Eric Andres
email: andres@sic.sp2mi.univ-poitiers.fr

The supercover of an m-flat is a discrete analytical object

Keywords: discrete geometry, computer graphics, supercover, m-flat, discrete analytical object, arbitrary dimension

The aim of this paper is to show that the supercover of an m-flat (i.e. a Euclidean affine subspace of dimension m) in Euclidean n-space is a discrete analytical object. The supercover of a Euclidean object F is a discrete object consisting of all the voxels that intersect F . A discrete analytical object is a set of discrete points that is defined by a finite set of inequalities. A method to determine the inequalities defining the supercover of an m-flat is provided.

Introduction

Classically, in computer graphics, a discrete primitive (such as a 2D line segment [START_REF] Bresenham | Algorithm for computer control of a digital plotter[END_REF], a 3D line segment [START_REF] Fujimoto | ARTS: Accelerated Ray-Tracing System[END_REF][START_REF] Amanatides | A Fast Voxel Traversal Algorithm for Ray Tracing[END_REF][START_REF] Cohen-Or | 3D Line Voxelization and Connectivity Control[END_REF], a 3D polygon [START_REF] Kaufman | An algorithm for 3D scan conversion of polygons[END_REF], etc.) is defined as the result of a local approximation algorithm applied to the corresponding Euclidean primitive. Such a generation algorithm typically defines the discrete primitive without any global analytical characterisation in the discrete domain. This approach has some drawbacks [START_REF] Bresenham | Ambiguities in Incremental Line Rastering[END_REF]. While the algorithms are designed to be efficient, it is difficult to study the properties of the generated discrete object. There is also no simple way to extend such primitives to higher dimensions and different discrete primitives are rarely coherent with each other. For instance, Kaufman's discrete 3D polygon [START_REF] Kaufman | An algorithm for 3D scan conversion of polygons[END_REF] is not a piece of a discrete 3D plane as defined by other authors [START_REF] Kim | Three-dimensional digital planes[END_REF][START_REF] Forchhammer | Digital plane and grid point segments[END_REF][START_REF] Reveillès | Géométrie Discrète, calculs en nombres entiers et algorithmique[END_REF][START_REF] Andres | Discrete Analytical Hyperplanes[END_REF] and its edges do not correspond to most notions of discrete 3D line segments [START_REF] Kim | Three-dimensional digital line segments[END_REF][START_REF] Fujimoto | ARTS: Accelerated Ray-Tracing System[END_REF][START_REF] Amanatides | A Fast Voxel Traversal Algorithm for Ray Tracing[END_REF][START_REF] Figueiredo | A Contribution to 3D Digital Lines[END_REF][START_REF] Cohen-Or | 3D Line Voxelization and Connectivity Control[END_REF][START_REF] Andres | Tunnel-free supercover 3D polygons and polyhedra[END_REF]. A more recent approach is the so-called discrete analytical geometry approach introduced by J-P. Reveillès [START_REF] Reveillès | Géométrie Discrète, calculs en nombres entiers et algorithmique[END_REF]. A discrete primitive is defined, in this approach, as a discrete analytical object, which is a discrete object defined by a finite set of discrete inequalities. A discrete inequality is an inequality, with coefficients in R, of which we retain only the integer coordinate solutions (in Z n ). The properties of such a discrete primitive are relatively easy to study because of its global analytical description. Extension to higher dimensions is often straightforward. The discrete analytical geometry approach has been used to define and study different classes of discrete primitives such as discrete 2D lines [START_REF] Reveillès | Géométrie Discrète, calculs en nombres entiers et algorithmique[END_REF], naive [START_REF] Debled-Renesson | A new approach to digital planes[END_REF][START_REF] Andres | Discrete Analytical Hyperplanes[END_REF] and standard 3D planes [START_REF] Andres | Le plan discret[END_REF][START_REF] Françon | Discrete Combinatorial Surfaces[END_REF][START_REF] Françon | Sur la topologie d'un plan arithmétique [About the topology of the arithmetical plane[END_REF][START_REF] Andres | Discrete Analytical Hyperplanes[END_REF], discrete 3D lines [START_REF] Figueiredo | A Contribution to 3D Digital Lines[END_REF][START_REF] Andres | Tunnel-free supercover 3D polygons and polyhedra[END_REF], discrete hyperplanes [START_REF] Stojmenovic | Digitization schemes and the recognition of digital straight lines, hyperplanes and flats in arbitrary dimensions[END_REF][START_REF] Veerlaert | On the flatness of digital hyperplanes[END_REF][START_REF] Andres | Discrete Analytical Hyperplanes[END_REF], discrete circles and hyperspheres [START_REF] Andres | The Discrete Analytical Hyperspheres[END_REF], discrete standard hyperplanes, m-flats, m-simplices [START_REF] Andres | Defining Discrete Objects for Polygonalization: the Standard Model[END_REF][START_REF] Andres | Discrete Analytical Objects: the Standard Model[END_REF] etc.

In this paper we examine, within the framework of the discrete analytical geometry approach, an old discretisation scheme [START_REF] Chassery | Discrete Convexity: Definition, Parametrization and Compatibility with Continuous Convexity[END_REF], called the supercover, which was studied by Cohen and Kaufman for volume graphics purposes [START_REF] Cohen-Or | Fundamentals of Surface Voxelization[END_REF]. The supercover of a Euclidean object F is the set of discrete points consisting of all the voxels that intersect F . The voxel associated with a discrete point, in Z n , is a unit hypercube, in R n , centered on the point with corresponding integer coordinates. In fact, supercovers have interesting properties under set operations [START_REF] Cohen-Or | Fundamentals of Surface Voxelization[END_REF]. It has been shown that the supercovers of a 2D line and a 3D plane [START_REF] Andres | Supercover 3D Polygon[END_REF], a 3D line, a 3D triangle and a 3D tetrahedron [START_REF] Andres | Tunnel-free supercover 3D polygons and polyhedra[END_REF][START_REF] Andres | Supercover of Straight Lines, Planes and Triangles[END_REF] are discrete analytical objects. The aim of this paper is to show that the supercover of an m-flat in dimension n is a discrete analytical object. An m-flat in a space of dimension n is a Euclidean affine subspace of dimension m. We first propose a simple discrete analytical characterisation, through discrete inequalities, of the supercover of a hyperplane (i.e. supercover of an (n -1)-flat) and the supercover of a point supercover (i.e. supercover of a 0-flat). We show that the supercover and the orthogonal projection commute. A similar result holds for the orthogonal extrusion. Using this result, we prove that the supercover of an m-flat F , for 0 ≤ m ≤ n -2, is equal to the intersection of the supercovers of orthogonal extrusions of F . From this we deduce a method of determining a set of discrete inequalities that characterise the supercover of an m-flat. This proves that the supercover of an m-flat is a discrete analytical object. The paper is organised as follows. In section 2 we present the notations and the main definitions used in this paper. We also present some straightforward results that will be useful in subsequent sections. In section 3, we introduce the notion of supercover and some of its general properties. The main results of this paper can be found in the fourth section. Section 4 is divided into 4 subsections. In subsection 1 we provide a discrete analytical characterisation of the supercover of a hyperplane (i.e. (n -1)-flat). In subsection 2 we do the same for the supercover of a point (i.e. 0-flat). In the third subsection, we prove a result that characterises the supercover of an m-flat. In a fourth part of section 4 we show that the supercover of an m-flat is a discrete analytical object. We provide a method that gives a discrete analytical characterisation of the supercover of an m-flat. We conclude in section 5 with some discussion and a conjecture on the discrete analytical characterisation of a simplex in dimension n.

Preliminaries

Basic notations

Let Z n be the subset of the nD Euclidean space R n that consists of all the integer coordinate points. A discrete (resp. Euclidean) point is an element of Z n (resp. R n ). A discrete (resp. Euclidean) object is a set of discrete (resp. Euclidean) points. A discrete inequality is an inequality with coefficients in R from which we retain only the integer coordinate solutions. A discrete analytical object is a discrete object defined by a finite set of discrete inequalities [START_REF] Andres | The Discrete Analytical Hyperspheres[END_REF]. An m-flat is a Euclidean affine subspace of dimension m. A Euclidean hyperplane is an (n -1)-flat. A Euclidean point is a 0-flat. We denote by p i the i-th coordinate of a point or vector p. For p a discrete point or vector in dimension n, we denote by gcd (p) the greatest common divisor of p 1 , . . . , p n . For 0 ≤ k ≤ n, Two discrete points p and q in dimension n are k-neighbours,

if |p i -q i | ≤ 1 and k ≤ n- n i=1 |p i -q i |.
A k-path is a sequence of discrete points such that two consecutive points of the sequence are k-neighbours. We denote by σ n the set of all the permutations of the set {1, 2, . . . , n}. We denote by J n m the set of all the strictly growing sequences of m integers all between 1 and n:

J n m = {j ∈ Z m |1 ≤ j 1 < j 2 < . . . < j m ≤ n }.
This defines a set of multi-indices. Let us consider an object F in the n-dimensional Euclidean space R n , with n > 1.

The orthogonal projection is defined by: π i (q) = (q 1 , . . . , q i-1 , q i+1 , . . . , q n ) , for 1 ≤ i ≤ n and q ∈ R n ;

π j (q) = (π j1 • π j2 • • • • • π jm ) (F ) and π j (F ) = {π j (q) |q ∈ F } for j ∈ J n m .
The orthogonal extrusion is defined by:

ε j (F ) = π -1 j (π j (F )) , for j ∈ J n m .
It is easy to see that, for two Euclidean objects X, Y ⊆ R n , the following three statements are equivalent (i)

π j (X) ∩ π j (Y ) = ∅ (ii) ε j (X) ∩ Y = ∅ (iii) X ∩ ε j (Y ) = ∅.
For any discrete or Euclidean object F , we have

π i • π -1 i (F ) = F while in general π -1 i • π i (F ) = ε i (F ) = F. Let σ n denote
the set of all permutations of {1, 2, . . . , n}. For each σ ∈ σ n , we define the corresponding axis permutation r σ by:

r σ : R n → R n x → x σ(1) , x σ(2) , . . . , x σ(n)
and the corresponding multi-index axis permutation r j by r j = r σj , for j ∈ J n m , where the permutation σ j ∈ σ n is defined by:

σ j (j i ) = i, for 1 ≤ i ≤ m, σ j (k i ) = i + m, for all 1 ≤ i ≤ n -m.
where

k 1 < k 2 < • • • < k n-m are the elements of {1, 2, . . . , n} \ {j i |1 ≤ i ≤ m }
in ascending order. Informally, we can say that r j (x) is obtained from x by placing the i th coordinate of x in the j th i position, the m + 2 nd coordinate of x in the next remaining position, and so on.

Lemma 1 : Let us consider a Euclidean object F in R n and a multi-index axis permutation r j , with j ∈ J n m . Then:

π j (F ) = π (1,2,...,m) r -1 j (F ) ε j (F ) = r j ε (1,2,...,m) r -1 j (F )
The proof of this lemma is straightforward. In fact we have designed r j to satisfy these two properties. This lemma will be helpful in section 4.

The voxel

V (p) ⊂ R n of a discrete nD point p is defined by V (p) = p 1 -1 2 , p 1 + 1 2 × • • • × p n -1 2 , p n + 1 2 .
In 1D and 2D a voxel is also called a pixel. For a discrete object

F , V (F ) = p∈F V (p).
It is easy to see that, for two discrete points p and q and a multi-index j ∈ J n m , we have

V (p) × V (q) = V (p × q), V (π j (p)) = π j (V (p)) and V (ε j (p)) = ε j (V (p)).
It is also easy to see that, for a discrete point p and an integer m that satisfies 1 ≤ m < n, we have

V (p) = j∈J n m ε j (V (p)).

General properties of the Supercover

In this section we will introduce the notion of supercover, recall some of its known properties [START_REF] Cohen-Or | Fundamentals of Surface Voxelization[END_REF] and present several original properties that seem not to have been mentioned in the literature.

Definition 2 : The supercover S (F ) ⊂ Z n of a Euclidean object F is defined by:

S (F ) = {p ∈ Z n |V (p) ∩ F = ∅ }
There exists an alternative way, proposed by Tajine et al. [START_REF] Tajine | Hausdorff discretization and its comparison to other discretization schemes[END_REF], of defining the supercover with the distance

d ∞ : S (F ) = p ∈ Z n d ∞ (p, F ) ≤ 1 2
for a Euclidean object F . The supercover has many basic properties [START_REF] Cohen-Or | Fundamentals of Surface Voxelization[END_REF]. Let us consider two Euclidean objects F and G. Then:

S (F ) = α∈F S (α), S (F ∪ G) = S (F ) ∪ S (G), S (F ∩ G) ⊆ S (F ) ∩ S (G) and if F ⊆ G then S (F ) ⊆ S (G).
One can say that the operators S and ∪ commute. There are several other known properties of the supercover such as the (n -1)-connectivity of a supercover, the separation property, etc. See [START_REF] Cohen-Or | Fundamentals of Surface Voxelization[END_REF] for more details. Let us examine now some original properties of the supercover: Proposition 3 : For any two Euclidean objects F, G, we have:

S (F × G) = S (F ) × S (G).
One can say that the operators S and × commute.

Proof. Let us consider two Euclidean objects

F ⊂ R m and G ⊂ R n . a) Is S (F ) × S (G) ⊂ S (F × G) true ?
Let us consider two discrete points p ∈ S (F ) and q ∈ S (G). The discrete point p belongs to S (F ) if and only if there exists a Euclidean point α belonging to V (p) ∩ F . The discrete point q belongs to S (G) if and only if there exists a Euclidean point β belonging to V (q)∩G. This means that (α, β) ∈ V (p) × V (q) and thus, since

V (p × q) = V (p) × V (q), we have (α, β) ∈ V (p × q). With (α, β) ∈ F × G, we have V (p × q) ∩ (F × G) = ∅ and therefore (p, q) ∈ S (F × G). This proves the first part of our proposition. b) Is S (F × G) ⊂ S (F ) × S (G) true ? Let us consider a discrete point r ∈ S (F × G).
We can write r in the form (p, q) ∈ Z m+n , with p ∈ Z m and q ∈ Z n . The discrete point r belongs to S (F × G) if and only if there exists a Euclidean point γ = (α, β) , with α ∈ R m and β ∈ R n , belonging to V (p × q) ∩ (F × G).

We have γ = (α, β) ∈ V (p × q) = V (p) × V (q) and (α, β) ∈ F × G. Therefore we have α ∈ V (p) ∩ F and β ∈ V (q) ∩ G. This proves that r = (p, q) ∈ S (F ) × S (G).

The following proposition shows that the supercover operator commutes with the operators π -1 , π and ε.

Proposition 4 : For any multi-index j ∈ J n m . Then:

(i) For F ⊆ R n-m ,we have π -1 j (S (F )) = S π -1 j (F ) . (ii) For F ⊆ R n , we have π j (S (F )) = S (π j (F )) and ε j (S (F )) = S (ε j (F )) .
Proof. We can suppose without loss of generality that j belongs to J n 1 , i.e. that j is an integer that satisfies 1 ≤ j ≤ n. If the assertions (i) and (ii) of proposition 4 hold for an arbitrary integer j then they hold for every multi-index j.

a) π -1 j (S (F )) = S π -1 j (F )
is an easy consequence of proposition 3. b) Let us show that S (π j (F )) = π j (S (F )). We can suppose without loss of generality that

j = n. S (π n (F )) = S α ∈ R n-1 |α = (α , α n ) ∈ F and thus S (π n (F )) = p ∈ Z n-1 |∀α = (α , α n ) ∈ F and α = π n (α) ∈ V (p ) . On the other hand π n (S (F )) = π n {p = (p , p n ) ∈ Z n |V (p) ∩ F = ∅ } and thus π n (S (F )) = p ∈ Z n-1 |∀α = (α , α n ) ∈ F and α = π n (α) ∈ V (p ) .
This proves that π n (S (F )) = S (π n (F )).

c) ε j (S (F )) = S (ε j (F )) is an immediate consequence of a) and b).

The following proposition is important. It shows that the supercover is independent of any axis permutation. There is no privileged direction. This is not true for most other digitisation schemes [START_REF] Bresenham | Algorithm for computer control of a digital plotter[END_REF][START_REF] Fujimoto | ARTS: Accelerated Ray-Tracing System[END_REF][START_REF] Amanatides | A Fast Voxel Traversal Algorithm for Ray Tracing[END_REF][START_REF] Kaufman | An algorithm for 3D scan conversion of polygons[END_REF][START_REF] Stojmenovic | Digitization schemes and the recognition of digital straight lines, hyperplanes and flats in arbitrary dimensions[END_REF][START_REF] Veerlaert | On the flatness of digital hyperplanes[END_REF][START_REF] Debled-Renesson | A new approach to digital planes[END_REF][START_REF] Figueiredo | A Contribution to 3D Digital Lines[END_REF].

Proposition 5 : Let F ⊆ R n and σ ∈ σ n . Then:

r σ (S (F )) = S (r σ (F ))
The proof of this proposition is obvious. One can say that the operators S and r σ commute.

The following proposition tells us that the problem of determining the supercover of an extrusion is essentially a problem of determining a supercover in the corresponding projection space.

Proposition 6 : Let F ⊆ R n and j ∈ J n m . Then:

S (ε j (F )) = r j (Z m × S (π j (F )))
Proof. We know from lemma 1 that ε j (F ) = r j ε (1,2,...,m) r -1 j (F ) for r j a multi-index axis permutation with j ∈ J n m . It is easy to see that ε

(1,...,m) (F ) = π -1 (1,...,m) π (1,...,m) (F ) = R m × π (1,...,m) (F ). Therefore, ε (1,...,m) r -1 j (F ) = R m × π (1,...,m) r -1 j (F ) .
We know from lemma 1 that π j (F ) = π (1,...,m) r -1 j (F ) . This leads to ε j (F ) = r j ε (1,2,...,m) r -1 j (F ) = r j (R m × π j (F )) and therefore S (ε j (F )) = S (r j (R m × π j (F ))). Since r j is also an axis permutation, we can apply proposition 5 and thus, S (ε j (F )) = r j (S (R m × π j (F ))). Now, because of proposition 3, we have S (ε j (F )) equal to r j (S (R m ) × S (π j (F ))). It is easy to see that S (R m ) = Z m , which proves the proposition.

The supercover of an m-flat

In this section we will study the supercover of an m-flat in dimension n. We will show that the supercover of an m-flat is a discrete analytical object and give a method that provides the corresponding discrete inequalities. This section is organized in four subsections. In the first subsection we show that the supercover of a hyperplane (i.e. (n -1)-flat) is described by 2 discrete inequalities. In the second subsection we show that the supercover of a point (i.e. 0-flat) is described by 2n discrete inequalities. In subsection 3, we show that the supercover of an m-flat is the intersection of supercovers of its orthogonal extrusions. Note that of these orthogonal extrusions is an l-flat for some l ≥ m. In subsection 4, we show that if m < n -1, the supercover of an m-flat is a discrete analytical object by providing a method that determines the discrete inequalities describing the supercover of an m-flat.

Supercover of a hyperplane

The following theorem characterises the supercover of a Euclidean hyperplane (i.e. a (n -1)-flat) as the set of discrete points that satisfy a double inequality.

Theorem 7 : Let H be a Euclidean hyperplane in R n defined by: H : e + n i=1 A i x i = 0, with A ∈ R n , e ∈ R. Then the supercover of H is:

S (H) = X ∈ Z n - n i=1 |A i | 2 ≤ e + n i=1 A i X i ≤ n i=1 |A i | 2
Proof. Let us suppose, without loss of generality, that A i+1 ≥ A i ≥ 0 for 1 ≤ i ≤ n -1. Here "if" follows from the fact that e + n i=1 A i p i -

≤ i ≤ n, p i -1 2 ≤ x i ≤ p i + 1 2 for all x ∈ V (p) : if e + n i=1 A i p i -1
2 > 0 then we would have that e

+ n i=1 A i x i > 0 for all x ∈ V (p), so that V (p) ∩ H = ∅; symmetrically we would have that V (p) ∩ H = ∅ if e + n i=1 A i p i -1 2 < 0. Corollary 8 :
The supercover of a hyperplane is a discrete analytical object.

When we compare the Supercover of a hyperplane to other discrete hyperplanes, we notice that it is not, in general, a Reveillès discrete analytical hyperplane [START_REF] Reveillès | Géométrie Discrète, calculs en nombres entiers et algorithmique[END_REF][START_REF] Andres | Le plan discret[END_REF][START_REF] Andres | Discrete Analytical Hyperplanes[END_REF]. There is a small difference in the definitions. A Reveillès discrete analytical hyperplane H is a discrete object satisfying a double inequality [START_REF] Reveillès | Géométrie Discrète, calculs en nombres entiers et algorithmique[END_REF][START_REF] Andres | Le plan discret[END_REF][START_REF] Andres | Discrete Analytical Hyperplanes[END_REF]:

H = p ∈ Z n α ≤ e + n i=1 A i p i < β ; where α, β, e ∈ R and A ∈ R n
One of the characteristics of a A Reveillès discrete analytical hyperplane is its arithmetical thickness defined by ω = β -α. There is however an interesting subset of hyperplanes, the rational hyperplanes, for which the supercover are Reveillès discrete analytical hyperplane [START_REF] Andres | Discrete Analytical Hyperplanes[END_REF]. A hyperplane or discrete analytical hyperplane is said to be rational if it has a normal vector whose components are all integers. See [START_REF] Reveillès | Géométrie Discrète, calculs en nombres entiers et algorithmique[END_REF] for a study of 2D rational discrete analytical straight lines, [START_REF] Andres | Le plan discret[END_REF][START_REF] Françon | Discrete Combinatorial Surfaces[END_REF][START_REF] Françon | Sur la topologie d'un plan arithmétique [About the topology of the arithmetical plane[END_REF] for a discussion on a specific class of 3D discrete analytical planes and [START_REF] Andres | Discrete Analytical Hyperplanes[END_REF] for a study of the rational discrete analytical hyperplanes in arbitrary dimensions. Theorem 7 implies that the supercover of any rational hyperplane is a discrete analytical hyperplane.

Corollary 9 : Let H : e + n i=1 A i x i = 0 be a rational hyperplane in R n with A ∈ Z n , e ∈ R. Then the supercover of H is equal to:

S (H) = X ∈ Z n - n i=1 |A i | 2 -e ≤ n i=1 A i X i ≤ n i=1 |A i | 2 -e
The proof of corollary 9 is immediate. Thus rational hyperplanes have a supercover that is characterised by Diophantine inequalities. One of the reasons why this is of interest is of course that, in a finite part of space, a discrete hyperplane is always rational.

Corollary 10 : The arithmetical thickness of the supercover of the rational hyperplane

H : e + n i=1 A i x i = 0 is ω = n i=1 |A i | or ω = n i=1 |A i | + 1.
The proof of this result is simple. The arithmetical thickness is equal to [START_REF] Françon | Discrete Combinatorial Surfaces[END_REF]. This is an important class of discrete analytical hyperplanes. See [START_REF] Andres | Discrete Analytical Hyperplanes[END_REF][START_REF] Françon | Discrete Combinatorial Surfaces[END_REF][START_REF] Françon | Sur la topologie d'un plan arithmétique [About the topology of the arithmetical plane[END_REF][START_REF] Andres | Defining Discrete Objects for Polygonalization: the Standard Model[END_REF][START_REF] Andres | Discrete Analytical Objects: the Standard Model[END_REF] for details on standard hyperplanes.

n i=1 |A i | + 1, when n i=1 |A i | is

Supercover of a point

The supercover of a Euclidean point in dimension n is given by: Proposition 11 : The supercover of a Euclidean point y ∈ Z n is equal to:

S (y) = Z n ∩ y 1 - 1 2 , y 1 + 1 2 × • • • × y n - 1 2 , y n + 1 2
Proof. We first consider the case in which y ∈ Z 1 . The point y is a 0-flat ∈ Z 1 . A normal vector of the 0-flat is A = 1 and therefore y is a rational 0-flat. We can apply corollary 9 (with n = 1, A 1 = 1 and e = -y) to deduce that the supercover of y is equal to

S (y) = x ∈ Z y -1 2 ≤ x ≤ y + 1 2
. This and Proposition 3 imply Proposition 11.

The supercover of a Euclidean point can be equal to more than one single discrete point. This is what we call a "bubble" [START_REF] Andres | Supercover of Straight Lines, Planes and Triangles[END_REF][START_REF] Andres | Defining Discrete Objects for Polygonalization: the Standard Model[END_REF][START_REF] Andres | Discrete Analytical Objects: the Standard Model[END_REF]. This is one of the reasons why, despite its properties, the supercover is not often used as a discretization scheme. Classically, in computer graphics [START_REF] Bresenham | Algorithm for computer control of a digital plotter[END_REF], one of the discrete points is, arbitrarily, chosen as the discretisation of the Euclidean point. This leads to asymmetries and the loss of many of the properties of the supercover [START_REF] Cohen-Or | Fundamentals of Surface Voxelization[END_REF]. The way to obtain a model without bubbles is given by the standard analytical discretization model [START_REF] Andres | Defining Discrete Objects for Polygonalization: the Standard Model[END_REF][START_REF] Andres | Discrete Analytical Objects: the Standard Model[END_REF].

Another way of considering proposition 11 is to see it as a special case of Theorem 13 (that is presented in the next section) since proposition 4 tell us that p ∈ S(F ) iff for all j ∈ J n n-(m+1) , π j (p) ∈ S (π j (F )).

Corollary 12 : The supercover of a point is a discrete analytical object. This is a straightforward corollary of proposition 11.

Supercover of an m-flat for m ≤ n -2

The m-flat supercover, for 0 ≤ m ≤ n -2, is given by: Theorem 13 : Let F be an m-flat in R n , where 0 ≤ m ≤ n -2. Then:

S (F ) = j∈J n n-(m+1) S (ε j (F ))
We have that S (F ∪ G) is equal to S (F ) ∪ S (G) but in general S (F ∩ G) is not equal to S (F ) ∩ S (G). Theorem 13 provides an example of a collection of sets for which the supercover of the intersection is the intersection of the sets' supercovers. Indeed, since F ⊆ ε j (F ) it is evident that

S (F ) ⊆ S j∈J n n-(m+1) ε j (F ) ⊆ j∈J n n-(m+1)
S (ε j (F )) and, in view of Theorem 13, this implies that S (F ) = S

j∈J n n-(m+1) ε j (F ) = j∈J n n-(m+1)
S (ε j (F )) holds.

Proof. Since F ⊆ ε j (F ) for any j, we see that S (F ) ⊂ S (ε j (F )) and therefore S (F ) ⊂ S (ε j (F )).

To prove the reverse inclusion, let p be any point in S (ε j (F )), we have that V (p)∩ε j (F ) = ∅ for every j ∈ J n n-(m+1) .

Equivalently, we have that

for all j ∈ J n n-(m+1) , ε j (V (p)) ∩ F = ∅ (1)
Now, if j and k are any subsequences of {1, 2, . . . , n} , then

ε j (V (p)) ∩ ε k (V (p)) = ε j∩k (V (p)).
Moreover, the intersection of any m + 1 distinct members of J n n-1 is a member of J n n-(m+1) . It follows from these two observations that the intersection of any m + 1 distinct members of the collections of sets ε j (V (p)) ∩ F j ∈ J n n-1 is a member of the collection of sets

ε j (V (p)) ∩ F j ∈ J n n-(m+1)
and therefore is non-empty, by [START_REF] Amanatides | A Fast Voxel Traversal Algorithm for Ray Tracing[END_REF].

Since ε j (V (p)) ∩ F j ∈ J n n-1
is a collection of convex sets in the m-dimensional affine set F , and we have just seen that every subcollection of m + 1 of these sets has a nonempty intersection, it follows from Helly's theorem [START_REF] Eggleston | Convexity[END_REF] that the entire collection of sets has a nonempty intersection. As the intersection is F ∩

j∈J n n-1 ε j (V (p)) = F ∩ V (p), we have that F ∩ V (p) = ∅. This proves that p ∈ S (F ) .
Corollary 14 : Let F be an m-flat in R n , where m ≤ n -2, and let l be an integer such that m < l < n. Then:

S (F ) = j∈J n l-m S (ε j (F ))
Proof. The non-trivial "⊇" part of corollary 14 follows immediately from theorem 13; indeed, if p lies in the intersection of corollary 14 then p lies in the intersection of theorem 13 since, if j is any subsequence of {1, 2, . . . , n} of length n -1 -m then there is a subsequence j of j of length l -m and we have that p ∈ S (ε j (F )) ⊆ S (ε j (F )).

Discrete analytical characterization of the supercover of an m-flat

Let us show that the supercover of an m-flat, for 0 ≤ m ≤ n -1, is a discrete analytical object. For that we will present a method that determines explicitely all the discrete inequalities characterising the supercover of an m-flat.

Recursive method : Let F be an m-flat in R n specified in a parametric form.

a) If F is a 0-flat in R n , we directly apply Proposition 11 which gives us 2n discrete inequalities that characterise S (F ).

b) If F is an (n -1)-flat in R n , we transform the parametric form of F into an analytical form. Then apply Theorem 7, which gives us 2 discrete inequalities that characterise S (F ).

c) If F is an m-flat in R n , with 0 < m < n -1, use Theorem 13 as follows. For each j ∈ J n n-1-m , we recursively apply the method with R m+1 in place of R n and with a parametric specification of π j (F ) in place of the parametric specification of F , to produce a discrete analytical characterisation of S (π j (F )). From the discrete analytical characterisation of S (π j (F )) in R m+1 , we use Proposition 6 to derive a discrete analytical characterisation of S (ε j (F )) for all j ∈ J n n-1-m ; it is a trivial matter to compute r j . From the discrete analytical characterisation of S (ε j (F )) for all j ∈ J n n-1-m , we use Theorem 13 to derive a discrete analytical characterisation of S (F ).

Theorem 15 : The supercover of an m-flat is a discrete analytical object.

Proof. Let us consider an m-flat F and let us apply the recursive method described above. Let us prove that the method terminates always with a discrete analytical description of S (F ). This will prove that the supercover of an m-flat is a discrete analytical object. In case a) and case b) we obtain an explicit discrete analytical description of S (F ) in R n . This is also true in case c), assuming that the recursive calls which are made in that case all terminate. Note that the dimension n of the ambient space must be greater than 2 in case c), the only case which makes recursive calls. Moreover, when case c) makes a recursive call it sets the dimension of the mabient space in the recursive call to m = 1, which is strictly less than the dimension n of the ambient space in the caller. It follows that the depth of recursion is bounded by N -2, where N is the dimension of the ambient space in the outermost call. So the recursive call must indeed terminate.

Conclusion

In this paper we propose to take a new look at an old discretisation method called the supercover. The supercover of a Euclidean object F is the set of discrete points consisting of all the voxels that intersect F . The supercover has the advantage, over many other discretisation schemes used in computer graphics, of being defined in an arbitrary dimension and for arbitrary objects. The principal aim of this paper is to prove that the supercover of an m-flat is a discrete analytical object. A method is proposed to determine a set of discrete inequalities that define the supercover of an m-flat. The next step will consist in looking at the supercovers of finite objects, such as simplices of dimension k in a space of dimension n. Finite discrete objects are, in practice, more useful in computer graphics than infinite discrete objects. They can be generated and an important part of the planned work on the supercovers of simplices will consist in designing efficient generation algorithms. In order to devise such generation algorithms, it would be helpful to have a discrete analytical characterisation of the supercover of a simplex. We will propose now in conclusion, as a conjecture, such a discrete analytical characterisation. Let us consider a simplex A of dimension k in a space of dimension n defined by k + 1 linearly independant Euclidean points: A = simp P 0 , . . . , P k with P i ∈ R n for 0 ≤ i ≤ k. Let us denote by A the k-flat defined by P 0 , . . . , P k . If n = k then, by definition, A = R k . Let us denote by E A, P i the half-space whose border is the (n -1)-flat defined by P 0 , . . . , P i-1 , P i+1 , . . . , P n , and which contains the point P i . S (ε j (A)) .

  even and e is an integer, or, when n i=1 |A i | is odd and e -1 2 is an integer. In all other cases the arithmetical thickness is equal to n i=1 |A i |. Discrete analytical hyperplanes that have an arithmetical thickness equal to n i=1 |A i | are called standard hyperplanes

S

  (ε j (F )). We need to show that p ∈ S (F ). Since p ∈ j∈J n n-(m+1)

Conjecture 16 :••

 16 For any symplex A = simp P 0 , . . . , P k ⊂ R n ,• If k = n then S (A) = n i=0 S E A, P i ∩ n j=1 S (ε j (A)) ; If k = n -1 then S (A) = S A ∩ n j=1 S (ε j (A)) ; If k ≤ n -2 then S (A) = j∈J n n-k-1

  and hence that V (p) (which is convex and therefore contains that segment) intersects H. A discrete point p belongs to S (H) if and only if V (p) ∩ H = ∅. We have V (p) ∩ H = ∅ for some discrete point p if and only if e +

	1 2 ≤ 0 and e + imply that the segment whose endpoints are p -1 2 , • • • , 1 2 and p + 1 n i=1 A i p i + 1 2 ≥ 0 would 2 , • • • , 1 2 must intersect
	H, n i=1 A i p i -1 2 ≤ 0 and e + follows from the fact that e + n i=1 A i p i -1 2 ≤ 0 and e + n i=1 A i p i + 1 n i=1 A i p i + 1 2 ≥ 0. Here "if" 2 ≥ 0 would imply that the segment whose endpoints are p -1 2 , • • • , 1 2 and p + 1 2 , • • • , 1 2 must intersect H, and
	hence that V (p) (which is convex and therefore contains that segment) intersects H. The "only
	if" part follows from the fact that, for 1

Acknowledgment: We would like to thank Gaelle Largeteau-Skapin, Jean Françon and the reviewers for their numerous past and present comments.

There is some information about this conjecture in dimensions 2 and 3 in [START_REF] Andres | Supercover 3D Polygon[END_REF][START_REF] Andres | Tunnel-free supercover 3D polygons and polyhedra[END_REF][START_REF] Andres | Supercover of Straight Lines, Planes and Triangles[END_REF].


Two-dimensional metal structures revealed by evolutionary computations: Pb/Al 13 Co 4 (100)s a case study F. Brix 1 and E. Gaudry * 1 Université de Lorraine, CNRS, Institut Jean Lamour, Campus Artem, 2 allée André Guinier, 54000 Nancy, France (*Electronic mail: Emilie.Gaudry@univ-lorraine.fr) (Dated: 25 November 2021)

We have combined extensive Density Functional Theory calculations with an evolutionary algorithm to investigate possible structural models for two-dimensional (2D) Pb films supported on the Al 13 Co 4 (100) quasicrystal approximant surface. The minimization of the total energy with the constraint of maximizing the atomic density in the layer leads to 2D atomic arrangement with pentagonal motifs, reflecting the symmetry of the substrate. Our findings show that the 2D structure can be interpreted as a stable structure, with 16 Pb atoms per surface cell in the film, in accordance with the measured coverage. This conclusion is also supported by the reasonable agreement between the experimental STM images and those simulated using this structural model. Alternatively, a metastable 2D film made of 15 Pb atoms fits with the experimental observations. This study opens a route towards the prediction of supported complex 2D films.

This paper is dedicated to the memory of Pat Thiel, who has been a guide for so many of us

I. INTRODUCTION
Two-dimensional (2D) metallic films have attracted the interest of scientists from a large diversity of fields, due to their fascinating physical and chemical properties, owing to their high aspect ratio, their unique structures and possible quantum-size effects. [START_REF] Fan | Thin Metal Nanostructures: Synthesis, Properties and Applications[END_REF][START_REF] Ling | Freestanding Ultrathin Metallic Nanosheets: Materials, Synthesis, and Applications[END_REF] They have potential applications in many domains such as catalysis, electronics/optoelectronic devices, sensing, and energy storage.

The close-packed structure of elemental metals, originating from the non-directional metallic bonding, is expected to be hardly compatible with 2D materials. However, many theoretical works predicted the existence of stable free-standing single-layer 2D metals (see [START_REF] Yu | Synthesis of Two-dimensional Metallic Nanosheets: From Elemental Metals to Chemically Complex Alloys[END_REF][START_REF] Bechstedt | Beyond graphene: Clean, hydrogenated and halogenated silicene, germanene, stanene, and plumbene[END_REF] and references therein). This is the case with lead, for which a buckled honeycomb structure with a potential topological character has been foreseen by Density Functional Theory (DFT). [START_REF] Rivero | Stability and properties of high-buckled two-dimensional tin and lead[END_REF][START_REF] Yu | From a normal insulator to a topological insulator in plumbene[END_REF][START_REF] Huang | Strain driven topological phase transitions in atomically thin films of group IV and V elements in the honeycomb structures[END_REF] Nevertheless, the experimental growth of two-dimensional metal layers is not straightforward, and 2D metallic films are generally synthesized over a substrate. [START_REF] Wang | Stability and synthesis of 2D metals and alloys: a review[END_REF] The latter is known to impact the atomic arrangements in 2D metal films. Focusing on ultrathin Pb films, several phases of Pb/Si(111) have been formed at room temperature with varying degrees of coverage up to 2 monolayers. [START_REF] Su | Quantum size effect on ultra-thin metallic films[END_REF] On Al(111), the large mobility of Pb adatoms and the weak interaction with the substrate leads to the formation of a ( √ 31 × √ 31)R8.95 o high-order commensurate structure (HOC structure). [START_REF] Deniozou | Aperiodic and modulated pb thin films on fivefold icosahedral Al-Cu-Fe and Al(111): Tailoring the structure of Pb[END_REF] Less trivial 2D arrangements of Pb atoms are found on intermetallic substrates, such as the graphene-like structure obtained recently by Yuhara et al. on Pd (1-x) Pb x (111). [START_REF] Yuhara | Graphene Latest Cousin: Plumbene Epitaxial Growth on a Nano WaterCube[END_REF] On quasicrystals, τ inflated quasiperiodic structures are generally observed. [START_REF] Deniozou | Aperiodic and modulated pb thin films on fivefold icosahedral Al-Cu-Fe and Al(111): Tailoring the structure of Pb[END_REF][START_REF] Mcgrath | The memory of surfaces: epitaxial growth on quasi-crystals[END_REF] Heterogeneous nucleation is a key factor that controls the epitaxial growth, through substrate-adsorbate interactions. Adsorbate-adsorbate and steric interactions, related to the size of the adsorbates and how they can be accommodated by the substrate, are also identified to be crucial. While these general principles are well known, the prediction of epitaxial structures is still challenging. Considerable progress have been made in crystal structure prediction these last years, through the development of efficient computer codes and the availability of large computational resources. Methods such as annealing, [START_REF] Schön | Possible Existence of Alkali Metal Orthocarbonates at High Pressure[END_REF][START_REF] Santoro | Theory of quantum annealing of an ising spin glass[END_REF] evolutionary, [START_REF] Oganov | Crystal structure prediction using evolutionary algorithms: principles and applications[END_REF][START_REF] Deaven | Molecular Geometry Optimization with a Genetic Algorithm[END_REF] swarm or basin/minima hopping [START_REF] Santoro | Theory of quantum annealing of an ising spin glass[END_REF][START_REF] Wang | CALYPSO: A method for crystal structure prediction[END_REF][START_REF] Schönecker | Thermal surface free energy and stress of iron[END_REF] based algorithms are able to escape the local minima to efficiently investigate the configuration space. These methods are increasingly used for bulk structures, but they are rarely employed for two-dimensional materials. [START_REF] Zhou | Two-dimensional magnetic boron[END_REF][START_REF] Mazitov | Evolutionary algorithm for prediction of the atomic structure of two-dimensional materials on substrates[END_REF] Though, they can bring useful insights to analyze the experimental observations, as explained in this article.

In this work, we performed evolutionary computations to model 2D lead films supported on the Al 13 Co 4 (100) quasicrystalline approximant surface. We selected this intermetallic substrate, motivated by previous experimental and theoretical studies. [START_REF] Addou | Lead adsorption on the Al 13 Co 4 (100) surface: Heterogeneous nucleation and pseudomorphic growth[END_REF][START_REF] Alarcó-Villaseca | Lead adsorption on the pseudo-tenfold surface of the Al 13 Co 4 complex metallic alloy: A first principle study[END_REF] When Pb deposition is achieved at 573 K, the ultrathin films are found to be quite well ordered, described by a periodic arrangement of irregular pentagons, in which the adsorbate-substrate interactions are conjectured to play a dominant role. [START_REF] Addou | Lead adsorption on the Al 13 Co 4 (100) surface: Heterogeneous nucleation and pseudomorphic growth[END_REF] The connected bipentagonal motifs, observed by scanning tunnelling microscopy (STM), are understood as being related to Al atomic arrangements on the clean surface. However, while the termination plane of the substrate contains four Al pentagonal motifs per surface cell, only two (larger) pentagonal features are visible in the experimental images. In addition, the STM images simulated from the structural model proposed so far, did not lead to to satisfactory agreement with the experimental data. [START_REF] Alarcó-Villaseca | Lead adsorption on the pseudo-tenfold surface of the Al 13 Co 4 complex metallic alloy: A first principle study[END_REF] In this article, we show that the combination of Density Functional Theory (DFT) with evolutionary computations lead to stable twodimensional Pb layers with pentagonal atomic arrangements, reflecting the symmetry of the substrate. The Pb atoms are not systematically located in most stable adsorption sites identified for atomic lead, but the structure of the Pb film results from the minimization of the total energy with the constrain of maximizing the atomic density in the film. Two structural models lead to simulated STM images in good agreement with the experimental ones: a stable film made of 16 Pb atoms per surface cell and a metastable 2D layer containing 15 Pb atoms per surface cell.

II. MATERIALS AND METHODS

A. Structural model of Al 13 Co 4 (100)

The Al 13 Co 4 orthorhombic structure (Pmn2 1 space group) is described as a pile of two types of planes, perpendicular to the [100] direction, labeled alternatively flat-type (F-type) and puckered-type (P-type) planes. [START_REF] Grin | Crystal Structure of Orthorhombic Co 4 Al 13[END_REF] The stacking sequence is F 0 P 0.25 F 0.5 P 0.75 and the interlayer distance is approximately 2 Å. On the other hand, the structure can be understood as a stacking of Henley-type clusters. [START_REF] Scheid | Bonding Network and Stability of Clusters: The Case Study of the Al 13 TM 4 Pseudo-10fold Surfaces[END_REF][START_REF] Chatelier | The pseudo-twofold surface of the Al 13 Co 4 catalyst: structure, stability, and hydrogen adsorption[END_REF] Several complementary techniques have been employed to provide a reliable model of the Al 13 Co 4 (100) surface. The combination of STM, Low Energy Electron Diffraction (LEED) and DFT calculations converged towards a surface terminating at puckered layers where on average all Al atoms are present and protruding Co atoms are missing. [START_REF] Shin | Structure of the Orthorhombic Al 13 Co 4 (100) Surface Using LEED, STM and Ab Initio Studies[END_REF] The combination of surface X-ray diffraction and DFT pointed towards the same surface model, with partial occupancies for surface Co sites slightly buried in the P-type plane. [START_REF] Gaudry | Structure of the Al 13 Co 4 (100) Surface: Combination of Surface X-Ray Diffraction and Ab Initio Calculations[END_REF] Specific atomic arrangements are found in the termination plane of Al 13 Co 4 (100). Aluminum atoms form pentagonal motifs (displayed with gray lines in Fig. 2), connected 2 to 2 by squared patterns and thus leading to two Al bipentagonal motifs (BMs) per surface cell. The two Al BMs are not equivalent: one is located slightly below the mean position of the termination plane (BM-empty, shown in light gray, Fig. 2), while the second one, decorated with Co atoms buried in the pentagonal Al hollow sites, lies slightly above the mean position of the termination plane (BM-Co, shown in dark gray, Fig. 2).

B. Genetic algorithm

Our methodology is based on the evolutionary algorithm USPEX, [START_REF] Glass | Uspex-evolutionary crystal structure prediction[END_REF][START_REF] Lyakhov | New developments in evolutionary structure prediction algorithm uspex[END_REF][START_REF] Oganov | How evolutionary crystal structure prediction works and why[END_REF] which has already been used to predict the crystal structure of two-dimensional materials. [START_REF] Zhou | Two-dimensional magnetic boron[END_REF][START_REF] Mazitov | Evolutionary algorithm for prediction of the atomic structure of two-dimensional materials on substrates[END_REF] Each structure was built with a 7-layer thick slab for the substrate (139 Al and 36 Co atoms in total, including 26 Al and 2 Co atoms in the termination plane), covered by a Pb film -thin region (2 Å thick) above the substrate -and a 11 Å void thickness. Several coverages have been considered : 14, 15, 16 and 17 Pb atoms per 1×1 surface cell. The positions of the atoms in the film were optimized, along with those in the topmost layer of the substrate. The initial structures of the film were generated randomly, and then built from variation operators that use the previous generation's best structures as parents, using operators such as Heredity (30%), Transmutation (10%) and Softmutation (10%). A few number of structures in a generation were produced randomly (50%) to diversify the population. All structures were relaxed by DFT (see below), using five consecutive relaxation steps with increasing accuracy. Total energies were used to rank the final structures. An example is given in Fig. 1 for the 15 Pb atom thin film structures. This process has been repeated cyclically until the best structures' set remains unchanged for 15 generations.

C. Computational details

Total energy calculations were based on the Density Functional Theory (DFT) using the Vienna ab initio simulation package (VASP) [START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF][START_REF] Kresse | Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set[END_REF][START_REF] Kresse | Efficient iterative schemes for ab initio totalenergy calculations using a plane-wave basis set[END_REF] . The interaction between the valence electrons and the ionic core was described using the spin polarized projector-augmented wave (PAW) method [START_REF] Blöchl | Projector augmented-wave method[END_REF][START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF] within the generalized gradient approximation of Perdew, Burke and Ernzerhof. [START_REF] Perdew | Generalized gradient approximation made simple[END_REF] The valences of Pb, Al and Co were taken as 6s 2 6p 2 , 3s 2 3p 1 and 3d 8 4s 1 , respectively. In the final relaxation step, the one-electron Kohn-Sham orbitals were expanded with a kinetic energy cutoff of 450 eV. The reciprocal space integration was approximated with a Monkhorst-Pack k-point grid of 2×2×1. This computational setup lead to lattice parameters for bulk Al 13 Co 4 in good agreement with the literature (a=8.195 Å, b=12.252Å, c=14.277Å). [START_REF] Addou | Lead adsorption on the Al 13 Co 4 (100) surface: Heterogeneous nucleation and pseudomorphic growth[END_REF][START_REF] Kandaskalov | Catalytic semihydrogenation of acetylene on the (100) surface of the o-al13co4 quasicrystalline approximant: density functional theory study[END_REF]

D. Adsorption energies

In a first step, the adsorption of a single Pb atom has been computed on 400 specific points of a regular grid covering the Al 13 Co 4 (100) surface. For each point, the coordinates of the adsorbed Pb atom in the direction normal to the surface, as well as all degrees of freedom of the uppermost layer of the substrate have been allowed to relax during the geometry optimization process. The in-plane coordinates of the Pb adsorbates have been frozen. For each point of the grid, the adsorption energy (E 1 ads ) has been calculated by:

E 1 ads = E Slab+Pb -E Slab -E SinglePb (1)
where E Slab+Pb , E Slab and E SinglePb are the total energies of the system {Slab+Pb}, of the clean slab and of a single Pb atom, respectively.

In a second step, the adsorption energies of Pb adlayers have been computed for several structures -single-layer thick Pb adlayer slab built by progressively filling all favorable adsorption sites, [START_REF] Villaseca | Lead adsorption on the pseudo-10-fold surface of the al13co4 complex metallic alloy: A first principle study[END_REF][START_REF] Anand | Non-Wetting Behavior of Al-Co Quasicrystalline Approximants Owing to their Unique Electronic Structures[END_REF] and models given by the structure prediction process -using the following equation:

E n ads = E slab+nPb -E slab -n E f ccPb 4 (2)
where n (n > 1) is the number of Pb atoms in the lead adlayer and E f ccPb is the cohesive energy of lead (E f ccPb = -2.96 eV). Our value for E f ccPb is in good agreement with other theoretical values calculated using the same methodology (-2.94 eV, -2.92 eV) [START_REF] Anand | Non-Wetting Behavior of Al-Co Quasicrystalline Approximants Owing to their Unique Electronic Structures[END_REF][START_REF] Lejaeghere | Error estimates for solid-state density-functional theory predictions: an overview by means of the ground-state elemental crystals[END_REF] and consistent with the experimental value (-2.03 eV). [START_REF] Kittel | Introduction to Solid State Physics[END_REF]

III. RESULTS

A. Surface energy landscape for a single Pb atom on Al 13 Co 4 (100)

The adsorption energy map, for a single Pb atom adsorbed on Al 13 Co 4 (100), is shown in Fig. 2. Overall, the interaction of atomic lead with the surface is favorable, all adsorption energies being negative, in the range between -2.7 eV and -1.2 eV. The most favorable adsorption sites, with adsorption energies between -2.6 eV and -2.7 eV, are hollow sites. They are located outside the pentagonal aluminum arrangements (8 sites per surface unit cell) and in the center of squared arrangements of Al atoms (center of the BMs, 2 sites per surface unit cell). Sites located at the center of the Al pentagonal arrangement are found to be favorable as well, except when surface Co atoms are present. By the way, all sites located on top of a substrate's surface atoms are unfavorable. The worse case is found on top of Al atoms belonging to the BMs filled with Co atoms and located slightly above the mean position of the substrate's termination plane (BMs-Co): the corresponding adsorption energies lie in the range [-1.7:-1.2] eV. (Fig. 2). The top sites of the other BMs motifs, not containing any Co atom and located slightly below the mean position of the substrate's termination plane (BMsempty), are less unfavorable, with adsorption energies close to -1.9 eV.

Thus, 12 favorable adsorption sites per surface unit cell are identified for atomic Pb. They are all located in hollow sites and thus are clearly driven by the topography of the substrate's termination plane. These results are consistent with the literature. Previous calculations indeed found the same most favorable adsorption sites, with similar adsorption energies (within 0.1 eV). [START_REF] Addou | Lead adsorption on the Al 13 Co 4 (100) surface: Heterogeneous nucleation and pseudomorphic growth[END_REF] These results are not specific to lead adsorption. Actually, a similar adsorption map plotted for atomic hydrogen on Al 13 Co 4 (100) identified distinct adsorption energies for hydrogen at the center of the Al pentagonal arrangements. However, no clear distinction between the two types of BMs has been noticed in this case. [START_REF] Chatelier | The pseudo-twofold surface of the Al 13 Co 4 catalyst: structure, stability, and hydrogen adsorption[END_REF][START_REF] Kandaskalov | Catalytic semihydrogenation of acetylene on the (100) surface of the o-al13co4 quasicrystalline approximant: density functional theory study[END_REF] B. Best structural models of supported Pb adlayers At low coverage (less than 0.5 monolayer), the experimental results have been successfully interpreted by decorating the preferential adsorption sites. [START_REF] Addou | Lead adsorption on the Al 13 Co 4 (100) surface: Heterogeneous nucleation and pseudomorphic growth[END_REF] However, such an approach cannot explain the experimental observations at higher coverages, i.e. when the number of Pb atoms in the twodimensional film is larger than the number of favorable adsorption sites. [START_REF] Alarcó-Villaseca | Lead adsorption on the pseudo-tenfold surface of the Al 13 Co 4 complex metallic alloy: A first principle study[END_REF] In the following, we show that in this case, evolutionary algorithms can provide realistic structural models.

The most stable Pb-adlayer structures containing 14, 15, 16 and 17 Pb atom per surface cell are gathered in Tab. I and Tab. II (appendix). Additional structures -metastable Pb-adlayers -are given in Tabs III-V (appendix). Our results show that the 1×1 surface cell can accommodate up to 16 Pb atoms. Indeed, the single layer character of the 2D Pb film is kept when n ≤ 16 (∆z ≤ 2Å), and a second layer starts to grow when n ≥ 17. The value of 16 Pb atoms per surface cell corresponds to the density of a fcc(111) plane. But the fcc arrangements of Pb atoms on Al 13 Co 4 (100) are not favored energetically, even if the average size mismatch between Pb(111) and Al 13 Co 4 (100) is small (the average value for the in-plane cell parameters is below 2.5%).

The superposition of the atomic adsorption map with the positions of the lead atoms in the 2D films shows that most favorable atomic adsorption sites are filled by Pb atoms. Thus, in all 2D films, Pb arrangements form distorted pentagonal motifs, driven by the local symmetries of the Al 13 Co 4 (100) termination. Less stable sites are also occupied by Pb atoms in the 2D metal films, such as the sites on top of surface Co atoms, as well as bridge sites in the squared motif in between two Al pentagonal arrangements.

The adsorption energies per surface area (E ads /A) are of the order of -30 meV/Å 2 for all structures with n ≤ 16 (Fig. 3). It means that the interaction between the 2D Pb-layer and the substrate is much stronger with structurally optimized structures than with structures obtained by progressively filling all favorable adsorption sites, as performed in a previous study ( -13 meV/Å 2 ). [START_REF] Anand | Non-Wetting Behavior of Al-Co Quasicrystalline Approximants Owing to their Unique Electronic Structures[END_REF] In addition, with increasing n, the interaction between the 2D-layer and the substrate becomes weaker and the Pb-Pb interaction gradually takes (more) control over the adlayer structure. For Pb atomic densities between 0.8 at/Å 2 and 0.9 at/Å 2 (14 to 16 at./surf. cell), the adsorption energy increases smoothly as a function of n. It suggests that part of most stable sites are filled first and that a compromise is then found to ensure a dense packing in the 2D Pb-layer while minimizing the total energy. For instance, the Pb atom located in the favorable hollow site labelled 'Init' (tab. I) in the 15 Pb atom film is shifted from its position in the 16 Pb atom film due to the presence of a close Pb atom (positions labelled 'Fin1' and 'Fin2'). The weakening of the adsorption is larger for n=17, consistently with the growth of a second Pb layer. The adhesion energy, i.e. the required energy (per unit area) to separate the slabs and create two free surfaces has been calculated for the stable 16 Pb atom layer using

E adh (nPb) = E slab+nPb -E slab -E nPb (3)
with E nPb the energy of the free-standing Pb film (E slab+nPb and E slab are defined in Section II.D). We find E adh (16Pb)=1.33 J/m 2 . This value is much larger than the one calculated for the exfoliation of one graphene layer from graphite (0.425 J/m 2 , 0.22 J/m 2 , 0.23 J/m 2 ) [START_REF] Han | Surface energies, adhesion energies, and exfoliation energies relevant to copper-graphene and copper-graphite systems[END_REF][START_REF] Wang | Atomic intercalation to measure adhesion of graphene on graphite[END_REF][START_REF] Koren | Adhesion and friction in mesoscopic graphite contacts[END_REF] . Thus, isolating a free-standing Pb layer with local 5fold symmetries, using a Pb layer grown on a quasicrystal approximant surface, seems unlikely.

C. Simulated STM images

The STM images have been simulated within the Tersoff-Hamman approximation (V bias = +1 V), [START_REF] Tersoff | Theory and Application for the Scanning Tunneling Microscope[END_REF][START_REF] Tersoff | Theory of the scanning tunneling microscope[END_REF] for a large number of structural models (Tabs. I-V). Overall, the simulated STM images are very diverse, the bright contrasts being very sensitive to the precise location of the Pb atoms (especially the height). Very few images show pentagonal features.

Distorted pentagonal bright features are visible in the STM image calculated from the most stable 16 Pb atom adlayer (dark gray motif shown in Tab. I). A better agreement with the experimental observations is observed for the STM image simulated from a metastable 15 Pb atom adlayer (Fig. 4). The averaged edge length of the pentagonal motifs is 4.02 Å, i.e. it is consistent with the value measured on the experimental image (4.83 Å). [START_REF] Addou | Lead adsorption on the Al 13 Co 4 (100) surface: Heterogeneous nucleation and pseudomorphic growth[END_REF] In both cases, the bright contrast of the pentagonal features is attributed to Pb atoms in the adsorption sites of the BM-Co motif, with the exception of the hollow site above the subsurface Co atom, which gives a dark contrast because of its lower height. Similarly, the favorable adsorption sites in the vicinity of the BM-empty motif, which are filled by Pb atoms, are not visible in the simulated STM images because they are positioned below the mean position of the plane.

The energy difference between the 15 Pb atom metastable structure and the 15 Pb atom most stable one is quite small (∆E 9 meV / Pb atom, Tab.IV). Such a small energy difference can be attributed to entropy. Vibrational contributions are known to have a stabilizing effect, whose magnitude is of the order of a few tens of percent of the surface energy -10% at 973 K for the low-index surfaces of GaAs and InAs, 47 30% at the melting point for the low-index surfaces of iron. [START_REF] Schönecker | Thermal surface free energy and stress of iron[END_REF] A few percent (5%) of the surface energy of lead (γ Pb(111) 17.2 meV/Å 2 ) [START_REF] Yu | First-Principles Study of Low-Index Surfaces of Lead[END_REF][START_REF] Yu | Orientation-dependent surface and step energies of Pb from first principles[END_REF] is thus enough to account for ∆E here.

Finally, we should notice that the periodic boundary conditions enforced in the calculations may have possible effects in the STM images. The surface of the substrate is periodic, but this periodicity do not necessarily applies in the Pb film. This could explain some differences between the simulated and observed experimental STM images.

IV. CONCLUSION

In this work, evolutionary computations have been performed to offer a surface model for the 2D metal quasicrystal approximant grown on Al 13 Co 4 (100). The minimization of the total energy with the constraint of maximizing the atomic density in the layer leads to 2D atomic arrangement with pentagonal motifs, reflecting the symmetry of the substrate. Adsorbate-adsorbate interactions play a non negligible role in the emergence of the ordered 2D metal structure, since several most favorable adsorption sites identified for the Pb atomic adsorption are not filled any more in the metal film, to optimize the film density. The pentagonal bright features in the experimental STM image are attributed to the Pb atoms located in the vicinity of the substrate BM-Co motif. Lead atoms adsorbed above the substrate BM-empty motif, appear with a dark contrast in the simulated STM image, because of their lower height.

Our study shows that it is possible to successfully predict the structure of 2D films with a rather good accuracy, based on extensive DFT calculations combined with evolutionary algorithms. This is very promising for the determination of more complex 2D films, such as 2D oxide films, made of two or more chemical elements. Energy differences (in meV per Pb atom) are given in each case, relatively to the most stable structure.

FIG. 1 .

 1 FIG. 1. Total energies of the 15 Pb atom thin film structures considered in this work.

FIG. 2 .

 2 FIG. 2. Adsorption energy map for atomic Pb adsorbates on Al 13 Co 4 (100). Color code: Al = light gray, Co = pink. The BM-Co and BM-empty pentagonal motifs are shown in dark gray and light gray, respectively.

FIG. 3 .

 3 FIG. 3. Adsorption energies of the most stable Pb films.

FIG. 4 .

 4 FIG. 4. (a) Simulated STM image of the metastable 15 Pb atom 2D film on Al 13 Co 4 (100) (V bias =+1 V). (b) Corresponding structural model, superimposed with the atomic adsorption map. Color code: Pb = dark gray, Al = light gray, Co = pink. (c) High-resolution STM image of the 2D Pb layer at 573 K on Al 13 Co 4 (100): 18 × 9 nm 2 and 2.6 × 3.2 nm 2 (inset). Adapted from R. Addou, A. K. Shukla, S. Alarcón-Villaseca, É. Gaudry, T. Deniozou, M. Heggen, M. Feuerbacher, R. Widmer, O. Gröning, V. Fournée, J.-M. Dubois, and J. Ledieu, "Lead adsorption on the Al13 Co4 (100) surface: Heterogeneous nucleation and pseudomorphic growth," New J. Phys. 13, 103011 (2011). ®IOP Publishing and Deutsche Physikalische Gesellschaft. Reproduced by permission of IOP Publishing. All rights reserved.

TABLE II .

 II Best structures for the 2D Pb film on Al 13 Co 4 (100).

	0	+4.07	+4.07
	+13.07	+14.43	+20.143
	+27.64	+51.29

TABLE III :

 III Several structures obtained in the last generation for a Pb-adlayer containing 14 atoms.Energy differences (in meV per Pb atom) are given in each case, relatively to the most stable structure.

	0	+0.2	+8.87
		+9.47	+13.87
	+19.00	+30.00	+30.47

TABLE IV :

 IV Several structures obtained in the last generation for a Pb-adlayer containing 15 atoms. Energy differences (in meV per Pb atom) are given in each case, relatively to the most stable structure.

	0	+0.06	+4.94
	+4.75	+5.81	+21.75
	+28.94	39.44

TABLE V :

 V Several structures obtained in the last generation for a Pb-adlayer containing 16 atoms.

ACKNOWLEDGMENTS

This work is supported by the European Integrated Center for the Development of New Metallic Alloys and Compounds. We acknowledge financial support through the COMETE project (COnception in silico de Matériaux pour l'EnvironnemenT et l'Énergie) co-funded by the European Union under the program FEDER-FSE Lorraine et Massif des Vosges 2014-2020. This work was granted access to the HPC resources of TGCC, CINES and IDRIS under the allocation 99642 attributed by GENCI (Grand Equipement National de Calcul Intensif). High Performance Computing resources were also partially provided by the EXPLOR centre hosted by the University de Lorraine (project 2017M4XXX0108).

DATA AVAILABILITY STATEMENT

The data that support the findings of this study are available from the corresponding author upon reasonable request.


Corentin Massonneau

Nadia Gauducheau

Matthieu Tixier

Matthieu Tixier Interpersonal

Interpersonal Relationships Development and Impression Management Online before IRL Encounters in an EBSN

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

This study explores the role of computer-mediated communication in intimate relationship development through Event-Based Social Network (EBSN), which are platforms where people can interact online before meeting "in real life" (IRL) for face-to-face group activities [START_REF] Ricken | Going online to meet offline: Organizational practices of social activities through meetup[END_REF]. In communication studies, the modality switching of communication has been identified as an important factor in relationship development [START_REF] Ramirez | When online meets offline: An expectancy violations theory perspective on modality switching[END_REF]. However, few researches has explored the relation between modality switching on EBSN, relationship development, online communication activities and impression management. We conducted a content analysis on the French EBSN "OnVaSortir" (OVS) in order to explore how the users manage their impression online through their profiles and interactions in the comments section before and after face-to-face meetings.

CONCLUSION

On this EBSN, users don't seem to use the platform to develop social links but mainly for the event coordination. The self-disclosure and the impression management are important factors for relationships development [START_REF] Altman | Social penetration: The development of interpersonal relationships[END_REF][START_REF] Hu | Achieving self-congruency? Examining why individuals reconstruct their virtual identity in communities of interest established within social network platforms[END_REF]. However, our results show that the occupied status (the "role") seems to be a determining factor in the choice of the strategy to adopt in order to develop interpersonal relationships. Organizers' strategy consists in reducing the uncertainties of potential participants through their profile and interactions (e. g., provide relevant information for a successful coordination). Participants' strategy consists in collecting information about people and events first and then self-disclose after the first face-to-face meeting in order to maintain the new social bonds. As we know that OVS is a way to create relationships [START_REF] Pharabod | Fréquenter des inconnus grâce à internet[END_REF], our future researches need to explore if EBSN user's use another online medium to develop interpersonal relationships and determine whether IRL events are the only way for this population to develop relationships.

RESULTS (1) Analysis of online conducted communication activities

We make a distinction between three kind of users given their role : organizer, participant to the event and other OVS user (who takes part to online exchanges but not to the IRL event) Before the event • We observe expression of self-disclosure from all kind of users but in a minimal way. • The comments feed is essentially used for the coordination of users (e. g., "Tickets OK." ; "I'm here with […]. We are next to the cash registers'," etc.). After the event • The post-event comments feed is essentially composed by acknowledgements to the attention of the organizer and other participants. • Participants only evaluate positively the course of the event.

• Participants express way more self-disclosure than other users after the face-to-face event.

• Relationships are more intimate after the face-to-face meeting (acknowledgments, positive feedbacks, etc.)

RESULTS (2) Analysis of users impression management

Through profiles • There is a quite normative use in the way personal information are voluntarily shared through the public profile among all users. • Organizers tend to self-disclose through profiles, unlike participants.

• Organizers profiles are more complete than participants ones because their role is to create an event for potential unknown people who need to be reassured (Berger & Calabrese, 1975). Through online interactions • Organizers' interactions on the platform are dedicated for the coordination of users in order to facilitate the meeting of everyone. They mostly answer to other users questions and notify them of any changes probably because of the importance of first impressions online (Bacev-Giles & Haji, 2017). • Participants generally do not interact online with the organizers before the event but when they do, this is in a minimal way (e. g. "I'll come."). They just collect relevant information about the event.

METHOD AND PROCEDURE

We randomly chose 41 OVS events who took place during the same week last year in the city of Paris. Then, we extracted the messages of the comments feed in order to analyze them and describe the communication dynamics between OVS users before and after the event. We also analyzed the strategies of impression management through the users profiles and their online interactions. For this study, 627 messages and 251 profile pages have been extracted and analyzed with two distinct analysis grids that we created.

CONTACT AND INFORMATIONS 32 nd International Congress of Psychology PRAGUE 18 -23 July 2021

  Corentin Massonneau, Nadia Gauducheau et Matthieu Tixier Université de Technologie de Troyes, France Interpersonal Relationships Development and Impression Management Online before IRL Encounters in an EBSN

Corentin MASSONNEAU corentin.massonneau@utt.fr PhD Student in Social Psychology Unité de Recherche LIST3N. Equipe de Recherche Tech-CICO Université de Technologie de Troyes.