# Recognition of affine-equivalent polyhedra by their natural developments

Victor Alexandrov  Sobolev Institute of Mathematics, Koptyug ave., 4,
Novosibirsk, 630090, Russia and Department of Physics, Novosibirsk State
University, Pirogov str., 2, Novosibirsk, 630090, Russia<EMAIL_ADDRESS>
###### Abstract.

The classical Cauchy rigidity theorem for convex polytopes reads that if two
convex polytopes have isometric developments then they are congruent. In other
words, we can decide whether two polyhedra are isometric or not by using their
developments only. In this article, we study a similar problem about whether
it is possible, using only the developments of two convex polyhedra of
Euclidean 3-space, to understand that these polyhedra are (or are not) affine-
equivalent.

Keywords: Euclidean 3-space, convex polyhedron, development of a polyhedron,
Cauchy rigidity theorem, affine-equivalent polyhedra, Cayley–Menger
determinant.

Mathematics subject classification (2010): 52C25, 52B10, 51M25.

##  1\. Introduction

The famous Cauchy rigidity theorem [1], [4, Chapter III], [33], [14, Section
6.4] can be formulated as follows: if the developments of two convex polyhedra
in $\mathbb{R}^{3}$ are isometric, then the polyhedra are congruent. Thus,
this theorem allows us to answer the question “are these convex polyhedra
congruent?”, operating only with the developments of these polyhedra. It seems
natural to us to pose the following problem: “is it possible to understand
that these convex polyhedra are (or are not) affine-equivalent if we are given
only developments of these polyhedra?”

We are not aware of the statement of this problem or any results about it in
the literature. In this article, we take the first steps in solving it.

Note that, apparently, the problem under study is related not only to the
Cauchy rigidity theorem and synthetic geometry, but also to the problems of
computer vision, pattern recognition, and image understanding. We refer the
interested reader, e. g., to the book [27], although in it, as in other books
and articles on computer vision known to us, the problem we are studying is
not studied directly.

Obviously, if the polyhedra $P$ and $P^{\prime}$ in $\mathbb{R}^{3}$ are
affine-equivalent, then their corresponding faces are also affine-equivalent
to each other. The converse, however, is not true. To verify this, consider
two convex triangular bipyramids $P$ and $P^{\prime}$ such that the length of
all edges of $P$ is equal to 1, and the length of one edge of $P^{\prime}$
connecting a vertex of valency 3 with a vertex of valency 4 is not equal to 1,
while the lengths of all other edges of $P^{\prime}$ are equal to 1. The
affine equivalence of the corresponding faces of $P$ and $P^{\prime}$
obviously follows from the fact that any two triangles are affine-equivalent
to each other. However, as is easy to see, $P$ and $P^{\prime}$ themselves are
not affine-equivalent.

Therefore, the main problem studied in this article can be reformulated as
follows: “what additional conditions on the developments of two convex
polyhedra (in addition to the condition of affine equivalence of the
corresponding faces) guarantee the affine equivalence of the polyhedra
themselves?”

The article is organized as follows. In Section 2, we refine the terminology
and formulate results of other authors in a form which is convenient for our
purposes. In Section 3, we prove that if polyhedra are simple (i. e., if
exactly three faces are incident to each vertex), then the affine equivalence
of the faces already implies the affine equivalence of the polyhedra
themselves (i. e., no additional conditions are needed). In Section 4, we
study our problem for suspensions, i. e., for polyhedra combinatorially
equivalent to regular $n$-gonal bipyramids. In Section 5, we study an
auxiliary local problem of affine equivalence of two polyhedra, each of which
is homeomorphic to a disk and contains only three faces. Finally, in Section
6, we describe some algorithm for recognizing affine-equivalent polyhedra from
their natural developments, based on the ideas developed in Sections 3–5,
suitable for polyhedra of any combinatorial structure. For some pairs of
polyhedra, it can certify that they are not affine-equivalent. Note that this
algorithm can be applied not only to convex or closed polyhedra.

##  2\. Refinement of the terminology

Throughout this article, a connected two-dimensional polyhedral surface in
Euclidean 3-space composed of a finite number of convex polygons is called
polyhedron and those convex polygons are called its faces. In this article,
the faces of a polyhedron are not necessarily triangular; a polyhedron itself
can either have a non-empty boundary or be closed, can be either convex (i.
e., coincide with the boundary or a part of the boundary of a convex set) or
non-convex, can have an arbitrary topological structure, and may have self-
intersections. All these possibilities are not excluded in our reasoning until
the corresponding restriction is explicitly formulated. On the other hand, we
always assume that any polyhedron under study is connected, i. e., that we can
go from any face to another by crossing edges, not vertices.

In this article, a finite collection $R$ of convex polygons on the plane is
called natural development of a polyhedron $P$ if $R$ is in one-to-one
correspondence with the collection of faces of $P$ such that each face of $P$
is congruent to the corresponding polygon of $R$ and two elements of $P$ are
incident to each other if and only if the corresponding two elements of $R$
are incident to each other. Here, for brevity, we call a face, edge or vertex
of $P$ an element of $P$, and a polygon, side of the polygon or vertex of the
polygon of $R$ an element of $R$ (sometimes we also call the latter a face,
edge or vertex of $R$, respectively). Natural developments $R$ and
$R^{\prime}$ are called isometric to each other if the set of polygons of $R$
is in one-to-one correspondence with the set of polygons of $R^{\prime}$, the
corresponding polygons are congruent to each other, and two elements of $R$
are incident to each other if and only if the corresponding two elements of
$R^{\prime}$ are incident to each other.

Using the notion of natural development of a polyhedron, we can formulate the
classical Cauchy rigidity theorem for convex polyhedra as follows:

######  Theorem 1.

Let $P$ and $P^{\prime}$ be convex closed polyhedra in Euclidean 3-space. If
the natural developments of $P$ and $P^{\prime}$ are isometric, then $P$ and
$P^{\prime}$ are congruent.

This theorem was first proved by Augustin-Louis Cauchy in 1813 [10]. It is
considered one of the most striking achievements of geometry. It has a
significant impact on the development of synthetic geometry and is well
presented in scientific [4, Chapter III], [12], [13, Section 23.1], [14,
Section 6.4], [40], educational [26, Addition K], [43, Chapter 24], and
popular science [1, Chapter 14], [15], [20, Theorem 24.1], [33, Chapter III,
§14] literature. Of recent articles using or generalizing Theorem 1, we
mention article [16], where the local rigidity of zonohedra is proved in
$\mathbb{R}^{3}$; articles [17], [18], [44], where the rigidity of a
polyhedron is proved in $\mathbb{R}^{3}$ provided that it is homeomorphic to
the sphere, torus or pretzel and all its faces are unit squares; article [9],
in which Theorem 1 carries over to the case of circular polytopes in the
2-sphere $\mathbb{S}^{2}$; and article [36], in which a rather unexpected
application of Theorem 1 is given to find a sufficient condition for a convex
polyhedron $P$ to realize by means of isometries of the ambient space
$\mathbb{R}^{3}$ all “combinatorial symmetries” of $P$, i. e., all maps of the
natural development of $P$ onto itself that preserve edge lengths.

A convex polyhedron can be regarded as a metric space if we put by definition
that the distance between any two points is equal to the infimum of the
lengths of the curves connecting these points and lying entirely on the
polyhedron. Each point of this metric space, other than a vertex of the
polyhedron, has a neighborhood isometric to a disk on the plane
$\mathbb{R}^{2}$. We can say that, after removing a finite number of points,
this space is locally Euclidean. It is convenient to represent such metric
space as a disjoint union of a finite number of convex polygons in
$\mathbb{R}^{2}$, the edges of which are identified so that each edge belongs
to exactly two polygons, and the lengths of any edge segment calculated in
each of these two polygons are the same. We call such a disjoint union of a
finite number of convex polygons in $\mathbb{R}^{2}$ together with the edge
identification rule connected if, starting from any polygon, it is possible to
pass to any other, successively crossing the sides (not vertices) of the
polygons of this disjoint union. Finally, a connected disjoint union of a
finite number of convex polygons in $\mathbb{R}^{2}$, together with the edge
identification rule, we call an abstract development of the polyhedron.

Obviously, an abstract development can be constructed for any polyhedron, not
just for a convex one. It is also obvious that any natural development of a
polyhedron is its abstract development. Fig. 1 shows that the converse is not
true. Indeed, in Fig. 1, a a regular tetrahedron $P$ with vertices
$x_{0},x_{1},x_{2},x_{3}$ is shown. In Fig. 1, b and Fig. 1, c two
developments of $P$ with vertices
$\widetilde{x}_{0},\widetilde{x}_{1},\widetilde{x}_{2},\widetilde{x}_{3}$ are
shown. Moreover, for each $j=0,1,2,3$, the vertex $x_{j}$ of $P$ corresponds
to the vertex $\widetilde{x}_{j}$ of its development. The development shown in
Fig. 1, b is natural. In contrast, the abstract development of $P$ shown in
Fig. 1, c, is not natural. The latter follows, e. g., from the fact that
triangle $\widetilde{x}_{1}\widetilde{x}_{2}\widetilde{x}_{0}$ on the abstract
development shaded in Fig. 1, c is located on two faces $x_{1}x_{2}x_{3}$ and
$x_{2}x_{3}x_{0}$ of $P$ simultaneously. In this case, it is said that the
face $\widetilde{x}_{1}\widetilde{x}_{2}\widetilde{x}_{0}$ and the edge
$\widetilde{x}_{1}\widetilde{x}_{0}$ of the abstract development “break” and
are not a true face and a true edge of $P$, respectively.

Figure 1. (a): a regular tetrahedron $P$. (b): a natural development of $P$.
(c): an abstract development of $P$, which is not natural.

A.D. Alexandrov proved [4, Chapter III] the following generalization of
Theorem 1:

######  Theorem 2.

Let $P$ and $P^{\prime}$ be convex closed polyhedra in Euclidean 3-space, and
let $R$ and $R^{\prime}$ be any of their abstract developments. If $R$ and
$R^{\prime}$ are isometric, then $P$ and $P^{\prime}$ are congruent.

In other words, Theorem 2 shows that in Theorem 1 there is no need to check
that the developments of the polyhedra $P$ and $P^{\prime}$ are natural.

Theorem 2 is interesting not only in itself, but also plays an essential role
in the proof of the following theorem on the existence of a convex polyhedron,
also proved by A.D. Alexandrov [4, Chapter IV]:

######  Theorem 3.

From any abstract development homeomorphic to the sphere and having the sums
of the angles at the vertices $\leqslant 2\pi$, one can glue a closed convex
polytope in $\mathbb{R}^{3}$.

The concept of the “sum of the angles at the vertex $\widetilde{x}$ of an
abstract development” used in Theorem 3 is intuitively obvious. To find this
sum, denoted by $\delta_{\widetilde{x}}$, you need to select flat angles with
the vertex $\widetilde{x}$ in all polygons of the abstract development and
calculate the sum of the values of these angles. The quantity
$2\pi-\delta_{\widetilde{x}}$ is called the curvature of the vertex
$\widetilde{x}$. Theorem 3 actually requires that the curvature of each vertex
be non-negative.

Note also that the example of an abstract development shown in Fig. 1, c shows
that if in the statement of Theorem 3 we replace the words “abstract
development” with the words “natural development”, then we get a wrong
statement. That is, we can say that Theorem 3 is true precisely because the
faces and edges of the development are allowed to break in the process of its
isometric realization in the form of a polyhedron.

For completeness, we mention that at present there are several substantially
different proofs of Theorem 3.

Initially Theorem 3 was published by A.D. Alexandrov in 1941 in a short note
[2], then in 1942 in a detailed article [3], and later was included in his
book [4], first published in 1950. A.D. Alexandrov deduced Theorem 3 from
Brouwer’s domain invariance theorem and Theorem 2. He also showed that, by
approximating any metric of positive curvature by polyhedral metrics of
positive curvature and then passing to the limit, we obtain the positive
solution to the generalized H. Weyl problem, i. e., that we can assert that
any two-dimensional sphere-homeomorphic manifold with an intrinsic metric of
positive curvature admits an isometric embedding into $\mathbb{R}^{3}$ by
means of a closed convex surface (which is not necessarily smooth and may even
degenerate into a doubly covered convex domain in the plane). Various aspects
of the Weyl problem (such as the existence, stability and smoothness of a
surface depending on the smoothness of the metric) are studied in books and
articles, see, e. g., [37], [48], [22], [25] and references given there.

In 1943 at the Steklov Mathematical Institute of the Academy of Sciences of
the USSR, L.A. Lyusternik gave a talk (see [5, p. 372]) in which he showed
that Theorem 3 follows from already known facts (see [32]) about the Weyl
problem for analytic metrics. Namely, he proposed to approximate any
polyhedral metric of positive curvature by analytic metrics of positive
curvature, then realize the latter by means of convex analytic surfaces, and
finally, passing to the limit, obtain the desired convex polyhedron which
realizes the original polyhedral metric. Therefore, the method proposed by
L.A. Lyusternik is usually called analytical; it is presented in detail in
[19, § 12].

Another proof of Theorem 3 was proposed by Yu.A. Volkov in his Ph.D. thesis,
defended at Leningrad State University in 1955 under the direction of A.D.
Alexandrov. The proof proposed by Yu.A. Volkov is usually called variational,
since it is based on the fact that the desired convex polyhedron delivers the
minimum to a function defined on a space of “3-dimensional developments” (the
latter was specially invented by Yu.A. Volkov for this proof). It is presented
in brief in [50] and [51], and in details in [53] and [52]. Among current
publications, we mention the articles [7] and [28], where proofs of Theorem 3
are proposed, which are also based on solving some variational problem and are
close in spirit to the proof proposed by Yu.A. Volkov.

Despite the significance of the concept of abstract development of a
polyhedron demonstrated above using the example of Theorem 3 and the Weyl
problem, in this article we only deal with natural developments of polyhedra.
The fact is that the problem of recognizing whether a given abstract
development is natural or not is in itself very difficult and we do not expect
to contribute here to its solution. To justify this fact, let us quote A.D.
Alexandrov: “To determine the structure of a polyhedron from an [abstract]
development, i. e., to indicate its genuine edges in the development, is a
problem whose general solution seems hopeless” [4, Section 2.3.3]. As far as
we know, even now there is very little progress in solving this problem with
“hopeless” general solution.

The only particular case we know of, in which it is possible to directly
indicate the true edges of a polyhedron on its abstract development, is
described in [45]. It is proved there that if an abstract development $R$
satisfies the conditions of Theorem 3 and there exists a simple edge cycle
$\gamma\subset R$ consisting of shortest paths connecting in series all
vertices of $R$ with positive curvature, and moreover for any two of vertices
with positive curvature that not connected by an edge of $\gamma$ there are
exactly two shortest paths connecting them, then the convex polyhedron $P$
existing by Theorem 3 is degenerate in the sense that it is located in a
plane, has only two faces, $\gamma$ is the common boundary of these faces, and
$\gamma$ consists of all true edges of $P$.

At the same time, researchers in various fields of science proposed a number
of computer algorithms for reconstructing the spatial shape of a convex
surface from its internal metric, see, e. g., [7], [29], [30], [38], [49].
Some of these algorithms work for polyhedra as well. Having computed the
spatial shape of a polyhedron, we, in a sense, can claim that we have found
its true edges as well. However, all of these algorithms are numerical. Among
them, the one proposed in [30] is distinguished by a particularly thorough
theoretical study. It is based on solving a differential equation, derived in
[7], it allows one to compute the spatial shape of a polyhedron with arbitrary
precision from its abstract development, and, in addition, a pseudopolynomial
bound on its running time is proved.

Summarizing what was said above about the relationship between the concepts of
abstract and natural developments, we can say that our desire to deal with
natural developments is by no means accidental.

Let us continue to clarify the terminology used in this article. We say that
two combinatorially equivalent polyhedra $P$ and $P^{\prime}$ in
$\mathbb{R}^{3}$ are affine-equivalent if there is a nondegenerate affine
transformation $\mathbb{R}^{3}\to\mathbb{R}^{3}$, which maps $P$ onto
$P^{\prime}$, and maps the vertices, edges and faces of $P$ onto the
corresponding (due to the combination equivalence of $P$ and $P^{\prime}$)
vertices, edges and faces of $P^{\prime}$. Affine-equivalent polygons are
defined similarly.

We can now clarify the main problem studied in this article:

######  Problem 1.

_What additional conditions on natural developments of two polyhedra (besides
the obvious condition of affine equivalence of the corresponding faces)
guarantee affine equivalence of the polyhedra themselves or, conversely,
guarantee that these polyhedra are not affine-equivalent?_

In Sections 3–6 below we obtain partial solutions to Problem 1 for some
classes of polyhedra in Euclidean 3-space. Wherein, we will use the so-called
Cayley–Menger determinants and their properties. Let us recall them in a
formulation and notation convenient for us.

Let $k\geqslant 2$ and $x_{0},x_{1},\dots,x_{k}$ be arbitrary points in
$\mathbb{R}^{k}$. We denote by $d_{ij}=d(x_{i},x_{j})$ the Euclidean distance
between $x_{i}$ and $x_{j}$, $i,j=0,1,\dots,k$. The determinant

$\textrm{cm}\,(x_{0},x_{1},\dots,x_{k})\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left|\begin{array}[]{ccccc}0&1&1&\dots&1\\\
1&0&d^{2}_{01}&\dots&d^{2}_{0k}\\\ 1&d^{2}_{10}&0&\dots&d^{2}_{1k}\\\
.&.&.&.&.\\\ 1&d^{2}_{k0}&d^{2}_{k1}&\dots&0\end{array}\right|$ (1)

is called the Cayley–Menger determinant of the points
$x_{0},x_{1},\dots,x_{k}$, see, e. g., [6, Formula 9.7.3.2]. However,
sometimes we will say that formula (1) defines the Cayley–Menger determinant
of the simplex $T$ with the vertices $x_{0},x_{1},\dots,x_{k}$, and denote it
by $\textrm{cm}\,(T)$.

We need two properties of the Cayley–Menger determinant:

(a) $k$-dimensional volume $\textrm{vol\,}(x_{0},x_{1},\dots,x_{k})$ of the
simplex with the vertices $x_{0},x_{1},\dots,x_{k}$ is related to the
Cayley–Menger determinant of the points $x_{0},x_{1},\dots,x_{k}$ by the
following formula [6, Lemma 9.7.3.3]:

$[\textrm{vol\,}(x_{0},x_{1},\dots,x_{k})]^{2}=\frac{(-1)^{k+1}}{2^{k}k!}\textrm{cm}\,(x_{0},x_{1},\dots,x_{k});$
(2)

(b) let $d_{ij}$ ($i,j=0,1,\dots,k$) be a collection of $k(k+1)/2$ arbitrary
positive real numbers such that $d_{ij}=d_{ji}$ and $d_{ii}=0$. The following
inequality

$(-1)^{k+1}\left|\begin{array}[]{ccccc}0&1&1&\dots&1\\\
1&0&d^{2}_{01}&\dots&d^{2}_{0k}\\\ 1&d^{2}_{10}&0&\dots&d^{2}_{1k}\\\
.&.&.&.&.\\\ 1&d^{2}_{k0}&d^{2}_{k1}&\dots&0\end{array}\right|>0$ (3)

is a necessary and sufficient condition for the existence of a nondegenerate
(i. e., not contained in any hyperplane) simplex (whose vertices we denote by
$x_{0},x_{1},\dots,x_{k}$) in $\mathbb{R}^{k}$ such that $d_{ij}$ is equal to
the Euclidean distance between $x_{i}$ and $x_{j}$, see [6, Theorem 9.7.3.4,
Remark 9.7.3.5 and Exercise 9.14.23].

Finally, let us agree on some more notation used in Sections 3–6 without
additional explanations. Given a polyhedron $P$, its development $R$ (it does
not matter natural or abstract), and a vertex $x$ of $P$, we denote by
$\widetilde{x}$ the vertex of $R$, corresponding to $x$. If $y$ is another
vertex of $P$, then we denote by $d(x,y)$ the Euclidean distance between the
points $x,y$ in $\mathbb{R}^{3}$, and by $\rho(\widetilde{x},\widetilde{y})$
the distance between $\widetilde{x},\widetilde{y}$ in $R$. If, besides $P$, a
polyhedron $P^{\prime}$ is given, which is combinatorially equivalent to $P$,
and $R^{\prime}$ is a development of $P^{\prime}$, then we denote by
$x^{\prime}$ the vertex of $P^{\prime}$ corresponding, by virtue of
combinatorial equivalence, to the vertex $x$ of $P$, and by
$\widetilde{x}^{\prime}$ the vertex of $R^{\prime}$, corresponding (due to the
construction of development) to the vertex $x^{\prime}$ of $P^{\prime}$. If
$y^{\prime}$ is another vertex of $P^{\prime}$, then by
$d(x^{\prime},y^{\prime})$ we denote the Euclidean distance between
$x^{\prime},y^{\prime}$ in $\mathbb{R}^{3}$, and by
$\rho^{\prime}(\widetilde{x}^{\prime},\widetilde{y}^{\prime})$ the distance
between $\widetilde{x}^{\prime},\widetilde{y}^{\prime}$ in $R^{\prime}$.

##  3\. Simple polyhedra

A closed convex polyhedron is called strictly convex if none of its dihedral
angles is equal to $\pi$. A strictly convex closed polyhedron is called simple
if any of its vertices is incident to exactly three edges (or, which is the
same, exactly three faces). Examples of simple closed polyhedrons are well
known: they are three of five Platonic solids (namely: tetrahedron, cube, and
dodecahedron), six of thirteen Archimedean solids (namely: truncated
tetrahedron, truncated cube, truncated octahedron, truncated cuboctahedron,
truncated icosahedron, and truncated icosidodecahedron, see [54] and [55]), as
well as every strictly convex $n$-gonal prism for any $n\geqslant 3$.

######  Theorem 4.

Let $P$ and $P^{\prime}$ be simple polyhedra in $\mathbb{R}^{3}$
combinatorially equivalent to each other. If the corresponding faces of
natural developments of $P$ and $P^{\prime}$ are affine-equivalent, then $P$
and $P^{\prime}$ themselves are affine-equivalent.

###### Proof.

Given a vertex $x$ of $P$, construct an affine transformation
$A_{x}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ as follows. Since $P$ is assumed to be
simple, there are exactly three of its vertices incident to $x$. Let us denote
them by $x_{1},x_{2},x_{3}$. Since $P$ is assumed to be strictly convex, the
linear span of three vectors $x_{1}-x,x_{2}-x,x_{3}-x$ coincides with
$\mathbb{R}^{3}$, i. e., these vectors are linearly independent. Similarly,
since $P^{\prime}$ is assumed to be strictly convex, vectors
$x^{\prime}_{1}-x^{\prime},x^{\prime}_{2}-x^{\prime},x^{\prime}_{3}-x^{\prime}$
are also linearly independent. Therefore, there is a unique affine
transformation $A_{x}:\mathbb{R}^{3}\to\mathbb{R}^{3}$ such that
$A_{x}(x_{j})=x^{\prime}_{j}$ for each $j=1,2,3$, and $A_{x}(x)=x^{\prime}$.

Let $F$ denote an arbitrary face of $P$ incident to $x$. For definiteness, let
$F$ is incident to $x,x_{1},x_{2}$. Let $F^{\prime}$ denote the face of
$P^{\prime}$ corresponding to $F$ by combinatorial equivalence. Then
$F^{\prime}$ is incident to $x^{\prime},x^{\prime}_{1},x^{\prime}_{2}$. Let us
verify that

$A_{x}(F)=F^{\prime}.$ (4)

Indeed, on the natural development of $P$ there is a polygon corresponding to
$F$. Let us denote it by $\widetilde{F}$. Due to the construction of natural
development, $F$ and $\widetilde{F}$ are congruent. Similarly, on the natural
development of $P^{\prime}$, there is a polygon $\widetilde{F}^{\prime}$
corresponding to $F^{\prime}$. It is congruent to $F^{\prime}$. By the
hypothesis of Theorem 4, $\widetilde{F}$ and $\widetilde{F}^{\prime}$ are
affine-equivalent. Hence the faces $F$ and $F^{\prime}$ are affine-equivalent.
This means that there is an affine mapping
$B_{F}:\textrm{aff}(F)\to\textrm{aff}(F^{\prime})$ that maps the affine hull
$\textrm{aff}(F)$ of $F$ into the affine hull $\textrm{aff}(F^{\prime})$ of
$F^{\prime}$ and is such that $B_{F}(F)=F^{\prime}$, $B_{F}(x)=x^{\prime}$,
$B_{F}(x_{1})=x^{\prime}_{1}$, and $B_{F}(x_{2})=x^{\prime}_{2}$. Since
$A_{x}(x)=x^{\prime}=B_{F}(x)$, $A_{x}(x_{1})=x^{\prime}_{1}=B_{F}(x_{1})$ and
$A_{x}(x_{2})=x^{\prime}_{2}=B_{F}(x_{2})$, the restriction of $A_{x}$ to
$\textrm{aff}(F)$ coincides with $B_{F}$. Thus,
$A_{x}(F)=B_{F}(F)=F^{\prime}$, i. e., (4) is proved.

Now let us make sure that if the vertices $x$ and $y$ of the polyhedron $P$
are connected by an edge, then

$A_{x}=A_{y}.$ (5)

For this, as before, we denote by $x_{1},x_{2},x_{3}$ the three vertices of
$P$ incident to $x$. For definiteness, we assume that $y=x_{3}$. Similarly,
let us denote by $y_{1},y_{2},y_{3}$ the three vertices of $P$ incident to
$y$. For definiteness, we assume that $x=y_{3}$ and the vertices
$x_{1},x=y_{3},y=x_{3},y_{2}$ are incident to a face of $P$, while the
vertices $x_{2},x=y_{3},y=x_{3},y_{1}$ are incident to another face of $P$.
Denote by $F_{1}$ the face of $P$ incident to $x_{1},x=y_{3},y=x_{3},y_{2}$,
and denote by $\textrm{aff}(F_{1})$ the affine hull of $F_{1}$. Denote by
$F^{\prime}_{1}$ the face of the polyhedron $P^{\prime}$ corresponding to
$F_{1}$ by virtue of combinatorial equivalence of $P$ and $P^{\prime}$. Denote
by $\textrm{aff}(F^{\prime}_{1})$ the affine hull of $F^{\prime}_{1}$. Arguing
as in the previous paragraph, we are convinced that there is an affine mapping
$B_{F_{1}}:\textrm{aff}(F_{1})\to\textrm{aff}(F^{\prime}_{1})$, which
coincides with both $A_{x}$ and $A_{y}$ on $F_{1}$. This implies that
$A_{x}(x_{1})=x^{\prime}_{1}=A_{y}(x_{1})$. In a similar way, using the face
$F_{2}$, which is incident to $x_{2},x=y_{3},y=x_{3}$, we make sure that
$A_{x}(x_{2})=x^{\prime}_{2}=A_{y}(x_{2})$. Thus, we made sure that the affine
transformations $A_{x}$ and $A_{y}$ coincide with each other at
$x,x_{1},x_{2},x_{3}$. Taking into account that the vectors
$x_{1}-x,x_{2}-x,x_{3}-x$ are linearly independent, we obtain (5).

Finally, let us make sure that the transformation $A_{x}$ is the same for all
vertices $x$ of $P$. Indeed, since $P$ is connected, any two of its vertices
can be connected by an edge path. If the vertex $y$ follows the vertex $x$ in
this edge path, then, as proved in the previous paragraph, $A_{y}=A_{x}$.
Hence, $A_{x}$ is the same at the initial and final vertices of the edge path,
i. e., is independent of $x$. Let us denote this transformation by $A$. Then
$A$ is an affine transformation that maps the polyhedron $P$ to the polyhedron
$P^{\prime}$ in such a way that the elements of $P$ are mapped to the
corresponding (due to combinatorial equivalence) elements of $P^{\prime}$.
Thus, $P$ and $P^{\prime}$ are affine-equivalent to each other. ∎

Theorem 4 shows that, for simple polyhedra, the affine equivalence of faces
already implies the affine equivalence of polyhedra. From the above proof of
Theorem 4, it is clear that, with a suitable refinement of the terminology,
similar statements can be formulated and proved both for some nonconvex
polyhedra and for projectively equivalent polyhedra. Of all the possible
generalizations of Theorem 4, we explicitly state only one that is applicable
not only to simple polytopes. Here it is:

######  Theorem 5.

Let strictly convex closed polyhedra $P$ and $P^{\prime}$ in $\mathbb{R}^{3}$
have the same combinatorial structure, and their corresponding faces are
affine-equivalent. And let there is an edge path on $\gamma$ $P$ such that

_(i)_ each vertex of $\gamma$ is incident to exactly three edges of $P$, and

_(ii)_ for every face of $P$, there exists a vertex of $\gamma$ incident to
this face.

Then $P$ and $P^{\prime}$ are affine-equivalent to each other.

The proof of Theorem 5 can be carried out similarly to the above proof of
Theorem 4. We leave it to the reader as an easy exercise. For every
$n\geqslant 4$, any $n$-gonal trapezohedron may serve as an example of a
polyhedron that satisfies the conditions of Theorem 5, but is not simple.
Recall that an $n$-gonal trapezohedron is the polyhedron dual to an $n$-gonal
antiprism, see [24], [42]. Fig. 2, a shows a 4-gonal antiprism; Fig. 2, b
shows a 4-gonal trapezohedron; and Fig. 2, c shows a 5-sided non-closed edge
path $\gamma$ on a 4-gonal trapezohedron satisfying the conditions of Theorem
5.

Figure 2. (a): A 4-gonal antiprism. (b): A 4-gonal trapezohedron. (c): A solid
broken line represents a 5-sided non-closed edge path $\gamma$ on a 4-gonal
trapezohedron, satisfying the conditions of Theorem 5.

Theorem 4 allows us to prove the following statement:

######  Theorem 6.

For $n\geqslant 4$, let $M_{n}$ be the set of all strictly convex closed
polyhedra with $n$ faces in Euclidean 3-space endowed with the Hausdorff
metric. There is an open dense subset $\Omega_{n}\subset M_{n}$ such that, for
any polyhedra $P,P^{\prime}\in\Omega_{n}$, the following conditions are
equivalent:

_(i)_ $P$ and $P^{\prime}$ are combinatorially equivalent, and every two
corresponding faces are affine-equivalent;

_(ii)_ $P$ and $P^{\prime}$ are affine-equivalent.

###### Proof.

Denote by $\Omega_{n}$ the set of all simple strictly convex closed polyhedra
in $\mathbb{R}^{3}$ with $n$ faces. $\Omega_{n}$ is an open everywhere dense
subset of $M_{n}$. To make sure of this, it suffice to slightly move the
planes of the faces parallel to themselves so that only three faces are
incident to each vertex. For polyhedra $P,P^{\prime}\in\Omega_{n}$, the
equivalence of conditions (i) and (ii) of Theorem 6 follows from Theorem 4. ∎

In a sense, Theorem 6 is similar to the statement that, in the space of all
closed polyhedra of a given combinatorial structure in $\mathbb{R}^{3}$, there
is an open dense set, each point of which corresponds to a rigid polyhedron.
The last statement was first proved by H. Gluck in [23]. Recently, Gluck’s
proof has been adapted to prove the rigidity of almost all triangulated circle
polyhedra, see [8].

##  4\. Suspensions

As stated in Section 1, our study of Problem 1 is mainly motivated by the
Cauchy rigidity theorem. The history of this theorem began in 1794, when in
the first edition of the famous textbook by A.M. Legendre [31, Note XII, pp.
321–334], it was formulated as a conjecture, while its proofs were given only
for some classes of polyhedra, including octahedra. In [41, Section 1], I.Kh.
Sabitov writes that in editions 2nd to 9th a part of Note XII related to the
rigidity theorem was removed by Legendre, and restored only in the 10th and
subsequent editions after Cauchy in 1813 proved the rigidity theorem in the
general case [10], making significant use of Legendre’s ideas. Therefore, we
consider it appropriate to investigate the different classes of polyhedra for
which we can make progress in solving Problem 1. In doing so, we would first
of all like to find arguments suitable for answering Problem 1 for octahedra.

In this Section we study suspensions, i. e., polyhedra combinatorially
equivalent to a regular $n$-gonal bipyramid for $n\geqslant 3$. Note that
suspensions are not necessarily convex or self-intersection free. From the
edges of a suspension $P$ one can form a closed cycle $L$, which passes
without repetition through all vertices of $P$, except for two. Denote the
vertices of $L$ by $x_{j}$, $j=1,\dots,n$, labeled in the cyclic order
prescribed by $L$. The cycle $L$ is called the equator of the suspension $P$.
Two vertices of $P$ that do not lie on the equator are denoted by $x_{0}$ and
$x_{n+1}$ and are called the southern and north poles of the suspension,
respectively. From what has been said, it is clear that each vertex of the
equator is incident to both the south and north poles, while the south and
north poles are not incident to each other. For $n=4$, a suspension is
combinatorially equivalent to an octahedron. A 5-gonal suspension is shown in
Fig. 3, a. Suspensions play an important role in the theory of flexible
polyhedra; for example, they were studied in articles [11], [39], [34], [47],
[35], [46], [21].

Figure 3. (a): An $n$-gonal suspension for $n=5$. (b): The tetrahedron $T_{j}$
with the vertices $x_{0}$, $x_{j}$, $x_{j+1}$, and $x_{n+1}$ of the suspension
is grayed out; the thin line segment $x_{0}x_{n+1}$ is an edge of $T_{j}$, but
is not an edge of the suspension.

Let $P$ be a suspension with $n$ vertices on the equator. Let $T_{j}$ denote a
tetrahedron with vertices $x_{0}$, $x_{j}$, $x_{j+1}$, and $x_{n+1}$, if
$j=1,2,\dots,n-1$ and with vertices $x_{0}$, $x_{n}$, $x_{1}$, and $x_{n+1}$,
if $j=n$, see Fig. 3, b. Note that although the segment $x_{0}x_{n+1}$ is an
edge of $T_{j}$, it is nevertheless not an edge of $P$. In other words,
$x_{0}x_{n+1}$ is the only edge of $T_{j}$ that we cannot find directly from
the natural development of $P$. Define polynomials $q_{j}(t)$,
$j=1,2,\dots,n$, by the formulas

$q_{j}(t)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\begin{cases}\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{0},\widetilde{x}_{j})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{j+1})]^{2}&t^{2}\\\
1&[\rho(\widetilde{x}_{j},\widetilde{x}_{0})]^{2}&0&[\rho(\widetilde{x}_{j},\widetilde{x}_{j+1})]^{2}&[\rho(\widetilde{x}_{j},\widetilde{x}_{n+1})]^{2}\\\
1&[\rho(\widetilde{x}_{j+1},\widetilde{x}_{0})]^{2}&[\rho(\widetilde{x}_{j+1},\widetilde{x}_{j})]^{2}&0&[\rho(\widetilde{x}_{j+1},\widetilde{x}_{n+1})]^{2}\\\
1&t^{2}&[\rho(\widetilde{x}_{n+1},\widetilde{x}_{j})]^{2}&[\rho(\widetilde{x}_{n+1},\widetilde{x}_{j+1})]^{2}&0\end{array}\right|,\
\textrm{if}\ j=1,\dots,n-1;\\\ \\\ \left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{0},\widetilde{x}_{n})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{1})]^{2}&t^{2}\\\
1&[\rho(\widetilde{x}_{n},\widetilde{x}_{0})]^{2}&0&[\rho(\widetilde{x}_{n},\widetilde{x}_{1})]^{2}&[\rho(\widetilde{x}_{n},\widetilde{x}_{n+1})]^{2}\\\
1&[\rho(\widetilde{x}_{1},\widetilde{x}_{0})]^{2}&[\rho(\widetilde{x}_{1},\widetilde{x}_{n})]^{2}&0&[\rho(\widetilde{x}_{1},\widetilde{x}_{n+1})]^{2}\\\
1&t^{2}&[\rho(\widetilde{x}_{n+1},\widetilde{x}_{n})]^{2}&[\rho(\widetilde{x}_{n+1},\widetilde{x}_{1})]^{2}&0\end{array}\right|,\
\textrm{if}\ j=n.\end{cases}$ (6)

Since $\rho$ denotes the distance in the natural development of $P$, for any
pair $\\{x,y\\}$ of vertices of $T_{j}$ other than the pair
$\\{x_{0},x_{n+1}\\}$, the equality $\rho(\widetilde{x},\widetilde{y})=d(x,y)$
holds true. Therefore, putting by definition $t_{*}=d(x_{0},x_{n+1})$ and
comparing formulas (6) and (1), we get

$q_{j}(t_{*})=\begin{cases}\textrm{cm}\,(x_{0},x_{j},x_{j+1},x_{n+1}),\
\textrm{if}\ j=1,2,\dots,n-1;\\\ \textrm{cm}\,(x_{0},x_{n},x_{1},x_{n+1}),\
\textrm{if}\ j=n.\end{cases}$ (7)

Now comparing formulas (7) and (2), for any $j=1,2,\dots,n$ we get the
following expression for the volume of $T_{j}$:

$[\textrm{vol\,}(T_{j})]^{2}=\frac{1}{2^{3}3!}q_{j}(t_{*}).$ (8)

Let $P^{\prime}$ be another suspension with $n$ vertices on the equator, for
which a combinatorial equivalence with the suspension $P$ is fixed, i. e., an
incidence-preserving mapping is given which maps the set of elements (i. e.,
vertices, edges and faces) of $P^{\prime}$ onto the set of elements of $P$. As
usual, we denote by $x^{\prime}_{j}$ the vertex of $P^{\prime}$, which
corresponds to the vertex $x_{j}$ of $P$ due to this combinatorial
equivalence. Let $T^{\prime}_{j}$ denote a tetrahedron with vertices
$x^{\prime}_{0}$, $x^{\prime}_{j}$, $x^{\prime}_{j+1}$, $x^{\prime}_{n+1}$, if
$j=1,2,\dots,n-1$, and with vertices $x^{\prime}_{0}$, $x^{\prime}_{n}$,
$x^{\prime}_{1}$, $x^{\prime}_{n+1}$, if $j=n$. Note that segment
$x^{\prime}_{0}x^{\prime}_{n+1}$ is the only edge of $T^{\prime}_{j}$, which
we cannot find directly from the natural development of $P^{\prime}$. We
define polynomials $q^{\prime}_{j}(t^{\prime})$, $j=1,2,\dots,n$, by the
formulas

$q^{\prime}_{j}(t^{\prime})\stackrel{{\scriptstyle\textrm{def}}}{{=}}\begin{cases}\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho^{\prime}(\widetilde{x}^{\prime}_{0},\widetilde{x}^{\prime}_{j})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{0},\widetilde{x}^{\prime}_{j+1})]^{2}&[t^{\prime}]^{2}\\\
1&[\rho^{\prime}(\widetilde{x}^{\prime}_{j},\widetilde{x}^{\prime}_{0})]^{2}&0&[\rho^{\prime}(\widetilde{x}^{\prime}_{j},\widetilde{x}^{\prime}_{j+1})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{j},\widetilde{x}^{\prime}_{n+1})]^{2}\\\
1&[\rho^{\prime}(\widetilde{x}^{\prime}_{j+1},\widetilde{x}^{\prime}_{0})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{j+1},\widetilde{x}^{\prime}_{j})]^{2}&0&[\rho^{\prime}(\widetilde{x}^{\prime}_{j+1},\widetilde{x}^{\prime}_{n+1})]^{2}\\\
1&[t^{\prime}]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{n+1},\widetilde{x}^{\prime}_{j})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{n+1},\widetilde{x}^{\prime}_{j+1})]^{2}&0\end{array}\right|,\
\textrm{if}\ j=1,\dots,n-1;\\\ \\\ \left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho^{\prime}(\widetilde{x}^{\prime}_{0},\widetilde{x}^{\prime}_{n})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{0},\widetilde{x}^{\prime}_{1})]^{2}&[t^{\prime}]^{2}\\\
1&[\rho^{\prime}(\widetilde{x}^{\prime}_{n},\widetilde{x}^{\prime}_{0})]^{2}&0&[\rho^{\prime}(\widetilde{x}^{\prime}_{n},\widetilde{x}^{\prime}_{1})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{n},\widetilde{x}^{\prime}_{n+1})]^{2}\\\
1&[\rho^{\prime}(\widetilde{x}^{\prime}_{1},\widetilde{x}^{\prime}_{0})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{1},\widetilde{x}^{\prime}_{n})]^{2}&0&[\rho^{\prime}(\widetilde{x}^{\prime}_{1},\widetilde{x}^{\prime}_{n+1})]^{2}\\\
1&[t^{\prime}]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{n+1},\widetilde{x}^{\prime}_{n})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{n+1},\widetilde{x}^{\prime}_{1})]^{2}&0\end{array}\right|,\
\textrm{if}\ j=n.\end{cases}$ (9)

Since $\rho^{\prime}$ denotes the distance in the natural development of
$P^{\prime}$, then for any pair $\\{x^{\prime},y^{\prime}\\}$ of vertices of
$T^{\prime}_{j}$ other than the pair $\\{x^{\prime}_{0},x^{\prime}_{n+1}\\}$,
the equality
$\rho^{\prime}(\widetilde{x}^{\prime},\widetilde{y}^{\prime})=d(x^{\prime},y^{\prime})$
is true. Therefore, putting by definition
$t^{\prime}_{*}=d(x^{\prime}_{0},x^{\prime}_{n+1})$ and comparing formulas (9)
and (1), we get

$q^{\prime}_{j}(t^{\prime}_{*})=\begin{cases}\textrm{cm}\,(x^{\prime}_{0},x^{\prime}_{j},x^{\prime}_{j+1},x^{\prime}_{n+1}),\
\textrm{if}\ j=1,2,\dots,n-1;\\\
\textrm{cm}\,(x^{\prime}_{0},x^{\prime}_{n},x^{\prime}_{1},x^{\prime}_{n+1}),\
\textrm{if}\ j=n.\end{cases}$ (10)

Now comparing formulas (10) and (2), for any $j=1,2,\dots,n$ we get the
following expression for the volume of $T^{\prime}_{j}$:

$[\textrm{vol\,}(T^{\prime}_{j})]^{2}=\frac{1}{2^{3}3!}q^{\prime}_{j}(t^{\prime}_{*}).$
(11)

Note that the polynomials $q_{j}$ and $q^{\prime}_{j}$, defined by formulas
(6) and (9), are uniquely determined by the natural developments of
suspensions $P$ and $P^{\prime}$. Using these polynomials we can formulate the
following partial answer to Problem 1 for suspensions:

######  Theorem 7.

Let $P$ and $P^{\prime}$ be suspensions in $\mathbb{R}^{3}$. Suppose $P$ and
$P^{\prime}$ have the same number of vertices $n\geqslant 3$ on the equator
and, for each $j=1,2,\dots,n$ polynomials $q_{j}$ and $q^{\prime}_{j}$ are
defined by formulas (6) and (9) via natural developments of $P$ and
$P^{\prime}$. Suppose also that the system of $n\geqslant 3$ algebraic
equations

$q^{\prime}_{j}(t^{\prime})=\delta q_{j}(t),\qquad j=1,2,\dots,n,$ (12)

with respect to three variables $\delta$, $t$, and $t^{\prime}$ has no
positive real solution (i. e. such a solution that $\delta>0$, $t>0$, and
$t^{\prime}>0$). Then $P$ and $P^{\prime}$ are not affine-equivalent.

###### Proof.

We argue by contradiction. Suppose that $P$ and $P^{\prime}$ are affine-
equivalent and denote by $A:\mathbb{R}^{3}\to\mathbb{R}^{3}$ an affine
transformation such that $P^{\prime}=A(P)$. By definition, put
$\delta_{*}=(\det A)^{2}$, $t_{*}=d(x_{0},x_{n+1})$, and
$t^{\prime}_{*}=d(x^{\prime}_{0},x^{\prime}_{n+1})$. According to formulas
(11) and (8), for every $j=1,2,\dots,n$ we have

$q^{\prime}_{j}(t^{\prime}_{*})=2^{3}\cdot
3!\cdot[\textrm{vol\,}(T^{\prime}_{j})]^{2}=2^{3}\cdot 3!\cdot(\det
A)^{2}[\textrm{vol\,}(T_{j})]^{2}=\delta_{*}q_{j}(t_{*}).$

Therefore, the triplet of positive real numbers $\delta_{*}$, $t_{*}$,
$t^{\prime}_{*}$ is a solution to the system (12). Thus, we have come to a
contradiction with the conditions of Theorem 7, which completes the proof. ∎

##  5\. Local realizability problem for a natural development

In [39], an algorithm is proposed to determine if a given suspension is
flexible. This gave us the idea to look for an algorithmic solution to Problem
1. In Section 6 we describe an algorithm that receives two combinatorially
equivalent natural developments at the input, and at the output either
guarantees that no polyhedra with such natural developments are affine-
equivalent, or answers that it cannot give such a guarantee. Section 5 is
devoted to the necessary preparatory work.

Recall that, in Section 2, we denoted by $\widetilde{x}$ the vertex of the
natural development of a polyhedron corresponding to the vertex $x$ of that
polyhedron. In Sections 5 and 6, we use the same notation even if only the
development is given and the polyhedron has yet to be found (or it has to be
proved that the required polyhedron does not exist).

A set $Z$ of three faces of an abstract development $R$ is called a patch of
$R$ incident to a vertex $\widetilde{x}_{0}$ of $R$ if each of the faces of
$Z$ is incident to $\widetilde{x}_{0}$ and $Z$ is a connected, i. e., that we
can go from any face of $Z$ to any other face of $Z$ by crossing edges of $Z$,
not vertices. The proofs of the following properties of patches are
straightforward and are left to the reader: every patch is a development;
every patch is homeomorphic to a disk; $\widetilde{x}_{0}$ can be either an
inner or a boundary point of a patch incident to $\widetilde{x}_{0}$.

Our approach to an algorithmic solution of Problem 1 is based on reducing it
to the following problem:

######  Problem 2.

_Let abstract developments $R$ and $R^{\prime}$ be combinatorially equivalent
to each other, $\widetilde{x}_{0}$ be a vertex of $R$, $Z$ be a patch of $R$
incident to $\widetilde{x}_{0}$, and $Z^{\prime}$ be the patch of $R^{\prime}$
corresponding to $Z$ due to the combinatorial equivalence of $R$ and
$R^{\prime}$. Suppose also that any face of $Z$ is affine-equivalent to the
corresponding by combinatorial equivalence face of $Z^{\prime}$. The problem
is whether there exist polyhedra $P$ and $P^{\prime}$ in $\mathbb{R}^{3}$ such
that (i) $P$ and $P^{\prime}$ are homeomorphic to the disc; (ii) $Z$ is the
natural development of $P$; (iii) $Z^{\prime}$ is the natural development of
$P^{\prime}$; (iv) $P$ and $P^{\prime}$ are affine-equivalent to each other._

For the convenience of speech, property (ii) is expressed by the words “a
natural development $Z$ is isometrically realized as a polyhedron $P$.” Note
that Problem 2 deals with the possibility of realizing only patches $Z$ and
$Z^{\prime}$ as affine-equivalent polyhedra, regardless of whether it is
possible to realize “ambient” abstract developments $R$ and $R^{\prime}$.
Therefore, Problem 2 can be called local.

Let $\widetilde{x}_{1},\widetilde{x}_{2},\widetilde{x}_{3}$ be sequentially
numbered vertices of a patch $Z$ incident to a vertex $\widetilde{x}_{0}$ of
an abstract development $R$, and let $n$ be the total number of vertices of
$R$ incident to $\widetilde{x}_{0}$. Let us study Problem 2 for each of the
following cases separately: $n=3$ (see Fig. 4, a); $n=4$ (see Fig. 4, b);
$n\geqslant 5$ (see Fig. 4, c).

Figure 4. Schematic representation of the star (grayed out) of a vertex
$\widetilde{x}_{0}$ of valency $n$ in an abstract development $R$ (part of
which is shown by solid lines) and a patch $Z$ incident to $\widetilde{x}_{0}$
(depicted in bold lines). The dotted lines show the segments that are not
edges of either $R$ or $Z$; their lengths are taken as “free parameters”
$r,s,t$. (a): The case $n=3$. (b): The case $n=4$. (c): The case $n\geqslant
5$.

Case $\bf{n=3}$. Suppose that there is a polyhedron $P$ in $\mathbb{R}^{3}$
for which $Z$ is a natural development. Then $P$ contains the star of a vertex
$x_{0}$ of a nondegenerate tetrahedron $T$ with vertices $x_{0}$, $x_{1}$,
$x_{2}$, $x_{3}$. As usual, let us denote by
$\rho(\widetilde{x},\widetilde{y})$ the distance in $R$ between points
$\widetilde{x}$ and $\widetilde{y}$. Since we have assumed that $Z$ is a
natural development of $P$, for any pair $\\{x,y\\}$ of vertices of $T$, the
equality $\rho(\widetilde{x},\widetilde{y})=d(x,y)$ holds true, where $d(x,y)$
denotes the Euclidean distance between the points $x,y\in\mathbb{R}^{3}$.
(There is no need to assume that $x$ and $y$ are connected by an edge of $P$;
it is sufficient that they are incident to one and the same face of $P$.)
Therefore, the Cayley–Menger determinant (1) of $T$ can be expressed in terms
of the distances on $R$:

$\textrm{cm}\,(x_{0},x_{1},x_{2},x_{3})=\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[d(x_{0},x_{1})]^{2}&[d(x_{0},x_{2})]^{2}&[d(x_{0},x_{3})]^{2}\\\
1&[d(x_{1},x_{0})]^{2}&0&[d(x_{1},x_{2})]^{2}&[d(x_{1},x_{3})]^{2}\\\
1&[d(x_{2},x_{0})]^{2}&[d(x_{2},x_{1})]^{2}&0&[d(x_{2},x_{3})]^{2}\\\
1&[d(x_{3},x_{0})]^{2}&[d(x_{3},x_{1})]^{2}&[d(x_{3},x_{2})]^{2}&0\end{array}\right|=$
$=\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{0},\widetilde{x}_{1})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{2})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{3})]^{2}\\\
1&[\rho(\widetilde{x}_{1},\widetilde{x}_{0})]^{2}&0&[\rho(\widetilde{x}_{1},\widetilde{x}_{2})]^{2}&[\rho(\widetilde{x}_{1},\widetilde{x}_{3})]^{2}\\\
1&[\rho(\widetilde{x}_{2},\widetilde{x}_{0})]^{2}&[\rho(\widetilde{x}_{2},\widetilde{x}_{1})]^{2}&0&[\rho(\widetilde{x}_{2},\widetilde{x}_{3})]^{2}\\\
1&[\rho(\widetilde{x}_{3},\widetilde{x}_{0})]^{2}&[\rho(\widetilde{x}_{3},\widetilde{x}_{1})]^{2}&[\rho(\widetilde{x}_{3},\widetilde{x}_{2})]^{2}&0\end{array}\right|\stackrel{{\scriptstyle\textrm{def}}}{{=}}{}^{3}q.$
(13)

Note that $T$ is non-degenerate because we have assumed that $Z$ is the
natural development of $P$ and because no two adjacent faces of any polyhedron
lie in the same plane. Therefore, according to (3), the number ${}^{3}q$
defined by formula (13) is positive. Moreover, in the case $n=3$, the
inequality ${}^{3}q>0$ is necessary and sufficient for $Z$ to be isometrically
realizable as the natural development of some polyhedron $P$, i. e., for
property (ii) from the statement of Problem 2 to be fulfilled.

Similarly, putting by definition

$\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho^{\prime}(\widetilde{x}^{\prime}_{0},\widetilde{x}^{\prime}_{1})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{0},\widetilde{x}^{\prime}_{2})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{0},\widetilde{x}^{\prime}_{3})]^{2}\\\
1&[\rho^{\prime}(\widetilde{x}^{\prime}_{1},\widetilde{x}^{\prime}_{0})]^{2}&0&[\rho^{\prime}(\widetilde{x}^{\prime}_{1},\widetilde{x}^{\prime}_{2})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{1},\widetilde{x}^{\prime}_{3})]^{2}\\\
1&[\rho^{\prime}(\widetilde{x}^{\prime}_{2},\widetilde{x}^{\prime}_{0})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{2},\widetilde{x}^{\prime}_{1})]^{2}&0&[\rho^{\prime}(\widetilde{x}^{\prime}_{2},\widetilde{x}^{\prime}_{3})]^{2}\\\
1&[\rho^{\prime}(\widetilde{x}^{\prime}_{3},\widetilde{x}^{\prime}_{0})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{3},\widetilde{x}^{\prime}_{1})]^{2}&[\rho^{\prime}(\widetilde{x}^{\prime}_{3},\widetilde{x}^{\prime}_{2})]^{2}&0\end{array}\right|\stackrel{{\scriptstyle\textrm{def}}}{{=}}{}^{3}q^{\prime},$
(14)

we make sure that the inequality ${}^{3}q^{\prime}>0$ is necessary and
sufficient for $Z^{\prime}$ to be isometrically realizable as the natural
development of some polyhedron $P^{\prime}$, i. e., for property (iii) from
the statement of Problem 2 to be fulfilled.

It was already noted above that tetrahedron $T$ with vertices $x_{0}$,
$x_{1}$, $x_{2}$, $x_{3}$ is non-degenerate, i. e., does not lie in any plane
in $\mathbb{R}^{3}$. Similarly, we can assert that tetrahedron $T^{\prime}$
with vertices $x^{\prime}_{0}$, $x^{\prime}_{1}$, $x^{\prime}_{2}$,
$x^{\prime}_{3}$ is also non-degenerate. Hence, there exists uniquely
determined affine transformation $A:\mathbb{R}^{3}\to\mathbb{R}^{3}$ such that
$A(x_{j})=x^{\prime}_{j}$ for all $j=0,\dots,3$. The fact that $A$ maps $P$
onto $P^{\prime}$ was already established by us in the proof of Theorem 4.
Moreover, we can find the absolute value of the Jacobian of $A$ by the formula

$|\det
A|=\textrm{vol\,}(T^{\prime}_{j})/\textrm{vol\,}(T_{j})=\sqrt{{}^{3}q^{\prime}/{}^{3}q}.$
(15)

This formula shows that, for $n=3$, the assignment of patches $Z$ and
$Z^{\prime}$ uniquely determines the value of $|\det A|$.

Thus, we have proved the following lemma:

######  Lemma 1.

In the case $n=3$, the answer to Problem  _2_ is positive if and only if both
numbers ${}^{3}q$ and ${}^{3}q^{\prime}$ given by formulas (13) and (14) are
positive.

###### Proof.

The “if” statement was proven above. To prove the opposite statement, one need
to do the same reasoning in reverse order. ∎

Case $\bf{n=4}$. The corresponding patch $Z$ is shown schematically in Fig. 4,
b. Suppose that there is a polyhedron $P$ in $\mathbb{R}^{3}$ for which $Z$ is
a natural development. Then $P$ contains the star of a vertex $x_{0}$ of an
octahedron $O$ for which $x_{0}$ is the south pole, and the points $x_{1}$,
$x_{2}$, $x_{3}$, $x_{4}$ lie on the equator. Since we assumed that $Z$ is a
natural development of $P$, then for any two-point set
$\\{x,y\\}\subset\\{x_{0},x_{1},x_{2},x_{3},x_{4}\\}$, except for the pairs
$\\{x_{1},x_{3}\\}$ and $\\{x_{2},x_{4}\\}$, the equality
$\rho(\widetilde{x},\widetilde{y})=d(x,y)$ holds true. (There is no need to
assume that $x$ and $y$ are connected by an edge of $P$; it is sufficient that
they are incident to one and the same face of $P$.) The segments $x_{1}x_{3}$
and $x_{2}x_{4}$ are usually called small diagonals of $P$. Their lengths in
$\mathbb{R}^{3}$ will be denoted by $t$ and $s$, respectively, and will be
called free parameters, since they are not expressed in terms of distances in
$R$.

For each $j=0,1,\dots,4$, we denote by $T_{j}$ the tetrahedron which is the
convex hull of the set
$\\{x_{0},x_{1},x_{2},x_{3},x_{4}\\}\setminus\\{x_{j}\\}$. In the
Cayley–Menger determinant (1) for $T_{j}$, replace the Euclidean distances
$d_{ij}=d(x_{i},x_{j})$ by the following expressions:

$d(x_{i},x_{j})=\begin{cases}\rho(\widetilde{x}_{i},\widetilde{x}_{j}),&\textrm{if\
}(i,j)\notin\\{(1,3),(2,4),(3,1),(4,2)\\};\\\ t,&\textrm{if\
}(i,j)\in\\{(1,3),(3,1)\\};\\\ s,&\textrm{if\
}(i,j)\in\\{(2,4),(4,2)\\}.\end{cases}$

The resulting polynomial depends on either $t$, or $s$, or both free
parameters $t$ and $s$ at once. As a result, we get:

${}^{4}q_{0}(t,s)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{1},\widetilde{x}_{2})]^{2}&t^{2}&[\rho(\widetilde{x}_{1},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{2},\widetilde{x}_{1})]^{2}&0&[\rho(\widetilde{x}_{2},\widetilde{x}_{3})]^{2}&s^{2}\\\
1&t^{2}&[\rho(\widetilde{x}_{3},\widetilde{x}_{2})]^{2}&0&[\rho(\widetilde{x}_{3},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{4},\widetilde{x}_{1})]^{2}&s^{2}&[\rho(\widetilde{x}_{4},\widetilde{x}_{3})]^{2}&0\end{array}\right|,$
(16)
${}^{4}q_{1}(s)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{0},\widetilde{x}_{2})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{3})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{2},\widetilde{x}_{0})]^{2}&0&[\rho(\widetilde{x}_{2},\widetilde{x}_{3})]^{2}&s^{2}\\\
1&[\rho(\widetilde{x}_{3},\widetilde{x}_{0})]^{2}&[\rho(\widetilde{x}_{3},\widetilde{x}_{2})]^{2}&0&[\rho(\widetilde{x}_{3},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{4},\widetilde{x}_{0})]^{2}&s^{2}&[\rho(\widetilde{x}_{4},\widetilde{x}_{3})]^{2}&0\end{array}\right|,$
(17)
${}^{4}q_{2}(t)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{0},\widetilde{x}_{1})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{3})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{1},\widetilde{x}_{0})]^{2}&0&t^{2}&[\rho(\widetilde{x}_{1},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{3},\widetilde{x}_{0})]^{2}&t^{2}&0&[\rho(\widetilde{x}_{3},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{4},\widetilde{x}_{0})]^{2}&[\rho(\widetilde{x}_{4},\widetilde{x}_{1})]^{2}&[\rho(\widetilde{x}_{4},\widetilde{x}_{3})]^{2}&0\end{array}\right|,$
(18)
${}^{4}q_{3}(s)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{0},\widetilde{x}_{1})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{2})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{1},\widetilde{x}_{0})]^{2}&0&[\rho(\widetilde{x}_{1},\widetilde{x}_{3})]^{2}&[\rho(\widetilde{x}_{1},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{2},\widetilde{x}_{0})]^{2}&[\rho(\widetilde{x}_{2},\widetilde{x}_{1})]^{2}&0&s^{2}\\\
1&[\rho(\widetilde{x}_{4},\widetilde{x}_{0})]^{2}&[\rho(\widetilde{x}_{4},\widetilde{x}_{1})]^{2}&s^{2}&0\end{array}\right|,$
(19)
${}^{4}q_{4}(t)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{0},\widetilde{x}_{1})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{2})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{3})]^{2}\\\
1&[\rho(\widetilde{x}_{1},\widetilde{x}_{0})]^{2}&0&[\rho(\widetilde{x}_{1},\widetilde{x}_{2})]^{2}&t^{2}\\\
1&[\rho(\widetilde{x}_{2},\widetilde{x}_{0})]^{2}&[\rho(\widetilde{x}_{2},\widetilde{x}_{1})]^{2}&0&[\rho(\widetilde{x}_{2},\widetilde{x}_{3})]^{2}\\\
1&[\rho(\widetilde{x}_{3},\widetilde{x}_{0})]^{2}&t^{2}&[\rho(\widetilde{x}_{3},\widetilde{x}_{3})]^{2}&0\end{array}\right|.$
(20)

For each $j=0,1,\dots,4$, denote by $T^{\prime}_{j}$ the tetrahedron which is
the convex hull of the set
$\\{x^{\prime}_{0},x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4}\\}\setminus\\{x^{\prime}_{j}\\}$.
By definition, put $d^{\prime}_{ij}=d(x^{\prime}_{i},x^{\prime}_{j})$. In the
Cayley–Menger determinant (1) for $T^{\prime}_{j}$, replace the Euclidean
distances $d(x^{\prime}_{i},x^{\prime}_{j})$ by the following expressions:

$d(x^{\prime}_{i},x^{\prime}_{j})=\begin{cases}\rho^{\prime}(\widetilde{x}^{\prime}_{i},\widetilde{x}^{\prime}_{j}),&\textrm{if\
}(i,j)\notin\\{(1,3),(2,4),(3,1),(4,2)\\};\\\ t^{\prime},&\textrm{if\
}(i,j)\in\\{(1,3),(3,1)\\};\\\ s^{\prime},&\textrm{if\
}(i,j)\in\\{(2,4),(4,2)\\}.\end{cases}$

Here $\rho^{\prime}$ denotes the distance in $R^{\prime}$, while $t^{\prime}$,
$s^{\prime}$ are new free parameters. As a result, we get polynomials
${}^{4}q^{\prime}_{0}(t^{\prime},s^{\prime})$,
${}^{4}q^{\prime}_{1}(s^{\prime})$, ${}^{4}q^{\prime}_{2}(t^{\prime})$,
${}^{4}q^{\prime}_{3}(s^{\prime})$, and ${}^{4}q^{\prime}_{4}(t^{\prime})$,
given by formulas (16)–(20), in which the letters $\rho$, $\widetilde{x}_{0}$,
$\widetilde{x}_{1}$, $\widetilde{x}_{2}$, $\widetilde{x}_{3}$,
$\widetilde{x}_{4}$, $t$, and $s$ are replaced by the same letters endowed
with a prime.

If the answer to Problem 2 is positive, then there exists uniquely determined
affine transformation $A:\mathbb{R}^{3}\to\mathbb{R}^{3}$ such that
$A(x_{j})=x^{\prime}_{j}$ for all $j=0,\dots,4$. This follows from our
assumptions: patch $Z$ is the natural development of polyhedron $P$; the faces
of a polyhedron are non-degenerate; and no adjacent faces lie in the same
plane. Therefore, we have

$\begin{cases}|\det
A|^{2}&=\dfrac{(\textrm{vol\,}T^{\prime}_{0})^{2}}{(\textrm{vol\,}T_{0})^{2}}=\dfrac{{}^{4}q^{\prime}_{0}(d^{\prime}_{13},d^{\prime}_{24})}{{}^{4}q_{0}(d_{13},d_{24})},\\\
|\det
A|^{2}&=\dfrac{(\textrm{vol\,}T^{\prime}_{1})^{2}}{(\textrm{vol\,}T_{1})^{2}}=\dfrac{{}^{4}q^{\prime}_{1}(d^{\prime}_{24})}{{}^{4}q_{1}(d_{24})},\\\
|\det
A|^{2}&=\dfrac{(\textrm{vol\,}T^{\prime}_{2})^{2}}{(\textrm{vol\,}T_{2})^{2}}=\dfrac{{}^{4}q^{\prime}_{2}(d^{\prime}_{13})}{{}^{4}q_{2}(d_{13})},\\\
|\det
A|^{2}&=\dfrac{(\textrm{vol\,}T^{\prime}_{3})^{2}}{(\textrm{vol\,}T_{3})^{2}}=\dfrac{{}^{4}q^{\prime}_{3}(d^{\prime}_{24})}{{}^{4}q_{3}(d_{24})},\\\
|\det
A|^{2}&=\dfrac{(\textrm{vol\,}T^{\prime}_{4})^{2}}{(\textrm{vol\,}T_{4})^{2}}=\dfrac{{}^{4}q^{\prime}_{4}(d^{\prime}_{13})}{{}^{4}q_{4}(d_{13})}.\end{cases}$
(21)

The above results can be summarized as follows:

######  Lemma 2.

Suppose $n=4$ and the answer to Problem  _2_ is positive. Then the system of
algebraic equations

$\begin{cases}{}^{4}q^{\prime}_{0}(t^{\prime},s^{\prime})&={}^{4}q_{0}(t,s)\alpha,\\\
{}^{4}q^{\prime}_{1}(s^{\prime})&={}^{4}q_{1}(s)\alpha,\\\
{}^{4}q^{\prime}_{2}(t^{\prime})&={}^{4}q_{2}(t)\alpha,\\\
{}^{4}q^{\prime}_{3}(s^{\prime})&={}^{4}q_{3}(s)\alpha,\\\
{}^{4}q^{\prime}_{4}(t^{\prime})&={}^{4}q_{4}(t)\alpha\end{cases}$ (22)

has a solution $\alpha,t,s,t^{\prime},s^{\prime}$ such that $\alpha>0$, $t>0$,
$s>0$, $t^{\prime}>0$, $s^{\prime}>0$.

###### Proof.

Put $\alpha=|\det A|^{2}$, $t=d_{13}$, $s=d_{24}$,
$t^{\prime}=d^{\prime}_{13}$, $s^{\prime}=d^{\prime}_{24}$ and use formulas
(21). ∎

Note that system (22) contains 5 equations for 5 unknowns
$\alpha,t,s,t^{\prime},s^{\prime}$. Therefore, we can be expect that in
“generic situation” it has only a finite number of solutions.

Case $\bf{n\geqslant 5}$. The corresponding patch $Z$ is shown schematically
in Fig. 4, c. Suppose that there is a polyhedron $P$ in $\mathbb{R}^{3}$ for
which $Z$ is a natural development. Then the star of the vertex $x_{0}$ of $P$
contains vertices $x_{j}$, $j=1,\dots,5$, and each segment $x_{0}x_{j}$ is an
edge of $P$. Since we assumed that $Z$ is a natural development of $P$, then
for any two-point set $\\{x,y\\}\subset\\{x_{0},x_{1},x_{2},x_{3},x_{4}\\}$,
except for the pairs $\\{x_{1},x_{3}\\}$, $\\{x_{2},x_{4}\\}$, and
$\\{x_{1},x_{4}\\}$ the equality $\rho(\widetilde{x},\widetilde{y})=d(x,y)$
holds true. (There is no need to assume that $x$ and $y$ are connected by an
edge of $P$; it is sufficient that they are incident to one and the same face
of $P$.) As usual, the segments $x_{1}x_{3}$, $x_{2}x_{4}$, and $x_{1}x_{4}$
are called small diagonals of $P$. Their lengths in $\mathbb{R}^{3}$ will be
denoted by $t$, $s$, and $r$ respectively, and will be called free parameters,
since they are not expressed in terms of distances in $R$.

For each $j=0,1,\dots,4$, we denote by $T_{j}$ the tetrahedron which is the
convex hull of the set
$\\{x_{0},x_{1},x_{2},x_{3},x_{4}\\}\setminus\\{x_{j}\\}$. In the
Cayley–Menger determinant (1) for $T_{j}$, replace the Euclidean distances
$d_{ij}=d(x_{i},x_{j})$ by the following expressions:

$d(x_{i},x_{j})=\begin{cases}\rho(\widetilde{x}_{i},\widetilde{x}_{j}),&\textrm{if\
}(i,j)\notin\\{(1,3),(2,4),(3,1),(4,2),(1,4),(4,1)\\};\\\ t,&\textrm{if\
}(i,j)\in\\{(1,3),(3,1)\\};\\\ s,&\textrm{if\ }(i,j)\in\\{(2,4),(4,2)\\};\\\
r,&\textrm{if\ }(i,j)\in\\{(1,4),(4,1)\\}.\end{cases}$

The resulting polynomial depends on one or few free parameters $t$, $s$, and
$r$. As a result, we get:

${}^{5}q_{0}(t,s,r)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{1},\widetilde{x}_{2})]^{2}&t^{2}&r^{2}\\\
1&[\rho(\widetilde{x}_{2},\widetilde{x}_{1})]^{2}&0&[\rho(\widetilde{x}_{2},\widetilde{x}_{3})]^{2}&s^{2}\\\
1&t^{2}&[\rho(\widetilde{x}_{3},\widetilde{x}_{2})]^{2}&0&[\rho(\widetilde{x}_{3},\widetilde{x}_{4})]^{2}\\\
1&r^{2}&s^{2}&[\rho(\widetilde{x}_{4},\widetilde{x}_{3})]^{2}&0\end{array}\right|,$
(23)
${}^{5}q_{1}(s)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{0},\widetilde{x}_{2})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{3})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{2},\widetilde{x}_{0})]^{2}&0&[\rho(\widetilde{x}_{2},\widetilde{x}_{3})]^{2}&s^{2}\\\
1&[\rho(\widetilde{x}_{3},\widetilde{x}_{0})]^{2}&[\rho(\widetilde{x}_{3},\widetilde{x}_{2})]^{2}&0&[\rho(\widetilde{x}_{3},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{4},\widetilde{x}_{0})]^{2}&s^{2}&[\rho(\widetilde{x}_{4},\widetilde{x}_{3})]^{2}&0\end{array}\right|,$
(24)
${}^{5}q_{2}(t,r)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{0},\widetilde{x}_{1})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{3})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{1},\widetilde{x}_{0})]^{2}&0&t^{2}&r^{2}\\\
1&[\rho(\widetilde{x}_{3},\widetilde{x}_{0})]^{2}&t^{2}&0&[\rho(\widetilde{x}_{3},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{4},\widetilde{x}_{0})]^{2}&r^{2}&[\rho(\widetilde{x}_{4},\widetilde{x}_{3})]^{2}&0\end{array}\right|,$
(25)
${}^{5}q_{3}(s,r)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{0},\widetilde{x}_{1})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{2})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{1},\widetilde{x}_{0})]^{2}&0&[\rho(\widetilde{x}_{1},\widetilde{x}_{3})]^{2}&r^{2}\\\
1&[\rho(\widetilde{x}_{2},\widetilde{x}_{0})]^{2}&[\rho(\widetilde{x}_{2},\widetilde{x}_{1})]^{2}&0&s^{2}\\\
1&[\rho(\widetilde{x}_{4},\widetilde{x}_{0})]^{2}&r^{2}&s^{2}&0\end{array}\right|,$
(26)
${}^{5}q_{4}(t)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left|\begin{array}[]{ccccc}0&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{0},\widetilde{x}_{1})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{2})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{3})]^{2}\\\
1&[\rho(\widetilde{x}_{1},\widetilde{x}_{0})]^{2}&0&[\rho(\widetilde{x}_{1},\widetilde{x}_{2})]^{2}&t^{2}\\\
1&[\rho(\widetilde{x}_{2},\widetilde{x}_{0})]^{2}&[\rho(\widetilde{x}_{2},\widetilde{x}_{1})]^{2}&0&[\rho(\widetilde{x}_{2},\widetilde{x}_{3})]^{2}\\\
1&[\rho(\widetilde{x}_{3},\widetilde{x}_{0})]^{2}&t^{2}&[\rho(\widetilde{x}_{3},\widetilde{x}_{3})]^{2}&0\end{array}\right|.$
(27)

Five points $x_{0},x_{1},x_{2},x_{3},x_{4}$ lying in $\mathbb{R}^{3}$ can be
considered as the vertices of a degenerate 4-dimensional simplex $T$. Its
four-dimensional volume is equal to zero. Therefore, replacing the edge
lengths in the Cayley–Menger determinant (1) for $T$ by either the
corresponding distances in the patch $Z$ or the free parameters $t$, $s$, $r$,
we obtain

${}^{5}q(t,s,r)\stackrel{{\scriptstyle\textrm{def}}}{{=}}\left|\begin{array}[]{cccccc}0&1&1&1&1&1\\\
1&0&[\rho(\widetilde{x}_{0},\widetilde{x}_{1})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{2})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{3})]^{2}&[\rho(\widetilde{x}_{0},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{1},\widetilde{x}_{0})]^{2}&0&[\rho(\widetilde{x}_{1},\widetilde{x}_{2})]^{2}&t^{2}&r^{2}\\\
1&[\rho(\widetilde{x}_{2},\widetilde{x}_{0})]^{2}&[\rho(\widetilde{x}_{2},\widetilde{x}_{1})]^{2}&0&[\rho(\widetilde{x}_{2},\widetilde{x}_{3})]^{2}&s^{2}\\\
1&[\rho(\widetilde{x}_{3},\widetilde{x}_{0})]^{2}&t^{2}&[\rho(\widetilde{x}_{3},\widetilde{x}_{2})]^{2}&0&[\rho(\widetilde{x}_{3},\widetilde{x}_{4})]^{2}\\\
1&[\rho(\widetilde{x}_{4},\widetilde{x}_{0})]^{2}&r^{2}&s^{2}&[\rho(\widetilde{x}_{4},\widetilde{x}_{3})]^{2}&0\end{array}\right|=0.$
(28)

For each $j=0,1,\dots,4$, we denote by $T^{\prime}_{j}$ the tetrahedron which
is the convex hull of the set
$\\{x^{\prime}_{0},x^{\prime}_{1},x^{\prime}_{2},x^{\prime}_{3},x^{\prime}_{4}\\}\setminus\\{x^{\prime}_{j}\\}$.
In the Cayley–Menger determinant (1) for $T^{\prime}_{j}$ and $T^{\prime}$,
replace the Euclidean distances
$d^{\prime}_{ij}=d(x^{\prime}_{i},x^{\prime}_{j})$ by the following
expressions:

$d(x^{\prime}_{i},x^{\prime}_{j})=\begin{cases}\rho^{\prime}(\widetilde{x}^{\prime}_{i},\widetilde{x}^{\prime}_{j}),&\textrm{if\
}(i,j)\notin\\{(1,3),(2,4),(3,1),(4,2)\\};\\\ t^{\prime},&\textrm{if\
}(i,j)\in\\{(1,3),(3,1)\\};\\\ s^{\prime},&\textrm{if\
}(i,j)\in\\{(2,4),(4,2)\\};\\\ r^{\prime},&\textrm{if\
}(i,j)\in\\{(1,4),(4,1)\\}.\end{cases}$

Here $\rho^{\prime}$ denotes the distance in $Z^{\prime}$, while $t^{\prime}$,
$s^{\prime}$, $r^{\prime}$ are new free parameters. As a result, we get
polynomials ${}^{5}q^{\prime}_{0}(t^{\prime},s^{\prime},r^{\prime})$,
${}^{5}q^{\prime}_{1}(s^{\prime})$,
${}^{5}q^{\prime}_{2}(t^{\prime},r^{\prime})$,
${}^{5}q^{\prime}_{3}(s^{\prime},r^{\prime})$,
${}^{5}q^{\prime}_{4}(t^{\prime})$, and
${}^{5}q^{\prime}(t^{\prime},s^{\prime},r^{\prime})$, given by formulas
(23)–(28), in which the letters $\rho$, $\widetilde{x}_{0}$,
$\widetilde{x}_{1}$, $\widetilde{x}_{2}$, $\widetilde{x}_{3}$,
$\widetilde{x}_{4}$, $t$, $s$, and $r$ are replaced by the same letters
endowed with a prime.

If the answer to Problem 2 for a patch $Z$ is positive, then there exists
uniquely determined affine transformation $A:\mathbb{R}^{3}\to\mathbb{R}^{3}$
such that $A(x_{j})=x^{\prime}_{j}$ for all $j=0,\dots,4$. This follows from
our assumptions: patch $Z$ is the natural development of polyhedron $P$; the
faces of a polyhedron are non-degenerate; and no adjacent faces lie in the
same plane. Therefore, we have

$\begin{cases}|\det
A|^{2}&=\dfrac{(\textrm{vol\,}T^{\prime}_{0})^{2}}{(\textrm{vol\,}T_{0})^{2}}=\dfrac{{}^{4}q^{\prime}_{0}(d^{\prime}_{13},d^{\prime}_{24})}{{}^{4}q_{0}(d_{13},d_{24})},\\\
|\det
A|^{2}&=\dfrac{(\textrm{vol\,}T^{\prime}_{1})^{2}}{(\textrm{vol\,}T_{1})^{2}}=\dfrac{{}^{4}q^{\prime}_{1}(d^{\prime}_{24})}{{}^{4}q_{1}(d_{24})},\\\
|\det
A|^{2}&=\dfrac{(\textrm{vol\,}T^{\prime}_{2})^{2}}{(\textrm{vol\,}T_{2})^{2}}=\dfrac{{}^{4}q^{\prime}_{2}(d^{\prime}_{13})}{{}^{4}q_{2}(d_{13})},\\\
|\det
A|^{2}&=\dfrac{(\textrm{vol\,}T^{\prime}_{3})^{2}}{(\textrm{vol\,}T_{3})^{2}}=\dfrac{{}^{4}q^{\prime}_{3}(d^{\prime}_{24})}{{}^{4}q_{3}(d_{24})},\\\
|\det
A|^{2}&=\dfrac{(\textrm{vol\,}T^{\prime}_{4})^{2}}{(\textrm{vol\,}T_{4})^{2}}=\dfrac{{}^{4}q^{\prime}_{4}(d^{\prime}_{13})}{{}^{4}q_{4}(d_{13})},\\\
{}^{5}q(t,s,r)&=0,\\\
{}^{5}q^{\prime}(t^{\prime},s^{\prime},r^{\prime})&=0.\end{cases}$ (29)

The above results obtained for the case $n\geqslant 5$ can be summarized as
follows:

######  Lemma 3.

Suppose $n\geqslant 5$ and the answer to Problem  _2_ is positive. Then the
system of algebraic equations

$\begin{cases}{}^{4}q^{\prime}_{0}(t^{\prime},s^{\prime})&={}^{4}q_{0}(t,s)\alpha,\\\
{}^{4}q^{\prime}_{1}(s^{\prime})&={}^{4}q_{1}(s)\alpha,\\\
{}^{4}q^{\prime}_{2}(t^{\prime})&={}^{4}q_{2}(t)\alpha,\\\
{}^{4}q^{\prime}_{3}(s^{\prime})&={}^{4}q_{3}(s)\alpha,\\\
{}^{4}q^{\prime}_{4}(t^{\prime})&={}^{4}q_{4}(t)\alpha,\\\
{}^{5}q(t,s,r)&=0,\\\
{}^{5}q^{\prime}(t^{\prime},s^{\prime},r^{\prime})&=0.\end{cases}$ (30)

has a solution $\alpha,t,s,r,t^{\prime},s^{\prime},r^{\prime}$ such that
$\alpha>0$, $t>0$, $s>0$, $r>0$, $t^{\prime}>0$, $s^{\prime}>0$,
$r^{\prime}>0$.

###### Proof.

Put $\alpha=|\det A|^{2}$, $t=d_{13}$, $s=d_{24}$, $r=d_{14}$,
$t^{\prime}=d^{\prime}_{13}$, $s^{\prime}=d^{\prime}_{24}$,
$r^{\prime}=d^{\prime}_{14}$ and use formulas (29). ∎

Note that system (30) contains 7 equations for 7 unknowns
$\alpha,t,s,r,t^{\prime},s^{\prime},r^{\prime}$. Therefore, we can be expect
that in “generic situation” it has only a finite number of solutions.

The results on Problem 2 obtained in Section 5 can be summarized as follows:

######  Theorem 8.

Let $P$ and $P^{\prime}$ be affine-equivalent polyhedra in $\mathbb{R}^{3}$,
let $A:\mathbb{R}^{3}\to\mathbb{R}^{3}$ be an affine mapping such that
$P^{\prime}=A(P)$, and let $R$, $R^{\prime}$ are natural developments of $P$,
$P^{\prime}$. Then, there exists a positive number $\alpha_{*}$ such that

_(i)_ For any patch $Z$ incident to an arbitrary vertex $\widetilde{x}_{0}$ of
valency 3 of $R$, the equality $\alpha_{*}=\sqrt{{}^{3}q^{\prime}/{}^{3}q}$
holds true, where the expressions ${}^{3}q$ and ${}^{3}q^{\prime}$ are given
by the formulas (13) and (14) respectively.

_(ii)_ For any patch $Z$ incident to an arbitrary vertex $\widetilde{x}_{0}$
of valency 4 of $R$, the system of algebraic equations (22) has a solution
$\alpha,t,s,t^{\prime},s^{\prime}$ such that $\alpha=\alpha_{*}$ and $t>0$,
$s>0$, $t^{\prime}>0$, $s^{\prime}>0$.

_(iii)_ For any patch $Z$ incident to an arbitrary vertex $\widetilde{x}_{0}$
of valency 5 of $R$, the system of algebraic equations (30) has a solution
$\alpha,t,s,r,t^{\prime},s^{\prime},r^{\prime}$ such that $\alpha=\alpha_{*}$
and $t>0$, $s>0$, $r>0$, $t^{\prime}>0$, $s^{\prime}>0$, $r^{\prime}>0$.

###### Proof.

By definition, put $\alpha_{*}=|\det A|$. Statement (i) follows from equality
(15). Moreover, according to Lemma 1, both numbers ${}^{3}q$ and
${}^{3}q^{\prime}$ are strictly positive. Statements (ii) and (iii) follow
from Lemmas 2 and 3, respectively. ∎

Note that in Theorem 8 polyhedra $P$ and $P^{\prime}$ are not assumed to be
convex or closed.

##  6\. Algorithmic solution to the problem of recognition of affine-
equivalent polyhedra by their natural developments

Let $P$ and $P^{\prime}$ be polyhedra in $\mathbb{R}^{3}$ that are not assumed
to be convex, closed, or affine-equivalent. And let $R$ and $R^{\prime}$ be
natural developments of $P$ and $P^{\prime}$. Let us organize an item-by-item
examination of all vertices $\widetilde{x}_{0}$ of $R$ and all patches $Z$ of
$R$ incident to $\widetilde{x}_{0}$.

$\bullet$ If $\widetilde{x}_{0}$ is a vertex of valency 3, then we put
$\alpha(\widetilde{x}_{0},Z)=\sqrt{{}^{3}q^{\prime}/{}^{3}q}$, where
expressions ${}^{3}q$ and ${}^{3}q^{\prime}$ are defined by formulas (13) and
(14), respectively. (Note that numerical values of both expressions ${}^{3}q$
and ${}^{3}q^{\prime}$ are strictly positive. According to Lemma 1, this
follows from our assumption that $P$ and $P^{\prime}$ do actually exist, and
$R$ and $R^{\prime}$ are their natural developments.)

$\bullet$ If $\widetilde{x}_{0}$ is a vertex of valency 4, then we distinguish
two cases: if the system of algebraic equations (22) has a solution
$\alpha,t,s,t^{\prime},s^{\prime}$ such that $\alpha>0$, $t>0$, $s>0$,
$t^{\prime}>0$, $s^{\prime}>0$, then we put
$\alpha(\widetilde{x}_{0},Z)=\alpha$; and if (22) has no solution with the
indicated properties, then we put
$\\{\alpha(\widetilde{x}_{0},Z)\\}=\varnothing$.

$\bullet$ If $\widetilde{x}_{0}$ is a vertex of valency $n\geqslant 5$, then
again we distinguish two cases: If the system of algebraic equations (30) has
a solution $\alpha,t,s,r,t^{\prime},s^{\prime},r^{\prime}$ such that
$\alpha>0$, $t>0$, $s>0$, $r>0$, $t^{\prime}>0$, $s^{\prime}>0$,
$r^{\prime}>0$, then we put $\alpha(\widetilde{x}_{0},Z)=\alpha$; and if (30)
has no solution with the indicated properties, then we put
$\\{\alpha(\widetilde{x}_{0},Z)\\}=\varnothing$.

$\bullet$ Finally, we find the set $\bigcap\\{\alpha(\widetilde{x}_{0},Z)\\}$,
i. e., the intersection of the sets $\\{\alpha(\widetilde{x}_{0},Z)\\}$, when
$\widetilde{x}_{0}$ runs through the set of all vertices of $R$ and $Z$ runs
through the set of all patches of $R$ incident to $\widetilde{x}_{0}$.

Based on Theorem 8, we assert that if
$\bigcap\\{\alpha(\widetilde{x}_{0},Z)\\}=\varnothing$, then $P$ and
$P^{\prime}$ are not affine-equivalent to each other.

##  7\. Concluding remarks

The results we have obtained on recognizing affine-equivalent polyhedra by
their developments are certainly far from the final solution. To obtain it,
apparently, fundamentally new ideas are required.

In 2016, at the suggestion of the author, several statements similar to
Theorems 4 and 5, were proved by A.V. Sherstobitov, who at that time was
planning to start postgraduate studies in geometry. However, this plan
remained unfulfilled, and the statements were not published.

The study was carried out within the framework of the state contract of the
Sobolev Institute of Mathematics (project no. 0314-2019-0006).

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