions
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{g}}(s,u)$
for the following system of PDEs with one constraint condition.

$\left\\{\begin{array}[]{ccc}\frac{\partial{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{g}}}{\partial
s}(s,u)\cdot{\bf b}(s)&=&0,\\\
\frac{\partial{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{g}}}{\partial
u}(s,u)\cdot{\bf b}(s)&=&0,\\\
\frac{\partial{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{g}}}{\partial
s}(s,u)\cdot{\bf t}(s)&=&0,\\\
\frac{\partial{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{g}}}{\partial
u}(s,u)\cdot{\bf t}(s)&=&0,\\\
\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\widetilde{g}}(s,u)-{\bf
r}(s)\right)\cdot{\bf b}(s)&=&0.\end{array}\right.$

By the above calculations, if $\widetilde{f}$ is a solution of the above
system of PDEs, then both $1+\frac{\partial\beta}{\partial s}(s,u)=0$ and
$\frac{\partial\beta}{\partial u}(s,u)=0$ must be satisfied. It follows
$\beta(s,u)=-s+c$ $(c\in\mathbb{R})$. It is easily seen that for any
$c\in\mathbb{R}$, the space curve $s\mapsto{\bf r}(s)+(-s+c){\bf t}(s)$ is a
solution of the above system of PDEs with one constraint condition. . Thus, in
this case, the system of PDEs with one constraint condition has uncountably
many solutions.

######  Example 4.3.

  1. (1)

(The shoe surface : Example 1 of [3]) In this example, along the general
theory developed in this paper, we start from making several general formulas
for the envelope created by the affine tangent plane family of the surface
having the form $\widetilde{\varphi}:\mathbb{R}^{2}\to\mathbb{R}^{3}$,
$\widetilde{\varphi}(x,y)=(x,\,y,\,\widetilde{\varphi}_{1}(x,y))$ such that
the origin $(0,0)$ is a singular point of the function
$\widetilde{\varphi}_{1}:\mathbb{R}^{2}\to\mathbb{R}$ and there are no other
singular points of $\widetilde{\varphi}_{1}$. Then, by calculating the
obtained general formulas in the case of the shoe surface
$\widetilde{\varphi}(x,y)=\left(x,\,y,\,\frac{1}{3}x^{3}-\frac{1}{2}y^{2}\right)$,
just by calculations, we confirm that the concrete representation form of the
envelope created by the affine tangent plane family of the shoe surface
$\widetilde{\varphi}$ is actually the shoe surface itself.

Let $\widetilde{\varphi}:\mathbb{R}^{2}\to\mathbb{R}^{3}$ be the mapping
having the form $\widetilde{\varphi}(x,y)=(x,y,\widetilde{\varphi}_{1}(x,y))$,
where the function $\widetilde{\varphi}_{1}:\mathbb{R}^{2}\to\mathbb{R}$ has a
unique singularity at the origin, namely
$\frac{\partial\widetilde{\varphi}_{1}}{\partial
x}(0,0)=\frac{\partial\widetilde{\varphi}_{1}}{\partial y}(0,0)=0$ and
$\left(\frac{\partial\widetilde{\varphi}_{1}}{\partial
x}(x,y),\,\frac{\partial\widetilde{\varphi}_{1}}{\partial
y}(x,y)\right)\neq(0,0)$ for any $(x,y)\in\mathbb{R}^{2}-\\{(0,0)\\}$. Then,
the mapping $\widetilde{\nu}:\mathbb{R}^{2}\to S^{2}$ defined by

$\widetilde{\nu}(x,y)=\frac{\frac{\partial\widetilde{\varphi}_{1}}{\partial
x}(x,y)\times\frac{\partial\widetilde{\varphi}_{1}}{\partial
y}(x,y)}{\parallel\frac{\partial\widetilde{\varphi}_{1}}{\partial
x}(x,y)\times\frac{\partial\widetilde{\varphi}_{1}}{\partial
y}(x,y)\parallel}=\frac{\left(-\frac{\partial\widetilde{\varphi}_{1}}{\partial
x},\,-\frac{\partial\widetilde{\varphi}_{1}}{\partial
y},\,1\right)}{\sqrt{\left(\frac{\partial\widetilde{\varphi}_{1}}{\partial
x}\right)^{2}+\left(\frac{\partial\widetilde{\varphi}_{1}}{\partial
y}\right)^{2}+1}}$

is a Gauss mapping of the tangent plane family of $\widetilde{\varphi}$. Here,
the tangent plane family of $\widetilde{\varphi}$ is
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$. Let
$(x_{0},y_{0})$ be an arbitrary point of $\mathbb{R}^{2}-\\{(0,0)\\}$. Then,
by the assumption on the function $\widetilde{\varphi}_{1}$, it follows that
$\widetilde{\nu}(x_{0},y_{0})\neq(0,\,0,\,1)$. Set

$\displaystyle{\bf v}_{0}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\widetilde{\nu}(x_{0},y_{0}),$ $\displaystyle{\bf
v}_{1}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\frac{(0,\,0,\,1)-\left((0,\,0,\,1)\cdot{\bf
v}_{0}(x_{0},y_{0})\right){\bf
v}_{0}(x_{0},y_{0})}{\parallel(0,\,0,\,1)-\left((0,\,0,\,1)\cdot{\bf
v}_{0}(x_{0},y_{0})\right){\bf v}_{0}(x_{0},y_{0})\parallel},$
$\displaystyle{\bf v}_{2}(x_{0},y_{0})$ $\displaystyle=$ $\displaystyle{\bf
v}_{0}(x_{0},y_{0})\times{\bf v}_{1}(x_{0},y_{0}).$

Then, $\langle{\bf v}_{0}(x_{0},y_{0}),{\bf v}_{1}(x_{0},y_{0}),{\bf
v}_{2}(x_{0},y_{0})\rangle$ is an orthonormal basis of $\mathbb{R}^{3}$, and
under the identification of two vector spaces $\mathbb{R}^{3}$ and
$T_{\widetilde{\nu}(x_{0},y_{0})}\mathbb{R}^{3}$, $\langle{\bf
v}_{1}(x_{0},y_{0}),{\bf v}_{2}(x_{0},y_{0})\rangle$ is an orthonormal basis
of the tangent vector space $T_{\widetilde{\nu}(x_{0},y_{0})}S^{2}$. Let
$\varepsilon$ be a sufficiently small positive number and denote the set
$\\{\Theta_{1}{\bf v}_{1}(x_{0},y_{0})+\Theta_{2}{\bf
v}_{2}(x_{0},y_{0})\,|\,-\varepsilon<\Theta_{1},\Theta_{2}<\varepsilon\\}$ by
$V^{\prime}$. Let $\exp:V^{\prime}\to S^{2}$ be the restriction of the
exponential mapping at $\widetilde{\nu}(x_{0},y_{0})$ to $V^{\prime}$ and set
$V=\exp(V^{\prime})$. Let $\left(V,\left(\Theta_{1},\Theta_{2}\right)\right)$
be the normal coordinate neighborhood at $\widetilde{\nu}(x_{0},y_{0})$
defined by $\exp^{-1}:V\to V^{\prime}$. Set

$\widetilde{\gamma}(x,y)=\widetilde{\varphi}(x,y)\cdot\widetilde{\nu}(x,y)=\frac{-x\frac{\partial\widetilde{\varphi}_{1}}{\partial
x}-y\frac{\partial\widetilde{\varphi}_{1}}{\partial
y}+\widetilde{\varphi}_{1}(x,y)}{\sqrt{\left(\frac{\partial\widetilde{\varphi}_{1}}{\partial
x}\right)^{2}+\left(\frac{\partial\widetilde{\varphi}_{1}}{\partial
y}\right)^{2}+1}}.$

Since $\widetilde{\nu}:\mathbb{R}^{2}\to S^{2}$ is a Gauss mapping of
$\widetilde{\varphi}:\mathbb{R}^{2}\to\mathbb{R}^{3}$, we have

$\displaystyle\frac{\partial\widetilde{\gamma}}{\partial x}(x_{0},y_{0})$
$\displaystyle=$
$\displaystyle\widetilde{\varphi}(x_{0},y_{0})\cdot\frac{\partial\widetilde{\nu}}{\partial
x}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{1}(x_{0},y_{0})\right)\frac{\partial\left(\Theta_{1}\circ\widetilde{\nu}\right)}{\partial
x}(x_{0},y_{0})+\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{2}(x_{0},y_{0})\right)\frac{\partial\left(\Theta_{2}\circ\widetilde{\nu}\right)}{\partial
x}(x_{0},y_{0})$

and

$\displaystyle\frac{\partial\widetilde{\gamma}}{\partial y}(x_{0},y_{0})$
$\displaystyle=$
$\displaystyle\widetilde{\varphi}(x_{0},y_{0})\cdot\frac{\partial\widetilde{\nu}}{\partial
y}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{1}(x_{0},y_{0})\right)\frac{\partial\left(\Theta_{1}\circ\widetilde{\nu}\right)}{\partial
y}(x_{0},y_{0})+\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{2}(x_{0},y_{0})\right)\frac{\partial\left(\Theta_{2}\circ\widetilde{\nu}\right)}{\partial
y}(x_{0},y_{0}).$

Thus, as the equality of $2$-dimensional cotangent vectors of
$T^{*}_{(x_{0},y_{0})}\mathbb{R}^{2}$, we have the following equality.

$\displaystyle d\widetilde{\gamma}$ $\displaystyle=$
$\displaystyle\frac{\partial\widetilde{\gamma}}{\partial
x}(x_{0},y_{0})dx+\frac{\partial\widetilde{\gamma}}{\partial
y}(x_{0},y_{0})dy$ $\displaystyle=$
$\displaystyle\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{1}(x_{0},y_{0})\right)d\left(\Theta_{1}\circ\widetilde{\nu}\right)+\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{2}(x_{0},y_{0})\right)d\left(\Theta_{2}\circ\widetilde{\nu}\right).$

Set $U=\mathbb{R}^{2}-\\{(0,0)\\}$ and assume that the singular set of
$\widetilde{\nu}$ is of Lebesgue measure zero. Then, since $(x_{0},y_{0})$ is
an arbitrary point of $U$, by Theorem 1 (1) and Theorem 2, it follows that
$\mathcal{H}_{(\widetilde{\varphi}|_{U},\widetilde{\nu}|_{U})}$ creates a
unique envelope. Set

$\widetilde{\omega}(x_{0},y_{0})=\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{1}(x_{0},y_{0})\right)d\left(\Theta_{1}\circ\widetilde{\nu}\right)+\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{2}(x_{0},y_{0})\right)d\left(\Theta_{2}\circ\widetilde{\nu}\right).$

Then, under the canonical identifications

$T^{*}_{\widetilde{\nu}(x_{0},y_{0})}S^{2}\cong
T_{\widetilde{\nu}(x_{0},y_{0})}S^{2}\subset
T_{\widetilde{\nu}(x_{0},y_{0})}\mathbb{R}^{3}\cong\mathbb{R}^{3},$

the $2$-dimensional cotangent vector

$\widetilde{\omega}(x_{0},y_{0})=\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{1}(x_{0},y_{0})\right)d\Theta_{1}+\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{2}(x_{0},y_{0})\right)d\Theta_{2}$

may be regarded as the following $3$-dimensional vector (denoted by the same
symbol $\widetilde{\omega}(x_{0},y_{0})$).

$\widetilde{\omega}(x_{0},y_{0})=\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{1}(x_{0},y_{0})\right){\bf
v}_{1}(x_{0},y_{0})+\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{2}(x_{0},y_{0})\right){\bf v}_{2}(x_{0},y_{0}).$

Therefore, by Theorem 1 (2), the envelope vector at $(x_{0},y_{0})$ must have
the following form:

$\displaystyle\widetilde{f}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\widetilde{\omega}(x_{0},y_{0})+\widetilde{\gamma}(x_{0},y_{0})\widetilde{\nu}(x_{0},y_{0})$
$\displaystyle=$ $\displaystyle\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{1}(x_{0},y_{0})\right){\bf
v}_{1}(x_{0},y_{0})+\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{2}(x_{0},y_{0})\right){\bf v}_{2}(x_{0},y_{0})$
$\displaystyle\qquad\qquad+\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{0}(x_{0},y_{0})\right){\bf v}_{0}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\widetilde{\varphi}(x_{0},y_{0}).$

By continuity, it follows that $\widetilde{f}=\widetilde{\varphi}$ is the
unique envelope created by the given plane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$.

Next, we apply the above formulas to the shoe surface. The shoe surface is the
image of $\widetilde{\varphi}:\mathbb{R}^{2}\to\mathbb{R}^{3}$ defined by
$\widetilde{\varphi}(x,y)=\left(x,y,\frac{1}{3}x^{3}-\frac{1}{2}y^{2}\right)$.
Set $\widetilde{\varphi}_{1}(x,y)=\frac{1}{3}x^{3}-\frac{1}{2}y^{2}$. Then,
the origin $(0,0)$ is a unique singular point of $\widetilde{\varphi}_{1}$.
For the given $\widetilde{\varphi}$, we have
$\widetilde{\nu}(x,y)=\frac{\frac{\partial\widetilde{\varphi}}{\partial
x}(x,y)\times\frac{\partial\widetilde{\varphi}}{\partial
y}(x,y)}{||\frac{\partial\widetilde{\varphi}}{\partial
x}(x,y)\times\frac{\partial\widetilde{\varphi}}{\partial
y}(x,y)||}=\frac{\left(-x^{2},\,y,\,1\right)}{\sqrt{x^{4}+y^{2}+1}}$. It is
easily confirmed that the set consisting of regular points of
$\widetilde{\nu}$ is dense. In fact, it is known that any singularity of
$\widetilde{\nu}$ is a fold singularity (see [3]). Set
$U=\mathbb{R}^{2}-\\{(0,0)\\}$ and take an arbitrary point $(x_{0},y_{0})$ of
$U$. For the shoe surface $\widetilde{\varphi}$, we set

$\displaystyle{\bf v}_{0}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\widetilde{\nu}(x_{0},y_{0})=\frac{\left(-x_{0}^{2},\,y_{0},\,1\right)}{\sqrt{x_{0}^{4}+y_{0}^{2}+1}},$
$\displaystyle{\bf v}_{1}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\frac{(0,\,0,\,1)-\left((0,\,0,\,1)\cdot{\bf
v}_{0}(x_{0},y_{0})\right){\bf
v}_{0}(x_{0},y_{0})}{\parallel(0,\,0,\,1)-\left((0,\,0,\,1)\cdot{\bf
v}_{0}(x_{0},y_{0})\right){\bf
v}_{0}(x_{0},y_{0})\parallel}=\frac{\left(x_{0}^{2},\,-y_{0},\,x_{0}^{4}+y_{0}^{2}\right)}{\sqrt{\left(x_{0}^{4}+y_{0}^{2}\right)\left(x_{0}^{4}+y_{0}^{2}+1\right)}},$
$\displaystyle{\bf v}_{2}(x_{0},y_{0})$ $\displaystyle=$ $\displaystyle{\bf
v}_{0}(x_{0},y_{0})\times{\bf
v}_{1}(x_{0},y_{0})=\frac{\left(y_{0},\,x_{0}^{2},\,0\right)}{\sqrt{x_{0}^{4}+y_{0}^{2}}}.$

By calculation, we have

$\displaystyle\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf v}_{1}(x_{0},y_{0})$
$\displaystyle=$
$\displaystyle\frac{x_{0}^{3}-y_{0}^{2}+\left(\frac{1}{3}x_{0}^{3}-\frac{1}{2}y_{0}^{2}\right)\left(x_{0}^{4}+y_{0}^{2}\right)}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{1}{2}}},$
$\displaystyle\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf v}_{2}(x_{0},y_{0})$
$\displaystyle=$
$\displaystyle\frac{x_{0}y_{0}+x_{0}^{2}y_{0}}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}}.$

Let $(V,\left(\Theta_{1},\Theta_{2}\right))$ be the normal cordinate
neighborhood of $S^{2}$ defined above. By calculations using the following two
identities

$\displaystyle\frac{\partial\widetilde{\nu}}{\partial x}(x_{0},y_{0})$
$\displaystyle=$ $\displaystyle{\bf
v}_{1}(x_{0},y_{0})\frac{\partial\left(\Theta_{1}\circ\widetilde{\nu}\right)}{\partial
x}(x_{0},y_{0})+{\bf
v}_{2}(x_{0},y_{0})\frac{\partial\left(\Theta_{2}\circ\widetilde{\nu}\right)}{\partial
x}(x_{0},y_{0}),$ $\displaystyle\frac{\partial\widetilde{\nu}}{\partial
y}(x_{0},y_{0})$ $\displaystyle=$ $\displaystyle{\bf
v}_{1}(x_{0},y_{0})\frac{\partial\left(\Theta_{1}\circ\widetilde{\nu}\right)}{\partial
y}(x_{0},y_{0})+{\bf
v}_{2}(x_{0},y_{0})\frac{\partial\left(\Theta_{2}\circ\widetilde{\nu}\right)}{\partial
y}(x_{0},y_{0}),$

we have the following.

$\displaystyle\frac{\partial\left(\Theta_{1}\circ\widetilde{\nu}\right)}{\partial
x}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\frac{-2x_{0}^{3}}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)},$
$\displaystyle\frac{\partial\left(\Theta_{2}\circ\widetilde{\nu}\right)}{\partial
x}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\frac{-2x_{0}y_{0}-2x_{0}y_{0}^{3}-2x_{0}^{5}y_{0}}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{3}{2}}},$
$\displaystyle\frac{\partial\left(\Theta_{1}\circ\widetilde{\nu}\right)}{\partial
y}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\frac{-y_{0}}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)},$
$\displaystyle\frac{\partial\left(\Theta_{2}\circ\widetilde{\nu}\right)}{\partial
y}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\frac{x_{0}^{2}}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{1}{2}}}.$

On the other hand, from the form
$\widetilde{\gamma}(x,y)=\widetilde{\varphi}(x,y)\cdot\widetilde{\nu}(x,y)=\frac{-\frac{2}{3}x^{3}+\frac{1}{2}y^{2}}{\sqrt{x^{4}+y^{2}+1}}$,
we have

$\displaystyle\frac{\partial\gamma}{\partial x}\left(x_{0},y_{0}\right)$
$\displaystyle=$
$\displaystyle\frac{-2x_{0}^{2}-2x_{0}^{2}y_{0}^{2}-x_{0}^{3}y_{0}^{2}-\frac{2}{3}x_{0}^{6}}{\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{3}{2}}},$
$\displaystyle\frac{\partial\gamma}{\partial y}\left(x_{0},y_{0}\right)$
$\displaystyle=$
$\displaystyle\frac{y_{0}+\frac{1}{2}y_{0}^{3}+\frac{2}{3}x_{0}^{3}y_{0}+x_{0}^{4}y_{0}}{\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{3}{2}}}.$

Thus, we have the following desired identity at $(x_{0},y_{0})$.

$\displaystyle d\widetilde{\gamma}$ $\displaystyle=$
$\displaystyle\frac{\partial\widetilde{\gamma}}{\partial
x}(x_{0},y_{0})dx+\frac{\partial\widetilde{\gamma}}{\partial
y}(x_{0},y_{0})dy$ $\displaystyle=$
$\displaystyle\frac{-2x_{0}^{2}-2x_{0}^{2}y_{0}^{2}-x_{0}^{3}y_{0}^{2}-\frac{2}{3}x_{0}^{6}}{\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{3}{2}}}dx+\frac{y_{0}+\frac{1}{2}y_{0}^{3}+\frac{2}{3}x_{0}^{3}y_{0}+x_{0}^{4}y_{0}}{\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{3}{2}}}dy$
$\displaystyle=$
$\displaystyle\left(\frac{x_{0}^{3}-y_{0}^{2}+\left(\frac{1}{3}x_{0}^{3}-\frac{1}{2}y_{0}^{2}\right)\left(x_{0}^{4}+y_{0}^{2}\right)}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{1}{2}}}\frac{-2x_{0}^{3}}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)}+\frac{\left(x_{0}y_{0}+x_{0}^{2}y_{0}\right)}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}}\frac{\left(-2x_{0}y_{0}-2x_{0}y_{0}^{3}-2x_{0}^{5}y_{0}\right)}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{3}{2}}}\right)dx$
$\displaystyle\quad+\left(\frac{x_{0}^{3}-y_{0}^{2}+\left(\frac{1}{3}x_{0}^{3}-\frac{1}{2}y_{0}^{2}\right)\left(x_{0}^{4}+y_{0}^{2}\right)}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{1}{2}}}\frac{-y_{0}}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)}+\frac{\left(x_{0}y_{0}+x_{0}^{2}y_{0}\right)}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}}\frac{x_{0}^{2}}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{1}{2}}}\right)dy$
$\displaystyle=$
$\displaystyle\left(\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{1}(x_{0},y_{0})\right)\frac{\partial\left(\Theta_{1}\circ\widetilde{\nu}\right)}{\partial
x}(x_{0},y_{0})+\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{2}(x_{0},y_{0})\right)\frac{\partial\left(\Theta_{2}\circ\widetilde{\nu}\right)}{\partial
x}(x_{0},y_{0})\right)dx$
$\displaystyle\quad+\left(\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{1}(x_{0},y_{0})\right)\frac{\partial\left(\Theta_{1}\circ\widetilde{\nu}\right)}{\partial
y}(x_{0},y_{0})+\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{2}(x_{0},y_{0})\right)\frac{\partial\left(\Theta_{2}\circ\widetilde{\nu}\right)}{\partial
y}(x_{0},y_{0})\right)dy$ $\displaystyle=$
$\displaystyle\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{1}(x_{0},y_{0})\right)d\left(\Theta_{1}\circ\widetilde{\nu}\right)+\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{2}(x_{0},y_{0})\right)d\left(\Theta_{2}\circ\widetilde{\nu}\right).$

Hence, by Theorem 1 (1) and Theorem 2, the plane family
$\mathcal{H}_{\left(\widetilde{\varphi}|_{U},\widetilde{\nu}|_{U}\right)}$ for
the shoe surface
$\widetilde{\varphi}(x,y)=\left(x,y,\frac{1}{3}x^{3}-\frac{1}{2}y^{2}\right)$
has a unique envelope
$\widetilde{f}:{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}U}\to\mathbb{R}^{3}$,
where $U=\mathbb{R}^{2}-\\{(0,0)\\}$. Then, under the canonical
identifications

$T^{*}_{\widetilde{\nu}(x_{0},y_{0})}S^{2}\cong
T_{\widetilde{\nu}(x_{0},y_{0})}S^{2}\subset
T_{\widetilde{\nu}(x_{0},y_{0})}\mathbb{R}^{3}\cong\mathbb{R}^{3},$

the $2$-dimensional cotangent vector

$\widetilde{\omega}(x_{0},y_{0})=\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{1}(x_{0},y_{0})\right)d\Theta_{1}+\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{2}(x_{0},y_{0})\right)d\Theta_{2}$

is identified with the following $3$-dimensional vector (denoted by the same
symbol $\widetilde{\omega}(x_{0},y_{0})$).

$\displaystyle\widetilde{\omega}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{1}(x_{0},y_{0})\right){\bf
v}_{1}(x_{0},y_{0})+\left(\widetilde{\varphi}(x_{0},y_{0})\cdot{\bf
v}_{2}(x_{0},y_{0})\right){\bf v}_{2}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\frac{\left(x_{0}^{3}-y_{0}^{2}+\left(\frac{1}{3}x_{0}^{3}-\frac{1}{2}y_{0}^{2}\right)\left(x_{0}^{4}+y_{0}^{2}\right)\right)}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{1}{2}}}\frac{\left(x_{0}^{2},\,-y_{0},\,x_{0}^{4}+y_{0}^{2}\right)}{{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{1}{2}}}}$
$\displaystyle\qquad+\frac{\left(x_{0}y_{0}+x_{0}^{2}y_{0}\right)}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}}\frac{\left(y_{0},\,x_{0}^{2},\,0\right)}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}}.$

Therefore, by Theorem 1 (2), the unique envelope $\widetilde{f}$ must have the
following desired parametric representation on $U=\mathbb{R}^{2}-\\{(0,0)\\}$.

$\displaystyle\widetilde{f}(x_{0},y_{0})$ $\displaystyle=$
$\displaystyle\widetilde{\omega}(x_{0},y_{0})+\widetilde{\gamma}(x_{0},y_{0})\widetilde{\nu}(x_{0},y_{0})$
$\displaystyle=$
$\displaystyle\frac{\left(x_{0}^{3}-y_{0}^{2}+\left(\frac{1}{3}x_{0}^{3}-\frac{1}{2}y_{0}^{2}\right)\left(x_{0}^{4}+y_{0}^{2}\right)\right)}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{1}{2}}}\frac{\left(x_{0}^{2},\,-y_{0},\,x_{0}^{4}+y_{0}^{2}\right)}{{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{1}{2}}}}$
$\displaystyle\qquad+\frac{\left(x_{0}y_{0}+x_{0}^{2}y_{0}\right)}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}}\frac{\left(y_{0},\,x_{0}^{2},\,0\right)}{\left(x_{0}^{4}+y_{0}^{2}\right)^{\frac{1}{2}}}+\frac{\left(-\frac{2}{3}x_{0}^{3}+\frac{1}{2}y_{0}^{2}\right)}{\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{1}{2}}}\frac{\left(-x_{0}^{2},\,y_{0},\,1\right)}{\left(x_{0}^{4}+y_{0}^{2}+1\right)^{\frac{1}{2}}}$
$\displaystyle=$
$\displaystyle\left(x_{0},\,y_{0},\,\frac{1}{3}x_{0}^{3}-\frac{1}{2}y_{0}^{2}\right)$
$\displaystyle=$ $\displaystyle\widetilde{\varphi}(x_{0},y_{0}).$

By continuity, it follows that the given shoe surface $\widetilde{\varphi}$
itself is the unique envelope created by the tangent plane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$.

The set called the parabolic line of
$\widetilde{\varphi}:\mathbb{R}^{2}\to\mathbb{R}^{3}$ consists of points
$(x,y)\in\mathbb{R}^{2}$ at which $\widetilde{\nu}$ is singular. For the shoe
surface, the parabolic line is the $y$-axis $\\{(0,y)\,|\,y\in\mathbb{R}\\}$.
Thus, as similar as the case of unit speed plane curves ${\bf
r}:\mathbb{R}\to\mathbb{R}^{2}$ with inflection points, the full discriminant
of the tangent plane family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ for the shoe
surface $\widetilde{\varphi}:\mathbb{R}^{2}\to\mathbb{R}^{3}$ is different
from the unique desired envelope $\widetilde{\varphi}$ itself, since the full
discriminant includes an affine tangent line
$\left\\{\left.\left(\lambda,y,-\frac{1}{2}y^{2}\right)\;\right|\;\lambda\in\mathbb{R}\right\\}$
at any point $(0,y)$. Therefore, even in the case of surfaces in
$\mathbb{R}^{3}$, by our method, one can distinguish the envelope in the sense
of Definition 1 and the full discriminant. This means that, in the case of
surfaces in $\mathbb{R}^{3}$ as well, our method has an advantage.

  1. (2)

(Example 4.1 of [14]) Let $\widetilde{\nu}:\mathbb{R}^{n}\to
S^{n}\subset\mathbb{R}^{n+1}$ be the mapping defined by
$\widetilde{\nu}\left(p_{1},\ldots,p_{n}\right)=\frac{1}{\sqrt{\sum_{i=1}^{n}p_{i}^{2}+1}}\left(p_{1},\ldots,p_{n},-1\right)$.
Then, $\widetilde{\nu}$ is non-singular and its inverse mapping
$\widetilde{\nu}^{-1}:\widetilde{\nu}\left(\mathbb{R}^{n+1}\right)\to\mathbb{R}^{n+1}$
is the central projection relative to the south pole $(0,\ldots,0,-1)$ of
$S^{n}$. Let $\widetilde{\varphi}:\mathbb{R}^{n}\to\mathbb{R}^{n+1}$ be an
arbitrary mapping. Set
$\widetilde{\gamma}(p)=\widetilde{\varphi}(p)\cdot\widetilde{\nu}(p)$ where
$p=\left(p_{1},\ldots,p_{n}\right)$ be a point of $\mathbb{R}^{n+1}$. Let
$\left(X=\left(X_{1},\ldots,X_{n}\right),Y\right)$ be a point of
$\mathbb{R}^{n}\times\mathbb{R}$. Since $J^{1}(\mathbb{R}^{n},\mathbb{R})$ and
$\mathbb{R}^{n}\times\mathbb{R}\times\mathbb{R}^{n}$ are identified, $(X,Y,p)$
may be regarded as the canonical coordinate system of
$J^{1}\left(\mathbb{R}^{n},\mathbb{R}\right)$. Since
$\frac{X_{i}\circ\widetilde{\nu}(p)}{Y\circ\widetilde{\nu}(p)}=-p_{i}$ for any
$i$ $(1\leq i\leq n)$ and any $p\in\mathbb{R}^{n+1}$, considering the first
order differential equation

$\left(\left(X,Y\right)-\widetilde{\varphi}(p)\right)\cdot\widetilde{\nu}(p)=0$

is exactly the same as considering the following Clairaut equation

$Y=\sum_{i=1}^{n}X_{i}p_{i}+\frac{\widetilde{\varphi}(p)\cdot\widetilde{\nu}(p)}{Y\circ\widetilde{\nu}(p)}.$

Thus, for each $x\in\mathbb{R}^{n}$ the hyperplane
$H_{\left(\widetilde{\varphi}(x),\widetilde{\nu}(x)\right)}$ is a complete
solution of the above Clairaut equation. Since $\widetilde{\nu}$ is non-
singular, by Theorem 1 and Theorem 2, the above Clairaut equation has a unique
singular solution $\widetilde{f}:\mathbb{R}^{n}\to\mathbb{R}^{n+1}$. By
Theorem 1 again,  the unique singular solution $\widetilde{f}$ has the
following expression where $x$ is an arbitrary point of $\mathbb{R}^{n}$ and
$\left(V,\left(\Theta_{1},\ldots,\Theta_{n}\right)\right)$ is a sufficiently
small normal coordinate neighborhood of $S^{n}$ at $\widetilde{\nu}(x)$.

$\widetilde{f}(x)=\sum_{i=1}\frac{\partial\left(\widetilde{\gamma}\circ\widetilde{\nu}^{-1}\right)}{\partial\Theta_{i}}\left(\widetilde{\nu}(x)\right)\frac{\partial}{\partial\Theta_{i}}+\widetilde{\gamma}(x)\widetilde{\nu}(x).$

By this expression, for instance, it is easily seen that when
$\widetilde{\gamma}(x)\equiv c(\neq 0)$ for any $x\in\mathbb{R}^{n+1}$, then
the unique singular solution $Y:U_{c}\to\mathbb{R}$ must be an explicit
solution with the following expression where $U_{c}=\\{X\;|\;\|X\|<|c|\\}$.

$Y(X)=\left\\{\begin{array}[]{rr}-\sqrt{|c|^{2}-\sum_{i=1}^{n}X_{i}^{2}}&(\mbox{
if }c>0)\\\ \sqrt{|c|^{2}-\sum_{i=1}^{n}X_{i}^{2}}&(\mbox{ if
}c<0).\end{array}\right.$

##  5\. Appendix: Alternative proof of Theorem 1 except for the assertion (3)
in the case $n=1$

Let $N$ be a $1$-dimensional manfold and let
$\widetilde{\varphi}:N\to\mathbb{R}^{2}$, $\widetilde{\nu}:N\to S^{1}$ be
mappings. Define the function $\widetilde{\Theta}:N\to\mathbb{R}$ by
$\widetilde{\nu}(t)=\left(\cos\widetilde{\Theta}(t),\sin\widetilde{\Theta}(t)\right)$.
Define also
$\widetilde{\tau}(t):=\left(\sin\widetilde{\Theta}(t),-\cos\widetilde{\Theta}(t)\right)$.
Then, the following trivially holds.

######  Fact 5.1.

For any $h:N\to\mathbb{R}^{2}$,

$h(t)=\left(h(t)\cdot\widetilde{\tau}(t)\right)\widetilde{\tau}(t)+\left(h(t)\cdot\widetilde{\nu}(t)\right)\widetilde{\nu}(t).$

We first show that the creative condition can be naturally obtained from an
envelope by introducing a gauge theoretic approach. Suppose that
$\widetilde{f}:N\to\mathbb{R}^{2}$ is an envelope created by the line family
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$. Then, we
have the following.

$\widetilde{\gamma}^{\prime}(t)=\left(\widetilde{f}(t)\cdot\widetilde{\nu}(t)\right)^{\prime}=\widetilde{f}^{\prime}(t)\cdot\widetilde{\nu}(t)+\widetilde{f}(t)\cdot\widetilde{\nu}^{\prime}(t)=0-\left(\widetilde{f}(t)\cdot\widetilde{\tau}(t)\right)\widetilde{\Theta}^{\prime}(t).$

Let $h:N\to N$ be a bijective mapping. Then, notice that

$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}=\mathcal{H}_{\left(\widetilde{\varphi}\circ
h,\widetilde{\nu}\circ h\right)}$

and

$\left(\widetilde{\gamma}\circ
h\right)^{\prime}(t)=-\left(\widetilde{f}(h(t))\cdot\widetilde{\tau}(h(t))\right)\widetilde{\Theta}^{\prime}(h(t))h^{\prime}(t).$

From these simple observations, we see that it is important to extract a
significant quantity which does not depend on the particular choice of $h$.
Then, we naturally reach the following setting.

$\widetilde{\omega}(t):=-\left(\widetilde{f}(t)\cdot\widetilde{\tau}(t)\right)d\widetilde{\Theta}.$

and we trivially have $d\widetilde{\gamma}=\widetilde{\omega}$. Take an
arbitrary point $t_{0}$ of $N$ and fix it. Let $(V,\Theta)$ be a normal
coordinate neighborhood of $S^{1}$ at $\widetilde{\nu}(t_{0})$ such that
$\Theta\left(\widetilde{\nu}(t_{0})\right)=0$ and
$\widetilde{\Theta}(t)=\left(\Theta\circ\widetilde{\nu}\right)(t)$ for any
$t\in\widetilde{\nu}^{-1}(V)$. In other words,
$\left(\Theta\circ\widetilde{\nu}\right)(t)$
$\left(t\in\widetilde{\nu}^{-1}(V)\right)$ is just the radian (or its
negative) between two unit vectors $\widetilde{\nu}(t_{0})$ and
$\widetilde{\nu}(t)$. By using the function $\Theta:V\to\mathbb{R}$, the
$1$-form $\widetilde{\omega}(t)$ may be written as follows.

$\widetilde{\omega}(t)=-\left(\widetilde{f}(t)\cdot\widetilde{\tau}(t)\right)\widetilde{\nu}^{*}d\Theta,$

where $\widetilde{\nu}^{*}d{\Theta}$ stands for the pullback of the $1$-form
$d\Theta$ by $\widetilde{\nu}$. Hence, we naturally reach the following
$1$-form which is denoted by the same symbol $\widetilde{\omega}$.

$\widetilde{\omega}(t)=-\left(\widetilde{f}(t)\cdot\widetilde{\tau}(t)\right)d\Theta.$

It is easily seen that for any $t\in\widetilde{\nu}^{-1}(V)$, under the
canonical identifications

$T^{*}_{\widetilde{\nu}(t)}S^{1}\cong T_{\widetilde{\nu}(t)}S^{1}\subset
T_{\widetilde{\nu}(t)}\mathbb{R}^{2}\cong\mathbb{R}^{2},$

the $1$-dimensional cotangent vector

$\widetilde{\omega}(t)=-\left(\widetilde{f}(t)\cdot\widetilde{\tau}(t)\right)d\Theta\in
T^{*}_{\widetilde{\nu}(t)}S^{1}$

is identified with the $2$-dimensional vector

$\widetilde{\omega}(t)=\left(\widetilde{f}(t)\cdot\widetilde{\tau}(t)\right)\widetilde{\tau}(t)\in\mathbb{R}^{2}.$

Since $t_{0}$ is an arbitrary point of $N$, we naturally see that the creative
condition is satisfied for
$\mathcal{H}_{\left(\widetilde{\varphi,\widetilde{\nu}}\right)}$ and the
following horizontal-vertical decomposition formula holds for any $t\in N$.

######  Fact 5.2.

$\widetilde{f}(t)=\left(\widetilde{f}(t)\cdot\widetilde{\tau}(t)\right)\widetilde{\tau}(t)+\left(\widetilde{f}(t)\cdot\widetilde{\nu}(t)\right)\widetilde{\nu}(t)=\widetilde{\omega}(t)+\widetilde{\gamma}(t)\widetilde{\nu}(t).$

Conversely, suppose that
$\mathcal{H}_{\left(\widetilde{\varphi},\widetilde{\nu}\right)}$ is creative.
Then, there exists a function $\alpha:N\to\mathbb{R}$ such that
$d\widetilde{\gamma}=\alpha d\widetilde{\Theta}$. Set
$\widetilde{\omega}=\alpha d\widetilde{\Theta}$. Let $t_{0}\in N$ be an
arbitrary point. Then, under the canonical identifications

$T^{*}_{\widetilde{\nu}(t)}S^{1}\cong T_{\widetilde{\nu}(t)}S^{1}\subset
T_{\widetilde{\nu}(t)}\mathbb{R}^{2}\cong\mathbb{R}^{2},$

the $1$-dimensional cotangent vector

$\widetilde{\omega}(t)=\alpha(t)d\Theta\in T^{*}_{\widetilde{\nu}(t)}S^{1}$

is identified with the $2$-dimensional vector

$\widetilde{\omega}(t)=-\alpha(t)\widetilde{\tau}(t)\in\mathbb{R}^{2},$

where $(V,\Theta)$ is a normal coordinate system of $S^{1}$ at
$\widetilde{\nu}(t_{0})$ such that $\Theta(\widetilde{\nu}(t_{0}))=0$ and
$t\in\widetilde{\nu}^{-1}(V)$. Set

$\widetilde{f}(t)=\widetilde{\omega}(t)+\widetilde{\gamma}(t)\widetilde{\nu}(t)=-\alpha(t)\widetilde{\tau}(t)+\widetilde{\gamma}(t)\widetilde{\nu}(t).$

Then, $\widetilde{f}$ clearly satisfies the condition (b) of Definition 1 for
any $t\in\widetilde{\nu}^{-1}(V)$. Moreover we have the following.

######  Lemma 5.1.

For any $t\in\widetilde{\nu}^{-1}(V)$,
$\widetilde{f}^{\prime}(t)\cdot\widetilde{\nu}(t)=0$ holds.

Proof of Lemma 5.1 We have

$\widetilde{\gamma}^{\prime}(t)=\left(\widetilde{f}(t)\cdot\widetilde{\nu}(t)\right)^{\prime}=\widetilde{f}^{\prime}(t)\cdot\widetilde{\nu}(t)-\left(\widetilde{f}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}(t)}\cdot\widetilde{\tau}(t)\right)\widetilde{\Theta}^{\prime}(t)=\widetilde{f}^{\prime}(t)\cdot\widetilde{\nu}(t)+\alpha(t)\widetilde{\Theta}^{\prime}(t).$

Thus, we have the following.

$\displaystyle\widetilde{\omega}(t)=d\widetilde{\gamma}=\widetilde{\gamma}^{\prime}(t)dt$
$\displaystyle=$
$\displaystyle\left(\widetilde{f}^{\prime}(t)\cdot\widetilde{\nu}(t)\right)dt+\alpha(t)\widetilde{\Theta}^{\prime}(t)dt$
$\displaystyle=$
$\displaystyle\left(\widetilde{f}^{\prime}(t)\cdot\widetilde{\nu}(t)\right)dt+\alpha(t)d\widetilde{\Theta}$
$\displaystyle=$
$\displaystyle\left(\widetilde{f}^{\prime}(t)\cdot\widetilde{\nu}(t)\right)dt+\widetilde{\omega}(t).$

It follows
$\left(\widetilde{f}^{\prime}(t)\cdot\widetilde{\nu}(t)\right)dt=0$. Since $t$
is a coordinate function on an open set $\widetilde{\nu}^{-1}(V)$ of $N$, for
any fixed $t\in\widetilde{\nu}^{-1}(V)$, the $1$-dimensional cotangent vector
$dt$ at $t$ is not zero. Therefore, the number
$\left(\widetilde{f}^{\prime}(t)\cdot\widetilde{\nu}(t)\right)$ is always zero
for any $t\in\widetilde{\nu}^{-1}(V)$. Since $t_{0}$ is an arbitrary point of
$N$, Theorem 1 (1) holds. By the above decomposition of $\widetilde{f}(t)$,
Theorem 1 (2) holds as well. ∎

## Acknowledgements

The author is most grateful to two anonymous reviewers. Their
comments/suggestions are hard to replace. The author would like to thank
Richard Montgomery for appropriate comments. His suggestions improved this
paper.

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