Laurent Garnier
email: laurent.garnier@cea.fr

Julien Dauchy

Valery Cervantes

Daniel Chatroux

ESPRIT project: an innovative electric car sharing solution

ESPRIT project is an important research project financed by Europe for 8.9 M€ in the framework of the program H2020 and coordinated by CEA (Commissariat à l'énergie atomique et aux énergies alternatives). The goal of ESPRIT project is to decrease the usage of individual cars in urban area and to complete the usage of public transport. It is ESPRIT deals with the development of some small electric cars which can be mechanically coupled to create road trains. This vehicles can be used within a carsharing system in city center and peri-urban area. These vehicles will allow for user to complete a trip done with public transport and perform the first or the last kilometer with a non-pollutant system. The aptitude to be coupled in road train allows the redistribution to secure the availability of vehicles in stations. That permits to propose a high reliability system with a limited investment cost. This paper shows the work done during the 3 years of the project. In 2016 a first paper [6] about this project was presented in PCIM: "Comparison between standard and innovative solutions to exchange energy between high energy storage systems". This first paper was dedicated to a sub-system inside the vehicle. In this second paper we are more focusing on the final results of the project.

ESPRIT project

Goal of project is to solve with technical innovation the current main issues of existing carsharing system.

Investment cost

First of issues is the investment cost. A classic carsharing system need to invest in stations with almost one point of charge for each vehicles. ESPRIT concept is to store vehicles in station in road train configuration (Figure 2). In this configuration, vehicles are electrically coupled. Thus an unique charging station allows to charge the complete road train (8 vehicles maximum for the moment). That decreases significantly the cost. Space of station in the street is reduced due to the road train configuration. The place needed by 8 ESPRIT vehicles in a road train is equavalent to the one needed by 3-4 conventional cars parked one behind the other. That is the second raison which explains a lower investment cost that conventional carsharing system.

Distribution cost

Reliability

The majority of 'one way' carsharing systems does not work at 100% of capability. Vehicles are not so used in comparison with the maximum possibility. This phenomena is mainly due to the bad confident in the system for the user. If somebody is late to an important meeting for job because of its unability to find a vehicle (the station near to home was empty), the next day, he will take another transport service. That's why the distribution is absolutely essential in a 'one way' carsharing system, to build customer loyalty.

In station, ESPRIT work like FIFO system.

The first vehicle will be used in first and the vehicle is restitute by the back. Operator can choose the number of vehicles he want to redistribute from one station in order to complete another station.

Technical innovations

Currently, the maximum number of vehicles in an ESPRIT road train during a distribution is limited to 8. This train is particular because the first vehicle can't tow alone the 8 vehicles. Firstly, because it has not enough power and secondly because, due to the homogenous repartition of mass oscillation phenomenon may appear. The phenomenon is greatly amplified if there are several trailers.

To solve these issues, all vehicles participate to the propulsion of the train. Each vehicle power supplies their own electrical motors, so there is no vehicle towed. To do this, a communication between vehicles is necessary. First vehicle is the master and the followers become slaves and execute orders given by first vehicle. It calculates in real time the oscillations of the road train with data provided from sensors placed in each vehicle. Algorithms determine the compensation needs and send orders to each vehicle independently.

To facilitate the oscillation compensation, ESPRIT vehicle is equipped with two motors. One on the back left and one other on the back right. With an intelligent command of torque on each motor, the oscillation can be controlled in real time. Another action can be apply on the front wheels. The drive of steering system (2 or 3 degrees) may also have a big impact on the oscillation compensation. A hydraulic damper is also used . The last possible action is to drive brakes by wires. An independent action on each brake of each wheel of the road train can also help to compensate the oscillation.

The detail of algorithms used to do this compensation is already exposed and is the subject of other publication [START_REF] Lucet | Stabilization of a road-train of articulated vehicles[END_REF].

From hardware point of view, the necessity to control each steering system, each motor and each brake of each vehicle means to be equipped with electrical components driven by a communication bus. For motors, and steering, that is not a big problem to find electronical components with communication bus. But for brakes, it was more complicated. A conventional brake system is piloted mechanically with the pedal. A hydraulic assistance is applied and distributed on each wheel (ABS system). For ESPRIT project, a specific brake by wire was designed by Continental. This brake can receive orders from CAN bus (controller area network) and manages independently the pressure in the brake of the four wheels.

Electrical architecture and power electronics 31. Electrical architecture

The electrical architecture (see Figures 4 and5) at the vehicle level is a standard one, in which we can find:

-Battery system, inverter and motor for the electric powertrain -Isolated DC/DC converter to the supply the 12V auxiliary network -And a 3kW embedded charger to charge the battery The electrical architecture at the road train level is more complicated because completely new functions must be implemented to ensure firstly the transfer of energy between vehicles (to charge batteries or to balance energy between batteries) and secondly the safety of the user.

Concerning the transfer of energy between vehicles, different solutions have been compared [START_REF] Garnier | Comparison between standard and innovative solutions to exchange energy between high energy storage systems[END_REF], but at the the preferred solution will be an AC bus associated with bi-directional chargers and contactors (see Figure 6) Concerning the safety, the most important is to ensure and monitor the ground continuity between vehicles.

There are different possibilities to exchange and transfer energy between vehicles (V2V) or from station to vehicles. For that it is possible to drive different embedded contactors and chargers located inside each vehicle. Vehicle number one is the master. It will be in charge of defining priorities, powers and thus the charging sequence.

The possible solutions (see fig. 6) are:

-Solution 1: Charge only one vehicle (red flow). Maximum charging power is 3kW -Solution 2: Charge N vehicles (red flows). Maximum charging power per vehicle is 7kW/N -Solution 3 : Exchange energy between vehicles (blue flow). Maximum power is 3kW

Power electronics and motors

At the beginning of the project, the design of the electric powertrain was one of the most important task. The main specifications can be summarized in the following items

For the battery:

-Minimum range of the single vehicle when loaded (between 600 and 800 kg) : 35 km -Life time objective without maintenance: 10 years -Small and specific volume due to the vehicle configuration -High voltage (300VDC) battery system to allow fast charge -Crash safety and waterproof

For the inverter, motor and gear:

-Loaded, the vehicle should reach the maximum speed of 65 km/h -Fully loaded the vehicle or the road train should be able to start in a slope of 13% -The vehicle or the road train should be able to be driven at 15 km/h in a slope of 20% during 2 minutes -Fully loaded the vehicle or the roadtrain should be able to reach a speed of 45km/h in less than 10 The battery pack Finally, the battery pack designed to fulfill the vehicle specifications was a Lithium Iron Phosphate battery system composed of an assembly of twelve modules in serial. Each module is also an assembly of 40 (8 serial 5 small cylindrical cells, with individual capacity of 2.3Ah. That gives a 300V 3.5kWh battery pack (see fig. 7 and8) allowing a maximum charging current of 35 A .

Inverter motor and gear

The choice of the best configuration for motors and inverters was performed thanks different tests at sub-systems level (see fig. 9 and 10). The objective was to measure power and thermal behaviors of different motors and inverters. Then thermal and losses models were created in order to evaluate maximum temperatures in function of power profiles, ambient temperature and also weight of the vehicle. With maximum temperatures limited around 115°C in the inverter and 150°C in the motor, we can optimize motor choice and gear ratio.

Project chronology -prototypes

Regarding the number of mechanical and electrical sub-systems to build this vehicle, the development was organized in several steps.

The first prototype contained the strict necessary elements to move the vehicle: battery pack, motors, inverters, charger, DC/DC converter, intelligent unit who manage the security and start auxiliaries, intelligent unit which manage motors, screen, "Drive Neutral Rear" buttons and a datalogger to monitor the communication bus. This first vehicle is constituted with a steel chassis (easy to realise) and had the objective to check and validate the concept and propulsion chain. This vehicle was sent in Germany in order to develop, check and validate the special brakes. This first vehicle was very important for project because it was the basis of the other vehicles, in which we made mainly adaptation and improvements.

A second prototype was done on the base of the first one with adaptations of mechanical parts. The assisted direction piloted by the intelligent unit which controls motors was added and tested. Then this vehicle was sent in Germany to include the automatic guiding system which helps the user to align the vehicle for coupling. Some mechanical modifications were done on central pivot.

A third prototype allows to develop the coupling device. Tests of traction was done in road train with prototype number 1 (updated to be setting in road train). Communication inter vehicle was also coded and validated.

The fourth prototype (see Fig. 13) allows to do a technological gap because chassis becomes in aluminum to replace steel and win mass. A road train (see Fig. 14) was done with prototypes 1, 3 and 4. This road train does tests of road holding and brake on a racetrack. To finish, prototype number six was delivered in last June. It integrates all functionalities developed until now. Prototypes 5 and 4 were updated to integrate all functionalities and allows to do a complete finish road train with 3 vehicles.

Fig. 1 :

 1 Fig. 1: ESPRIT concept

Fig. 5 :Fig. 3 :Fig. 4 :

 534 Fig. 5: -Synoptic of electrical architecture

Fig. 6 :

 6 Fig. 6: Energy transfer in the road train

Fig. 7 :Fig. 9 :Fig. 10 :Fig. 8 :

 79108 Fig. 7: Battery pack design

  of auto body and corrects some electrical and software bugs. Other equipments like lights, klaxon, and wiper are integrated and validated. For all of these equipment the communication between vehicles in road train is necessary. For

Fig. 12 :

 12 Fig. 12: Prototypes 1 and 3 in road train

Fig. 11 :Fig. 14 :

 1114 Fig. 11: First prototype

Demonstration

Since last august, several demonstrations were done in different European cities: Lyon, Glasgow and Hospitalet de Llobregat. Many medias [1], [2], [3], [4], [5] cover these events (Euronews, BBC news, Le progrès, France 3, Lyon première, The Herald, La tribune, Intelligent transport ou encore Bref Eco, …)


Olivier Devillers

Charles Duménil

Stochastic Analysis of Empty-Region Graphs

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

We start by defining the notion of empty-region graph [START_REF] Cardinal | Empty region graphs[END_REF]: Definition 1 For each pair (p, q) ∈ R d × R d , let R(p, q) be a family of regions. Consider a locally finite point set X ⊂ R d . We denote by G ∅ R (X) the graph on X in which p is a neighbor of q if and only if there exists an empty region in R(p, q). This notion unifies the classical Delaunay triangulation [START_REF] Delaunay | Sur la sphere vide[END_REF] where R(p, q) is the set of disks whose boundaries contains p and q, the Gabriel graph [START_REF] Gabriel | A new statistical approach to geographic variation analysis[END_REF] where R(p, q) is reduced to the disk of diameter pq, the βskeleton [START_REF] Kirkpatrick | A framework for computational morphology[END_REF][START_REF] Amenta | The crust and the β-skeleton: Combinatorial curve reconstruction[END_REF], the empty-ellipse graph [START_REF] Devillers | Empty-ellipse graphs[END_REF], the nearest neighbor-graph, the Θ-graphs, and the Yao graphs [START_REF] Yao | On constructing minimum spanning trees in k-dimensional spaces and related problems[END_REF].

In this paper, we will assume that X is a Poisson point process in the plane and compute quantities like the expected degree of a point p ∈ X in G ∅ R (X) or the probability that p has neighbors further than some threshold. Computing such quantities when R(p, q) is a singleton, as for the Gabriel graph, is much easier than when it is a bigger set, as for Delaunay triangulation. To this aim, it is interesting to try to get upper and lower bounds by comparing empty-region graphs. This idea was already used by Devroye, Lemaire and Moreau [START_REF] Devroye | Expected time analysis for delaunay point location[END_REF] to bounds the size of the Delaunay triangulation by the sizes of the Gabriel graph and the half-moon graph.

In this paper we formalize the process through two lemmas: the Combination lemma and the Partition lemma, and we illustrate these tools with empty-ellipse graphs. In a forthcoming paper we apply these results to equations of higher degree appearing when parameterizing 3D surfaces.

2 First Example: Delaunay and Gabriel Graphs

Delaunay Triangulation

The Delaunay triangulation is the empty-region graph where R(p, q) = {D(p, q, r); r ∈ R 2 } and D(p, q, r) is the open disk with p, q, and r on its boundary.

Although the expected degree of a random point in any kind of triangulation is well known to be 6 using Euler formula, we prove it using stochastic tools to illustrate the complexity of such a computation: Theorem 2 Let X be a Poisson point process with intensity λ in R 2 and p a point of R 2 . The expected degree E [deg(p, Del)] of p in the Delaunay triangulation Del(X ∪ {p}) is 6.

Proof. Without loss of generality, we assume that p is at the origin. Let D(p, q, r) denote the open disk with p, q, and r on its boundary. The number of neighbors of p in Del(X ∪ {p}) is the number of distinct sets {q, r} in X 2 with q = r such that D(p, q, r) does not contain any point of X. It is given by the random value: deg (p, Del) = 1 2 q∈X r∈X\{q} 1 [D(p,q,r)∩X=∅] , where the factor 1 2 corrected the double counting of each set {q, r} in the sum. We compute the expectation of this formula:

E [deg (p, Del)] = E   1 2 q∈X r∈X\{q} 1 [D(p,q,r)∩X=∅]   = 1 2 R 2 R 2 λ 2 P [D(p, q, r) ∩ X = ∅] drdq by Slivnyak-Mecke Theorem [8] = 1 2 R 2 R 2
λ 2 e -λ|D(p,q,r)| drdq by definition of Poisson point process.

The computation of this integral is a bit technical and is given in appendix. It involves a Blaschke-Petkantschin like variables substitution to turn the cartesian coordinates of q and r in the coordinate of the center of D(p, q, r) and two angles to place q and r on the boundary of D(p, q, r).

It finally turns out that the value of this integral is 6, as anticipated.

Gabriel Graph and Half-Moon Graph

We now turn our interest to cases where R(p, q) is a singleton. We consider the three following possibilities: R(p, q) = {Gab(p, q)} the disk of diameter pq, R(p, q) = {hm r (p, q)} the half-disk of diameter pq to the right of pq, and R(p, q) = {hm (p, q)} the half-disk of diameter pq to the left of pq. Then G ∅ {Gab} is the Gabriel graph, G ∅ {hmr} is the right half-moon graph, and

G ∅ {hm } is the left half-moon graph. The half-moon graph is G ∅ {hmr} ∪ G ∅ {hm } .
Lemma 3 Let X be a Poisson point process with intensity λ in R 2 and p a point of R

G ∅ {Gab} (X) is 4.
Proof.

E deg p, G ∅ {Gab} = E   q∈X 1 [Gab(p,q)∩X=∅]   = q∈R 2 λ P [Gab(p, q) ∩ X = ∅] dq = R 2 λe -λ| Gab(p,q)| dq = R + 2π 0 λe -λ πρ 2 4 ρ dθdρ = 4.
Lemma 4 Let X be a Poisson point process with intensity λ in R 2 and p a point of R 2 . Then

E deg p, G ∅ {hmr} = E deg p, G ∅ {hm } = 8.
Proof. By symmetry, we only do the computation for

E deg p, G ∅ {hmr} . E deg p, G ∅ {hmr} = E   q∈X 1 [hmr(p,q)∩X=∅]   = R 2 λe -λ|hmr(p,q)| dq = R + 2π 0 λe -λ πρ 2 8 ρ dθdρ = 8.
As one can see, the fact that R(p, q) is a singleton made the computation much simpler than in the case of the Delaunay triangulation.

Graph Relations

The following relations between the graphs are straightforward since any disk with p and q on its boundary contains either hm r (p, q) or hm (p, q) (see Figure 1):

G ∅ {Gab} = G ∅ {hmr} ∩ G ∅ {hm } , G ∅ {Gab} ⊂ Del ⊂ G ∅ {hmr} ∪ G ∅ {hm } = G ∅ {hmr,hm } .
From this, we deduce

deg p, G ∅ {Gab} ≤ deg (p, Del) ≤ deg p, G ∅ {hmr,hm } 4 ≤ deg (p, Del) ≤ 8 + 8 -4 = 12.
This result is weaker than the exact bound of Theorem 2 but the computations are much simpler. It also illustrates that, given regions of similar areas, the degree remains of equal order of magnitude. In that case, for two points p and q, Gab(p, q) and hm r (p, q) or hm (p, q) have both an area quadratic in the distance between p and q, and this induces a constant expected degree.

General Method

We propose a general method that both formalizes and generalizes the half-moon method to link the degree in general empty-region graphs to the degree in emptyregion graphs defined by singletons. We formalize the following facts: (i) the Delaunay disks can be parameterized by their center on the bisector of pq, (ii) this bisector can be partitioned in two rays at the midle of pq, and (iii) each half-moon is contained in all disks centered on one of the rays.

In a more general setting, the general idea is (i) to identify a parameter space in R k defining the regions, (ii) to partition this space in convex domains, and (iii) have inclusion relations for regions at the vertices of the partition.

The following lemma is instrumental for proving that if a set of region depends on k parameters and if the k-tuple of parameters belongs to a convex polyhedron P of R k then, if we want to prove that all regions parameterized by P contain a given region, it is enough to prove this inclusion for the regions parameterized by the vertices of P . If P is not bounded, we can extend the lemma to limit points at infinity: for a point c going to infinity along some ray of R k the region r c has a limit. The result also holds using this limit regions. We will show below as a didactic example how this lemma can be applied on Delaunay disks.

Lemma 5 (Combination Lemma) Let c ∈ R k and E c : R d → R such that for any x ∈ R d , c → E c (x)
is an affine function, and let r c be the region {x ∈ R d , E c (x) < 0}. Let P be a subset of R k , if c ∈ P , then r v v∈X (P )

⊂ r c , where X (P ) denotes the extreme points of the convex hull of P .

Proof. Consider two points a, b

∈ P ⊂ R k . Let x ∈ r a ∩ r b and, for t ∈ [0, 1], c t = (1 -t)a + tb be a point on [ab]. The function f : t → E ct (x) verifies f (0) = E a (x) < 0 and f (1) = E b (x) < 0. Since f is affine, for any t ∈ [0, 1], E ct (x) = f (t) = (1 -t)f (0) + tf (1) < 0, so x ∈ r ct . Thus r a ∩ r b ⊂ r ct for any c t on the edge [ab].
The extension from an edge [ab] to the convex hull of P follows directly from the its convexity.

We now show, as an example, that any Delaunay disk contains one of the two half-moons using the Combination lemma: Corollary 6 Let p, q two points in the Euclidean plane and D a disk with p and q on its boundary, then hm r (p, q) ⊂ D or hm (p, q) ⊂ D Proof. We choose the coordinate system so that p is the origin and q = (x q , y q ) with y q = 0. A disk D with p and q on its boundary can be parameterized by the inequality E c (x, y) :

x 2 -2xx c + y 2 -2yy c < 0 where c verifies y c = x 2 q -2xqxc+y 2 q 2yq
. Since this is actually the equation of the bisector line of [pq], the centers c = (x c , y c ) are the actual geometric centers of the disks. That provides a 1-dimensional family of disks parameterized by x c . In that parameterization, x c → E c (x, y) is an affine function.

Then we can consider the center c Gab of the Gabriel disk and the center c r at the infinity of the bisector line to the right of -→ pq; their associated regions are the Gabriel disk Gab(p, q) and the half-plane hp r (p, q) to the right of -→ pq. Since the ray [c Gab , c r ) is convex, we can apply the Combination lemma with (k, d) = (1, 2) to ensure that any disk whose center belongs to [c Gab , c r ) contains hm r (p, q); indeed hm r (p, q) = Gab(p, q) ∩ hp r (p, q). We apply the same reasoning for hm (p, q) to conclude that if a disk has p and q on its boundary, it contains either hm r (p, q) or hm (p, q) depending on the position of its center on the bisector line.

After the Combination lemma, the second ingredient of our demonstration scheme is the Partition lemma: Lemma 7 (Partition Lemma) Let G ∅ R be an emptyregion graph with R(p, q) = {r c ; c ∈ P ⊂ R k } a set of regions parameterized by c. Let (P i ) 1≤i≤n be a convex subdivision of P , the parameter space. Let R * i (p, q) = {r * i (p, q)} be n singletons. If ∀c ∈ P

i ; r * i (p, q) ⊂ r c then G ∅ R is a subgraph of ∪ 1≤i≤n G ∅ R * i and deg p, G ∅ R ≤ 1≤i≤n deg p, G ∅ R * i .
Proof. If pq is an edge of G ∅ R (X), according to Definition 1, there exists c ∈ P such that r c (p, q) ∩ X = ∅. Using the convex subdivision, there is some j such that c ∈ P j and r * j (p, q) ⊂ r c (p, q) by the hypothesis in the lemma. Thus r * j (p, q) ∩ X = ∅ and pq is also an edge of

G ∅ R * j .
Using these two lemmas, the general idea of the method we apply to compute an upper bound on the degree of a point in a given empty-region graph of a Poisson point process can be outlined as follows: (i) find a good affine parameterization of the regions to be able to apply the Combination lemma, (ii) find a good partition of the parameter space to be able to apply the Partition lemma, and (iii) analyze the size of the relevant empty-singleton-region graphs.

Empty Axis-Aligned Ellipse Graphs

In this section, we analyze empty-region graphs where the regions are axis-aligned ellipses. By "axis-aligned", we mean that their axes of symmetry are parallel to the x and y axes. We then call aspect ratio, the ratio of the lengths of the vertical axis to the horizontal axis of the ellipse.

Some Features of Axis-Aligned Ellipses

We give some explanations on the expression of ellipses we consider, and some properties that will be used thereafter. In R 2 , we consider an axis-aligned ellipse with the origin p on its boundary. We denote the ellipse r since it is seen as a region. Such an ellipse has three degrees of freedom, that can be set by considering a positive number α and a point c = (x c , y c ), so that r can be defined by the inequality:

r : α 2 x 2 -2xx c + y 2 -2yy c < 0.
In that parameterization, c is the affine parameter of r, and α its aspect ratio. To ensure that the boundary of the ellipse passes through a second point q, the three parameters x c , y c and α must satisfy: α 2 x 2 q -2x q x c + y 2 q -2y q y c = 0. Expressing α in terms of c and q, we define

E c (x, y) :=α 2 x 2 q x 2 -2xx c + y 2 -2yy c , (1)
with

α 2 = 2xqxc-y 2 q +2yqyc x 2 q .
The inequality E c (x, y) < 0 is an affine parameterization of r c (p, q), the only axis-aligned ellipse passing through p and q with c for parameter. We stress that c is not the usual geometric center of the ellipse.

In some proofs, we bring the expression back to its canonical form namely x 2 a 2 + y 2 b 2 -1 = 0, in which the ellipse has aspect ratio b a and area πab.

Proposition 8 For a given q ∈ R 2 , the parameters c of the ellipses r c (p, q) with same aspect ratio lie on a line perpendicular to (pq).

Proof. The aspect ratio is given by the coefficient of x 2 in Equation ( 1). The set of points c = (x c , y c ) that yields to a constant aspect ratio defines a line parallel to L : xx q + yy q = 0, by multiplying by

x 2 q
2 and omitting the constant terms in the expression of α 2 . This line is perpendicular to (pq).

Proposition 9 For a given q ∈ R 2 and for α ∈ R + , consider the ellipse r c (p, q) parameterized by c = (α 2 xq 2 , yq 2 ). The geometric center of r c (p, q) is the midpoint of [pq], and its area is π

4 αx 2 q + y 2 q α .
Proof. Transforming the equation E c (x, y) < 0 of r c (p, q) we get, its canonical form:

4α 2 α 2 x 2 q + y 2 q (x - x q 2 ) 2 + 4 α 2 x 2 q + y 2 q (y - y q 2 ) 2 -1 < 0.
We identify, with that expression, that r c (p, q) is the translated copy of an ellipse of center p, and area

π 4 αx 2 q + y 2 q α
by the vector 1 2 -→ pq. Details can be found in appendix.

Unbounded Aspect Ratio: Right-Triangle Graph

In this section, we prove, using our framework, a logarithmic bound for the empty axis-aligned ellipse graph of a Poisson point process in a bounded domain. A similar result was proven for a uniform distribution instead of a Poisson distribution [START_REF] Devillers | Empty-ellipse graphs[END_REF]. For two points p and q in R 2 , we consider the family Ell(p, q) of all axis-aligned ellipses with p and q on their boundaries. Assuming that p is the origin, we show that the expected degree of p in the associated emptyregion graph G ∅ Ell (X) is Θ(ln λ) when X is a Poisson process of intensity λ. In order to identify an upper bound, we consider the graph G ∅ {∆r,∆ } where ∆ r (p, q) (resp. ∆ (p, q)) denotes the axis-aligned right triangle with hypotenuse [pq] on the right (resp. left) side of -→ pq.

Lemma 10 G ∅ {∆r,∆ } is a super-graph of G ∅ Ell . c c r c ∞ L 0 L r, P r P c 0 c p q Figure 2:
Partition of space of parameters into {P , P r }.

Proof. For each region r c (p, q) ∈ Ell(p, q), we consider the parameterization r c (p, q) : E c (x, y) < 0 where E c is defined in Equation ( 1).

The space P ⊂ R 2 where c lives is delimited by the inequality: 2x q x c -y 2 q + 2y q y c > 0 that is the half-plane whose boundary is the line L 0 perpendicular to (pq) passing through c 0 = (0, yq 2 ) and that does not contain p (if c is not in this half-plane, the equation for α 2 has no positive solutions). With a small abuse of notation, we define two points at infinity at the two extremities of L 0 : c r to the right of -→ pq and c to its left. (see Figure 2). On the boundary L 0 of P , elliptic regions degenerate into parabolas. At point c 0 it degenerates to the horizontal strip r c0 = {|y-

yq 2 | < |yq| 2
} that can be seen as an ellipse with aspect ratio 0 (see Figure 3). At the limit when going at infinity along L 0 the parabola degenerates in two half-planes: r cr and r c bounded by (pq). This ensures, by the Partition lemma, that

G ∅ Ell ⊂ G ∅ {∆r,∆ } = G ∅ {∆r} (X) ∪ G ∅ {∆ } (X).
Now, we bound from above the expected degree of p in G ∅ {∆r} (X) or G ∅ {∆ } (X) when X is a Poisson point process with intensity λ.

The area of both triangles ∆ r and ∆ is |xqyq| 2 . Unfortunately, for any positive λ, R R e -λ|xy| dydx does not converge. In that case, we assume that X is distributed in a rectangle D = [-L, L] × [-l, l] for positive L and l.

Lemma 11 Let X be a Poisson point process with intensity

λ in D = [-L, L] × [-l, l]. The expected de- gree E deg p, G ∅ {∆r} of the origin p in G ∅ {∆r} (X) is Θ(ln λ + ln L + ln l) .
Proof. Let t be a positive number such that tLl > 1, we start by bounding from above the following integral:

I L,l (t) = L 0 l 0 e -txy dydx = L 0 lx 0 e -tu x dudx with u = xy = L 0 1 -e -tlx tx dx = 1 t tLl 0 1 -e -v v dv with v = tlx = 1 t 1 0 1 -e -v v dv + tLl 1 1 -e -v v dv ≤ 1 t 1 0 dv + tLl 1 1 v dv since 1-e -v ≤ min(1, v) = 1 t (1 + ln(tLl)) .
And bounding from below: has the smallest area among ellipses passing through p and q so that a far point keeps chance to be a neighbor of p as long as it is close to an axis.

I L,l (t) = 1 t tLl 0 1 -e -v v dv ≥ 1 t tLl 0 1 v + 1 dv since 1 -e -v v ≥ 1 v + 1 if v ≥ 0 = 1 t (ln(tLl + 1)) > ln(tLl) t .
Then we can give an upper bound on the expected degree:

E deg p, G ∅ {∆r} = E   q∈X 1 [∆r(q)∩X=∅]   = D λ(P [∆ r (q) ∩ X = ∅])dq = L -L l -l λe -λ|∆r(q)| dydx = 4λ L 0 l 0 e -λ xy 2 dydx = 4λI L,l ( λ 2 ) = Θ(ln(λLl)).
We can compare with the Delaunay triangulation, the Gabriel and the half-moon graphs. In those cases, the empty region had an area quadratic in the distance pq and the expected degree was constant. In this new case, the "xy" area provides a logarithmic degree.

Lemmas 10 and 11 give an upper bound on the degree of a point in the empty-axis aligned-ellipse graph. To get a lower bound, we will exhibit a subgraph of G ∅ Ell . In order to get a tight bound, the area of the chosen region must be Θ(x q y q ).

To find such an ellipse, that we name xy-ell(p,q), we use Proposition 9, with α = yq xq . We define xy-ell(p,q) to be the ellipse parameterized by c xy = y 2 q 2xq , yq 2 (see Figure 4), then according to Proposition 9, the area of this ellipse is π 2 x q y q .

Lemma 12 G ∅ {xy-ell} is a subgraph of G ∅ Ell .

Proof. Straightforward because xy-ell ∈ Ell.

Lemma 13 Let X be a Poisson point process with intensity

λ in D = [-L, L] × [-l, l]. The expected degree E deg p, G ∅ {xy-ell} of the origin p in G ∅ {xy-ell} (X) is Θ(ln λ + ln L + ln l).
Proof. As said above, for any q ∈ R 2 , the area of xy-ell(p, q) is π 2 x q y q . Then we can express the expected degree:

E deg p, G ∅ {xy-ell} = E   q∈X 1 {xy-ell(p,q)∩X=∅}   = D λe -λ|xy-ell(p,q)| dp = 4λ L 0 l 0 e -λπ xy 2 dydx = Θ(ln(λLl)).
The above lemmas allow to conclude:

Theorem 14 Let X be a Poisson point process with intensity λ in D = [-L, L] × [-l, l] and p the origin:

E deg p, G ∅ Ell = Θ(ln λ + ln L + ln l).

Bounded Aspect Ratio: Rhombus Graph

In the previous part, we proved that when the aspect ratio is not bounded, neither is the expected degree.

One can wonder what happens when the aspect ratio ranges between two finite numbers. For two points p and q in R 2 and a number β ∈ (0, 1), we consider the family Ell [β,1] (p, q) of horizontal elliptic regions with p and q on their boundary and whose aspect ratio ranges between β and 1. An important fact to be considered is that, when the aspect ratio is not bounded, a point q far from p could be a neighbor of p as long as it is close enough to the axis, since in that case, ellipses passing through p and q may have a small area, and that leads to a logarithmic bound. When the aspect ratio is bounded, all ellipses preserve an area Ω(x 2 q +y 2 q ), so that we expect a constant bound on the expected degree. In this section we will prove that it is actually the case and detail how this constant depends on β.

In order to apply the same method as above, we search for simple geometrical regions that fit inside the whole family Ell [β,1] (p, q).

A good choice is the following: as before, we consider the intersec-

p q hm [β,1] hm [β,1]
r tion of ellipses that are centered on the midpoint of [pq], and we cut the intersection along (pq). The remaining regions hm [β,1] r (p, q) and hm [β,1] (p, q) look like two axis-aligned right triangles with rounded sides for almost all q (see above figure). , hm

[β,1] } is a super- graph of G ∅ Ell [β,1] . Proof.
The proof is very similar to the one of Lemma 10, so we just spell out the important points. For each r c (p, q) ∈ Ell [β,1] (p, q), we consider the parameterization r c (p, q) : E c (x, y) < 0 defined in Equation ( 1) with β ≤ α ≤ 1.

The space P ⊂ R 2 where c lives is delimited by the inequality:

β 2 ≤ 2xqxc-y 2 q +2yqyc x 2
q ≤ 1 that is the strip perpendicular to (pq) whose boundary are the lines L β and L 1 , where L α = {(x, y), α 2 x 2 q = 2x q x -y 2 q + 2y q y}. We consider the segment defined by y = yq 2 inside P and its extremities c β on L β and c 1 on L 1 . We partition P into P r and P where P r is the part of P on the right of [c β , c 1 ), and P the part on its left (see Figure 5). c β and c 1 have for regions the ellipses r c β and r c1 with respectively β and 1 for aspect ratio. Furthermore, any parameter c r in P at infinity on the right of -→ pq has its region that degenerates into the half-plane bounded by (pq) on the right side of -→ pq (and the same holds for c and the left side). By the Combination lemma, if c ∈ P r then hm 6). By the Partition lemma, an edge of

[β,1] r := r c1 ∩ r c β ∩ r cr ⊂ r c and if c ∈ P then hm [β,1] := r c1 ∩ r c β ∩ r c ⊂ r c (see Figure
G ∅ Ell [β,1] is an edge of G ∅ hm [β,1] or G ∅ hm [β,1]
. The problem now is that it may be complicated to compute an integral involving the area of hm [β,1] r (p, q) or hm [β,1] (p, q). To solve this issue, we consider a strictly smaller region. We could have used the axis-aligned right triangles ∆ r and ∆ but their areas do not respect the order of magnitude (as illustrated by Figure 7). A more suitable region is what we call the half-rhombus. We define the rhombus Rh β (p, q) as the one whose vertices are the horizontal extreme points of r c1 (p, q) and the vertical extreme points of r c β (p, q). Then we separate it into two halves Rh β r (p, q) and Rh β (p, q), delimited by (pq). By convexity, it is clear that Rh β r (p, q) ⊂ (p, q) and Rh β (p, q) ⊂ hm [β,1] (p, q) (see Fig-

ure 6). Finally, we can say that

G ∅ {Rh β r } is a super- graph of G ∅ {hm [β,1] r } and that G ∅ {Rh β } is a super-graph of G ∅ {hm [β,1] } .
Before proceeding to the computation of the expected degree, we introduce a lemma that provides properties on the involved integral.

Lemma 16 Let t > 0, β ∈ (0, 1) and

I β (t) = R R e -t √ (x 2 +y 2 )(β 2 x 2 +y 2 ) dydx, I β (t) = 1 t I β (1) ≤ π t 1 + ln( 1 β ) .
Proof. The computations are in appendix.

Lemma 17 Let X be a Poisson point process with intensity λ in R 2 , and β ∈ (0, 1).

E deg p, G ∅ {Rh β r } = O(ln 1 β ).
Proof. We first compute the area of the rhombus Rh β (p, q). We identify its width and height as being respectively x 2 q + y 2 q and β 2 x 2 q + y 2 q so that the value of its area is given by 1 2

x 2 q + y 2 q β 2 x 2 q + y 2 q .

E deg p, G ∅ {Rh β r } = E   q∈X 1 [Rh β r (p,q)∩X=∅]   = R 2 λ P Rh β r (p, q) ∩ X = ∅ = R 2 λe -λ 1 2 |Rh β (p,q)| dq = R R λe -λ 4 (x 2 q +y 2 q )(β 2 x 2 q +y 2 q ) dydx = λI β λ 4
≤ 4π (1 -ln(β)) by Lemma 16.

We obtain a tight lower bound, when β goes to 0, by identifying, for each q, a particular region, named β-ell(p, q), such that β-ell(p, q) is or contains an element of Ell [β,1] (p, q). To achieve this, we partition the plane into two parts (see Figure 8):

1. if q ∈ D β := {(x, y), β|x| < |y| < |x|}, then,

as in

Lemma 12, we define β-ell(p, q) = xy-ell(p,q), 2. otherwise β-ell(p, q) = R 2 , that is another way to say that q is not a neighbor of p.

Lemma 18 G ∅ {β-ell} is a subgraph of G ∅ Ell [β,1] .
Proof. If q ∈ D β , we have to prove that β-ell(p, q), i.e. xy-ell(p,q), is in Ell [β,1] . This is true because the aspect ratio of if xy-ell(p,q) is | yq xq |, and verifies

β < | yq xq | < 1 if β|x q | < |y q | < |x q |.
Otherwise, it is clear that β-ell(p, q), i.e. R 2 , is larger than any other ellipse from Ell [β,1] .

Lemma 19 Let X be a Poisson point process with intensity λ in R 2 .

The expected degree

E deg p, G ∅ {β-ell} of the origin p in G ∅ {β-ell} (X) is Ω(ln 1 β ).
Proof. β-ell(p, q) is actually chosen to simplify the computation.

Recall that D β is the domain {(x, y), β|x| < |y| < |x|};

E deg p, G ∅ {β-ell} = E q∈X 1 [β-ell(p,q)∩X=∅] = R 2 λ P [β-ell(p, q) ∩ X = ∅] dp = D β λ P [xy-ell(p, q) ∩ X = ∅] dp + R 2 \D β 0 dp = D β λe -λ|xy-ell(p,q)| dp = 4 ∞ 0 x βx λe -λπ xy 2 dydx = 4 π 4 tan -1 (β) ∞ 0 λρe -λπ ρ 2 cos(θ) sin(θ) 2 dρdθ = 4 π 4 tan -1 (β) λ 1 λπ cos(θ) sin(θ) dθ = 4 π ln(tan( π 4 ))-ln(β) since d dθ ln(tan(θ)) = 1 cos(θ) sin(θ) = 4 π ln( 1 β ).
We can finally conclude using Lemmas 15 to 19:

Theorem 20 Let X be a Poisson point process with intensity λ in R 2 . The expected degree

E deg p, G ∅ Ell [β,1] of the origin p in G ∅ Ell [β,1] (X ∪ {p}) is Θ ln( 1 β ) .
If β > 1 we have by symmetry,

E deg p, G ∅ Ell [1,β] = E deg p, G ∅ Ell [ 1 β ,1]
= Θ(ln β).

Non-Axis-Aligned Ellipses

We turn our interest to the case of non-axis-aligned ellipses. We consider the graph in which two points p and q are neighbors if there exists an empty ellipse passing through p and q whose aspect ratio is between β and 1, for β ∈ [0, 1]. For two points p and q, we define the family Ell (p, q) of all ellipses with such aspect ratios passing through p and q, and G ∅ Ell The case where β = 1 corresponds to the Delaunay triangulation, and the case where β = 0 corresponds to the complete graph, since we can consider that a segment between two points is an ellipse with aspect ratio 0 and random points are in general position. Thus we assume that β ∈ (0, 1). Consider two points, p at the origin and q, and an ellipse E passing through p and q. Since E is not anymore axis-aligned but has its great axis in some direction θ, we can consider the regions hm

p q s s r hm [β,1] ,θ hm [β,1] r,θ E θ C β
[β,1] r,θ
and hm

[β,1] ,θ
as in the previous sections but parameterized by direction θ. Clearly the circle C β centered at the midpoint of pq and of diameter β|pq| is inside hm

[β,1] r,θ ∪ hm [β,1]
,θ

(see Figure 9). Consider the isosceles triangles pqs r and pqs such that s , s r ∈ C β with s r on the right of -→ pq and s on its left. Then pqs r ⊂ hm

[β,1]
r,θ and pqs ⊂ hm

[β,1]
,θ . Since this is true for any ellipse, we can assume that any ellipse whose aspect ratio is between β and 1 and passing through p and q contains either pqs r or pqs . Notice that these triangles are independent of the direction θ. So we can apply the Partition lemma to yield that G ∅

Ell [β,1] * is a subgraph of G ∅ {pqsr,pqs } .
Now we consider a Poisson point process X of intensity λ, and we compute an upper bound on the expected degree of p in G ∅

Ell

[β,1] * (X ∪ {p}). E deg p, G ∅ {pqsr,pqs } ≤ 2 E   q∈X 1 [pqsr∩X=∅]   = 2 R 2 λP [pqs r ∩ X = ∅]dq = 2 R 2 λe -λ|pqsr| dq = 2 2π 0 ∞ 0 λe -λ 8 βρ 2 ρ dρdθ = 16π β .
On the other hand, among the ellipses passing through p and q, we can choose the ellipse E β * whose great axis is [p, q] and has aspect ratio

β to obtain a subgraph of G ∅ Ell [β,1] * .
The expected degree of p in this graph is

E deg p, G ∅ {E β * } = E q∈X 1 [E β * ∩X=∅] = R 2 λ P [E β * ∩ X = ∅] dq = R 2 λe -λ|E β * | dq = 2π 0 ∞ 0 λe -λ π 4 βρ 2 ρ dρdθ = 4 β
We deduce the following theorem:

Theorem 21 Let X be a Poisson point process in R 2 .

The expected degree of the origin

p in G ∅ Ell [β,1] * (X ∪ {p}) is Θ 1 β .

Probability of Existence of Far Neighbors

At some point, for a given graph G and a positive number t, we may be interested in computing the probability for p to have a neighbor in G at a distance greater than t.

As before, for illustration on a simple case, we start by the Delaunay triangulation:

Lemma 22 Let X be a Poisson point process with intensity λ in R 2 , p a point of R 2 , and t a positive number. The probability that p has some Delaunay neighbor at a distance greater than t is smaller than 8e -λ

√ 2 8 t 2 .
Proof. If q is a Delaunay neighbor of p, Let σ be an empty disk whose boundary passes through p and q. If q is at distance greater than t from p, then the diameter of σ is obviously also greater than t, so its homothet σ toward p that has exactly diameter t is included in σ and consequence empty. Consider the triangle with vertices p, ( √ 2 2 t, 0), and ( 1 2 t, 1 2 t) and its seven adjacent copies around p (see Figure 10). We name them τ i for i ∈ {1, . . . , 8}. Their area is |τ 1 | = √ 2 8 t 2 . One can notice that, at least one triangle is included in σ : the one whose angular sector from p contains the center of σ .

So we get:

P [∃q ∈ X; [pq] ∈ Del(X ∪ {p}) | |pq| > t] ≤ P [∃i ∈ [1, . . . , 8], τ i ∩ X = ∅] ≤ i=1...,8 P [τ i ∩ X = ∅] = 8 P [τ 1 ∩ X = ∅] = 8e -λ √ 2 8 t 2 .
We establish in the next lemma a similar bound for the empty axis-aligned ellipse graph with bounded aspect ratio in [β, 1]. We are mainly interested in the behavior of the probability when β is small, thus we assume β < 1 2 .

Figure 10: If |pq| > t, any disk passing through p and q contains one of the 8 triangles.

Lemma 23 Let X be a Poisson point process with intensity λ in R 2 , p a point of R Proof. The proof idea is similar to the previous one, except that we apply a homothety on the empty ellipse σ until its image σ fits inside the axis-aligned square inscribed in the circle of radius t (see Figure 11). We consider eight triangles (tr i ) 1≤i≤8 , that have the property that for any ellipse σ, σ contains one of them.

To this aim we define the four points

v 1 =( 1 2 t, 0), v 2 =( √ 2 4 t, √ 2 4 βt), v 3 =( √ 2β 4 t, √ 2
4 βt), v 4 =(0, 1 2 βt).

The triangles tr 1 and tr 2 are respectively pv 1 v 2 and pv 3 v 4 . Their respective areas are √ 2 16 βt 2 and √ 2 16 β 3 2 t 2 . We will show that any ellipse tangent to the square in the upper right quadrant contains tr 1 or tr 2 . We complete the set of triangles by their symmetrical copies with respect to the x-axis, to the y-axis and to the point p, and name them according to the trigonometric order Figure 11: If |pq| > t, any ellipse passing through p and q contains one of the 8 triangles. from tr 1 to tr 8 to cover the ellipses tangent to other parts of the square.

Without loss of generality, we assume that the center c of σ is in the upper right quadrant. In such a case, the right most point of σ has abcissa √ 2 2 t, its left most point has negative abcissa, and its center verifies 0 ≤ x c ≤ √ 2 4 t. As long as x c ≥ 1 4 t, using the symmetry of the ellipse with respect to its vertical axis, v 1 is between p and the symmetric of p, and thus is inside σ 0 . We prove that such ellipses, with x c ≥ 1 4 t, also contain v 2 . Actually v 2 is chosen as the highest point of the thinnest ellipse of center c = ( √ 2 4 0) (in yellow on Figure 11), with aspect ratio β. Since the abscissa of v 2 is between p and v 1 , moving the center c upward or to the left or increasing β imply that v 2 remains inside σ . So as long as x c ≥ 1 4 t, the triangle tr 1 is inside σ . Suppose now that x c ≤ 1 4 t. The ellipse σ , if its aspect ratio is α, has equation:

α 2 x 2 -2α 2 xx c + y 2 -2yy c ≤ 0.
For a fixed α, the lowest center is reached when x c = 1 4 t and since σ is tangent to the right side of the square at ( √ 2 2 t, y c ), by substitution we have:

1 2 α 2 t 2 - √ 2α 2 t 1 4 t -y 2 c = 0.
Thus y c is minimized for α = β, and so:

y c = 1 2 2 - √ 2 βt 0.383βt.
We can deduce, by symmetry with respect to the horizontal axis of σ , that all those ellipses contain the segment between p and (0, 2 -√ 2 βt), including v 4 . To prove that v 3 ∈ σ , we make a distinction between the side of tangency of σ . We call contact point of an ellipse, the point of the ellipse in which it is tangent to the square, for σ we name it q . Suppose first that σ is tangent to the right side of the square. We consider the two extreme ellipses σ high and σ low , with highest and lowest contact points q high and q low , at respectively v3 ∈ σ low :

β 2 √ 2β 4 t 2 -β 2 √ 2β 4 t t 4 + √ 2 4 βt 2 -2 √ 2 4 βt √ 2- √ 2 2 βt = β 2 t 2 β 8 - √ 2β 16 + 1 8 - √ 4-2 √ 2 8 ≤ 0 for β ≤ 0.6 v3 ∈ σ high : √ 2β 4 t 2 -0 + √ 2 4 βt 2 -2 √ 2 4 βt √ 2 4 t = βt 2 1 8 + β 8 -2 8 ≤ 0 since β ≤ 1
We call bottom part of the ellipse, the counterclockwise arc from p to the contact point, and top part the following arc from the contact point to the intersection with the y-axis. We show that the bottom part of σ is below the bottom part of σ high . We apply a vertical affine transformation that flattens σ high until its contact point becomes q . The new ellipse clearly has its bottom part lower since the transformation lowered every point. Then we shift horizontally the center into c , maintaining the points p and q . Since that makes the aspect ratio grow, here again we lowered the bottom part. So the bottom part of σ is below the bottom part of σ high . On the other hand we apply a homothetic transformation on σ low centered on its contact point such that the length of the horizontal axis is the same as the length as σ , followed by a vertical translation until the contact point coincides with q , finally completed by a vertical affine transformation that makes it reach the correct aspect ratio, that is greater. All these transformations make the upper part of the ellipse go upward. We deduce that any ellipse tangent to the right side of the square and whose center has abscissa smaller than 1 4 t contains tr 2 . Then we can go to ellipses tangent to the top side of the square. The proof is quite identical so we do not develop it but keep in mind that the important point is that v 3 belongs to circle centered at (0, √ 2 4 t) because v 3 lies on the parabola y = 2 √ 2 t x 2 , that is above the circle for y < √ 2 4 t. Above arguments proved that any ellipse whose center is in the upper right corner of the triangle contains either (pv 1 v 2 ) or (pv 3 v 4 ). By extension, we deduce that any ellipse contains at least one of the 8 triangles tr i .

So we get:

P [pq] ∈ G ∅ Ell [β,1] (X) | |pq| > t ≤ P [∃i ∈ [1, . . . , 8], tr i ∩ X = ∅] = 4 (P [tr 1 ∩ X = ∅] + P [tr 2 ∩ X = ∅]) = 4 e -λ √ 2 16 βt 2 + e -λ √ 2 16 β 3 2 t 2 = Θ(e -λ √ 2β 3 2 ( t 4 ) 2 )
We factorize by the right factor, = ρ 3 (-cos ϕ cos θq -cos 2 θq -sin ϕ sin θq -sin 2 θq)

• (-sin ϕ cos θr + cos ϕ sin θr) + (cos ϕ cos θr + cos 2 θr + sin ϕ sin θr + sin 2 θr)

• (-sin ϕ cos θq + cos ϕ sin θq) = ρ 3 (-cosϕ cosθq -sinϕ sinθq -1) (-sinϕ cosθr +cosϕ sinθr) +(cosϕ cosθr +sinϕ sinθr +1) (-sinϕ cosθq +cosϕ sinθq) .

We distribute the 1, develop, and many terms cancel each other, = ρ 3 sin θq cos θr -cos θq sin θr + sin ϕ cos θr -cos ϕ sin θr -sin ϕ cos θq + cos ϕ sin θq . Dividing by α 2 x 2 q +y 2 q 4

, another equation of rc(p, q) is: 4α 2 α 2 x 2 q + y 2 q (x -xq 2 ) 2 + 4 α 2 x 2 q + y 2 q (y -yq 2 ) 2 -1 < 0.

We identify, with that expression, that rc(p, q) is the translation by the vector 1 2 -→ pq of the ellipse defined by: 4α 2 α 2 x 2 q + y 2 q

x 2 + 4 α 2 x 2 q + y 2 q y 2 -1 = 0, whose center is p, and area is π

4 αx 2 q + y 2 q α .
Lemma 16 Let t > 0, β ∈ (0, 1) and

I β (t) = R R e -t (

Figure 1 :

 1 Figure 1: The Gabriel graph, on the left, is included in the Delaunay triangulation, in the middle, itself included in the half-moon graph, on the right.

Figure 3 :

 3 Figure3: Ellipses corresponding to points in Figure2. They all contain either ∆ r or ∆ . Regions whose parameter are on L r, are in purple, they range from r c0 (in yellow), to rc ∞ . Any region whose parameter is in P , like r c , contains ∆ .

Figure 4 :

 4 Figure 4: Left: Some xy-ell ellipses whose color depends of the aspect ratio. Right: An instance of G ∅{xy-ell} where p is the central (red) point. For any point q, xy-ell(p, q) has the smallest area among ellipses passing through p and q so that a far point keeps chance to be a neighbor of p as long as it is close to an axis.

Figure 5 :

 5 Figure 5: Partition of space of parameters into {P , P r } in the bounded aspect ratio case.

Figure 6 :

 6 Figure 6: The green ellipse with parameter to the right of -→ pq contains Rh β r .

Figure 7 :

 7 Figure 7: The area of ∆ r and Rh β r can have different order of magnitude.

Figure 8 :

 8 Figure 8: Some β-ell ellipses for points in D β and an instance of G ∅ {β-ell} where p is the red point. A far point reduces strongly its probability to be a neighbor of p because it cannot anymore be close to the axes.

Figure 9 :

 9 Figure 9: Non-axis-aligned ellipses.

  . They both contain v 3 :

Finally we apply the 2 sin θq -ϕ 2 sin ϕ-θr 2 ,λ 2 eR 2ρ 3 λ 2 e 2 sin 2 × 2π × 3π = 6 . 9 9 2 -2xxc + y 2 -2yyc with α 2 = 2xq xc-y 2 q +2yq yc x 2 q. 2 ) 2 + (y - yq 2 ) 2 -α 2 x 2 q 4 - y 2 q 4 .

 22222226992222222244 formulae: cos a sin b -cos b sin a = sin(a -b), on the three well-chosen pairs of terms, = ρ 3 (sin(θq -θr) + sin(θq -ϕ) + sin(ϕ -θr)) = ρ 3 (sin(π -(θq -θr)) + sin(θq -ϕ) + sin(ϕ -θr)) = 4ρ 3 sin π-(θq -θr ) where the last line derives from the formula: sin a + sin b + sin c = 4 sin a 2 sin b 2 sin c 2 when a + b + c = π. So that we get: -λπρ 2 |det (J(ρ, ϕ, θq, θr))| dθrdθqdϕdρ = translation (θq, θr) → (θq -π -ϕ, θrπ -ϕ) applied in the (2π, 2π)-periodic function (θq, θr) → sin π-(θq -θr ) Proposition For a given q ∈ R 2 and for α ∈ R + , consider the ellipse rc(p, q) parameterized by c = (α 2 xq 2 , yq 2 ). The geometric center of rc(p, q) is the midpoint of [pq], We note Ec(x, y) := α 2 x If yc = yq 2 , then α 2 x 2 q -2xqxc = 0, and so xc = α 2 xq 2 . Ec(x, y) = α 2 x 2 -2xxc + y 2 -2yyc = α 2 (x 2 -xxq) + y 2 -yyq = α 2 (x -xq

(x 2 +y 2 )(β 2 x 2 +y 2 (x 2 +y 2 )(β 2 x 2 +y 2 re -r 2 √ β 2 cos 2 θ+sin 2 θ drdθ = 2 π 2 0β 2 π 2 ], β 2 cos 2 θ + sin 2 θ - 1 2 is smaller than both 1 β 1 2 decreases from 1 β to 1 , 1 2≥

 22222222211111 cos 2 θ + sin 2 θ and π 2θ ; on the one hand, because β 2 cos 2 θ + sin 2 θ on the other hand, because β 2 cos 2 θ + sin 2 θ sin θ ≥ 2 π θ, so that: π (1 -ln(β)) .

  2 , t and β two positive numbers with β <1 2 . The probability that p has some neighbor in G ∅ Ell [β,1] (X) at a distance greater than t is smaller than 4 e -λ

	√ 2 16 βt 2 + e -λ	√ 2 16 β	3 2 t 2 .

  x 2 +y 2 )(β 2 x 2 +y 2 ) dydx, (x 2 +y 2 )(β 2 x 2 +y 2 ) dydx

	I β (t) =	1 t	I β (1) ≤	π t	1 + ln( 1 β ) .
	Proof. of Lemma 16
	We apply, in the integral, the variables substitution:
	(x, y) = ( 1 √ t X, 1 √ t Y ) with Jacobian determinant 1 t .
	I β (t) = -t = R R e R R 1 t e

-(X 2 +Y 2 )(β 2 X 2 +Y 2

Acknowledgements

The authors thank Sylvain Lazard for its relevant suggestions.

This research has been funded by grant ANR-17-CE40-0017 of the French National Research Agency (Aspag)

Appendix

Proof. Integral for Theorem 2

We have to compute

λ 2 e -λ|D(p,q,r)| drdq.

We use a Blaschke-Petkantschin like variables substitution [8, Theorem 7.2.7] from R 4 to R + × [0, 2π) 3 , to express the parameterization of q and r into (ρ, ϕ, θq, θr) where (ρ, ϕ) denotes the polar coordinates of the center c of the circle circumscribing p, q, and r, and θq and θr denote the angles of the points q and r from c to the horizontal line (see Figure 12). xq = ρ(cos ϕ + cos θq), yq = ρ(sin ϕ + sin θq), xr = ρ(cos ϕ + cos θr), yρ = r(sin ϕ + sin θr).

The Jacobian matrix J of the transformation can be written:

and has the following determinant: Figure 12: The Blaschke-Petkantchin variables substitution converts the Cartesian coordinates of q and r into polar coordinates related to the circle circumscribing p, q and r.


                                  
Le questionnaire quanti-qualitatif s'intéresse au développement formel de 31 projets ENR codév. Ex. : 1 e réunion publique. La recherche CITENR s'intéresse au processus de coopérations multi-acteurs, en considérant les dimensions territoriales et de régulation des projets ENR codéveloppement sur les phases allant du développement jusqu'à la construction. Nous ne traitons pas l'étape de l'exploitation. La couleur du contour du triangle représente la posture affirmée publiquement (dans le conseil de gestion). La couleur intérieure concerne l'autre étiquette moins affirmée.

1.Émergence

                                                                                         
                                         
Cet exercice est tiré du manuel de Recherche Action Participative de J. Chevalier page 69. Le but de l'outil est de comprendre les facteurs qui contribuent à maintenir une situation et ceux qui en atténuent la portée qu'il s'agisse d'un problème, d'une activité ou d'un projet.

   

L'Analyse de domaine ou de construits est un outil de co-construction en groupe de sens. L'outil en question, est adapté afin de conférer toute sa force à la recherche-action collaborative.

  qualitative (entretiens + monographies) porte sur le développement informel et formel de 3 projets ENR codév. Ex. Rencontre de 2 acteurs lors d'une foire.

  mise en oeuvre d'un projet d'énergies renouvelables se déroule schématiquement en quatre étapes, sachant que la taille des flèches n'est pas représentative du temps nécessaire à chaque étape. EP, juin 2020, 282 projets. Source : BDD CITENR, juin 2020, 42 projets.



  Il ne s'agit pas d'une lecture chronologique ou linéaire des évolutions du champ des EnR. & Spécialiste des ENR Directeur de la PME en charge du codéveloppement Les postures ambivalentes des personnes sont indiquées par des triangles bicolores. La couleur du contour du triangle représente la posture affirmée publiquement (dans le conseil de gestion). La couleur intérieure concerne ldes personnes sont indiquées par des triangles bicolores.

  des partenaires d'un projet d'EnR en codév 3 ....

Figure 1 Figure 21 :

 121 Figure 1 Schéma des acteurs impliqués dans projets d'EnR de codév311 Figure 2 : La diversité des formes de codéveloppement ..................................................................... 13 Figure 3: Une analyse chronologique de l'évolution du champ des EnR multi-acteurs entre 2000 et 2020 ....................................................................................................................................................... 15 Figure 4 Méthodologies de recherche mixte croisant recherche conventionnelle, participative et en ligne ....................................................................................................................................................... 21 Figure 5 : Phases d'analyse des projets ENR multi-acteurs ................................................................. 31 Figure 6 Les 3 piliers de la démarche de RAP et les moyens habiles ou savoirs de la RAP. .............. 35 Figure 7 Chronologie des ateliers participatifs organisés pour le projet CIT'EnR de février à décembre 2020 ....................................................................................................................................................... 40 Figure 8 Répartition de la participation aux ateliers RAP par type d'acteurs pour la totalité des ateliers et par grands groupes d'atelier (1,2, 3 et 4). ......................................................................................... 41 Figure 9 Répartition des acteurs par localisation géographique. (sur la base des inscriptions). .......... 42 Figure 10 Nom et nature des projets représentés dans les ateliers. .................................................... 43 Figure 11 Combinaison des 3 outils numériques pour organiser des ateliers en distanciel.. ............... 43 Figure 12 Aperçu des comptes rendus dynamiques réalisés pour les 7 ateliers RAP.. ....................... 44 Figure 13 Graphique des visites sur le site recoupant le nombre de (nouveaux) visiteurs, le nombre de connexions et le nombre de pages consultées. Source : Statcounter -statistique des connexions sur le site du projet CIT'EnR -2020 ...................................................................................................... 44 Figure 14 Guide de prise en main du support MURAL.. ....................................................................... 45 Figure 15 Extrait du tutoriel d'accompagnement à l'utilisation des outils interactifs collaboratifs CIT'EnR.. ............................................................................................................................................... 45 Figure 16 L'état d'avancement des projets EnR.................................................................................... 48 Figure 17 : Ecosystème d'acteurs du Projet 1....................................................................................... 61 Figure 18 : Ecosystème d'acteurs du Projet 2....................................................................................... 62 Figure 19 : Ecosystème d'acteurs du Projet 3....................................................................................... 63 Figure 20 Régions concernées par les ateliers interrégionaux. Et noms des projets représentés à ces ateliers.. ................................................................................................................................................. 80

Figure 38 :

 38 Figure 21 : Origines des subventions bénéficiant aux projets codév3 .................................................. 84 Figure 22 Les dispositifs publics favorables aux projets codév3 ......................................................... 84 Figure 23 : Les relations réciproques entre projets codév3 et dispositifs publics ................................. 85 Figure 24 Freins et leviers au déploiement du codév3 qui ont émergé des ateliers de la région Occitanie et regroupés en familles de facteurs. ................................................................................... 86 Figure 25 : Les freins/leviers au déploiement du codév3 dans une vision interrégionale. .................... 88 Figure 26 : Principaux freins et leviers interrégionaux sur lesquels les acteurs indiquent avoir un degré de contrôle élevé.. ................................................................................................................................. 89 Figure 27 : Thématiques prioritaires à traiter en plan d'action (vote des participants) ......................... 90 Figure 28 : Plan d'actions pour la région Occitanie -par groupes thématiques ................................. 91 Figure 29 -Plan d'Actions produit en atelier Interrégional pour développer le codév 3 et complété à partir du plan d'action Occitanie ............................................................................................................ 92 Figure 30 Plan d'action .......................................................................................................................... 93 Figure 31 -Synthèse des enjeux territoriaux (spécificités régionales, conditions d'émergence et déploiement, freins et leviers du codév3) .............................................................................................. 94 Figure 32 Kit de familiarisation aux outils co-produits en atelier et permettant de se scorer en ligne.. 95 Figure 33 Les 4 outils co-construits dans les ateliers et mis à disposition sur le site pour « scoring » et enrichissement. ..................................................................................................................................... 96 Figure 34 : Fiche d'explication de la roue socratique issue du tool kit de Jacques Chevalier. ........... 96 Figure 35 : La roue d'autoévaluation finale du Codév3. ........................................................................ 97 Figure 36 : Présentation des critères et de leur déclinaison en sous critères constituant la roue du codév3 ................................................................................................................................................... 98 Figure 37 : Construction de la roue du Codév3 (à gauche). Du remue-méninge à la formalisation d'une roue socratique. (à droite). .................................................................................................................... 99 Figure 38 : Roues remplies, notées et commentées par les porteurs de projet. ................................ 101 Figure 39 Séquence d'auto-évaluation des projets présents -atelier inter-régional n° 4 ................... 102 Figure 40 Autoévaluation en ligne sur Mural via le site Internet par 3 porteurs de projet. ................ 103 Figure 41 Le champ de force. Fiche technique issue du guide de J. Chevalier. ................................ 104 Figure 42 Champ de force de la région Occitanie construit lors de l'atelier 2A2B. ............................. 105 Figure 43 Champ de force généré par les participants des ateliers 2A2B3A et 3B -périmètre interrégional (7 régions françaises). ........................................................................................................... 106 Figure 44 Nouvelles colonnes de facteurs positifs à gauche ou négatifs à droite créées par les participants autres que venant de l'Occitanie...................................................................................... 107 Figure 45 Organisation du champ de force. Nombre de postit par colonne de facteurs. ................... 108 Figure 46 : Atelier Occitanie. Degré de contrôle perçu par les participants sur chacun des facteurs. Réalisé à partir de l'exercice Champ de Force ou freins et des leviers .............................................. 109 Figure 47 le plan d'action tel que présenté dans le kit de démonstration. .......................................... 110 Figure 48 Le plan d'action final. Qui fusionne le plan de l'atelier Occitanie et celui de 'interrégional. En vert les leviers et en rouge les freins déclinés en action et sous actions. ........................................... 110 Figure 49 Actions et sous actions développées dans les groupes thématiques « potentiel et ancrage dans la coopération ». et dans « implication locale et défiance .......................................................... 111 Figure 50 Analyse de Domaine. Fiche technique ............................................................................... 112 Figure 51 : Matrice des construits produite en atelier Occitanie. Scores fictifs en vue d'une démo. 113 Figure 52 : Analyse de construit pour CIT ENR. Carte, cluster et dendrogramme produit avec le logiciel RepGrid. .................................................................................................................................. 114 Figure 53 Les critères de la roue codév3 ............................................................................................ 115 Figure 54 Analyse des familles ........................................................................................................... 116 Figure 55 -Grandes étapes du dernier atelier n°4 .............................................................................. 117 Figure 56 : Une approche multiniveau du secteur de l'énergie : les projets EnR multi-acteurs comme marges instituantes ............................................................................................................................. 120

Implication de la société d'aménagement de l'agglomération, du syndicat d'énergie et de la société d'économie mixte rattachée à l'AREC

	Les opposant.es : Association locale AMAP Collectif de Transition Ce qui a freiné le codéveloppement Comité éolien Mairie Association locale 1 éolienne Pôle d'Equilibre Ce qui a favorisé le codéveloppement Territorial et Rural Propriétaires terriens qui ont accepté offre développeur privé-industriel AMAP : Association pour le Maintien de l'Agriculture Paysanne AREC Projet de 5 éoliennes Communauté ECLR EPI EPA URScop Abréviations ECLR : Energies Citoyennes Locales et Renouvelables en Occitanie URScop : Union Régionale des Sociétés Coopératives et Participatives EPI : Energie Partagée Investissement EPA : Energie Partagée Association Enercoop Soutien au projet, Accompagnement AREC : Agence Régionale Energie Climat Société de Projet (SAS développeur) Incubateur d'innovation sociale en Midi-Pyrénées 1 PME d'agglomérations Les habitant.es du territoire Entreprises locales Société de projet Enercoop Les opposant.es : Association locale de riverain.es SEM MPEI AREC Syndicat d'énergie Société d'aménagement de la communauté d'agglomération Coopératives agricoles locales Grande entreprise du CAC 40 -propriétaire Société citoyenne Parc naturel régional ECLR Agence locale de l'énergie et du climat URScop Précisions légendaires Raccordement Légende Catégories d'acteurs Collectivités territoriales Développeur privé industriel Légende déterminant Enedis Autres acteurs Catégories d'acteurs Collectivités territoriales Société civile Accompagnement/habitant.es Associations professionnelles locales Client Société de Projet (SAS) Liens Financement Gouvernance Financement & gouvernance Accompagnement Syndicat d'énergie Développeur privé industriel Société civile Accompagnement/habitant.es Financeurs Société de Projet (SAS) Liens Financement Gouvernance Accompagnement Collectivités publiques : participe au projet Accompagnement/habitant.es : ne participe plus au projet Société citoyenne : il est prévu qu'elle soit créée Acteur semi-public Chambre d'agriculture départementale Abréviations ECLR : Energies Citoyennes Locales et Renouvelables en Occitanie URScop : Union Régionale des Sociétés Coopératives et Participatives PME : Petites & Moyennes Entreprises SEM : Société d'Economie Mixte MPEI : Midi-Pyrénées Energies Investissements dividendes AREC : Agence Régionale Energie Climat Un projet d'énergie photovoltaïque au sol Société civile Habitants hors du département Financement participatif, actionnaire Opposants en association Actions militantes contre le projet & sabotage Opposants en association Actions militantes contre le projet & Entreprises locales Soutien au projet Associations environnement, transition, circuits Entreprises CAC 40 Acteur financeur Enerfip données actionnaires, versement Membre association Financement participatif, gestion des  Propriétaire du terrien, Accord, URScop Soutien, Aide juridique  courts Soutien au projet Coopératives agricoles locales ; Chambre d'agriculture Soutien, Implication, Financement, Membres association accompagnement Energie partagée et investissement Soutien, accompagnement, notoriété, financement Incubateur innovation sociale Soutien, accompagnement, aide juridique ECLR Enercoop  Soutien, achat d'électricité  habitants  Formation, Notoriété, Participation  temps Urscop Soutien et accompagnement un Acteur Entreprises locales Soutien au projet AREC Coordination comité éolien Agence locale de l'énergie et du climat pendant un temps mais acteur exclu  Soutien et accompagnement  sabotage + Recours juridiques ECLR Formation, Notoriété, Participation habitants ECLR Formation, Notoriété, Participation habitants financement, actionnaire SPV SEM MPEI AREC Soutien, participation au  & gouvernance développeur industriel Ademe Financement Financement Chambre d'agriculture départementale Association du projet Société familiale de terriens projet citoyen versus accord avec territoire gouvernance, actionnaire (à venir) Région Occitanie Financement Entreprises locales Financement participatif Propriétaires Choix d'attribuer son terrain à un Habitants du Financement participatif, Communauté de communes communes pour/contre d'agglomération association  Autre Communauté de Evoplution de son positionnement : Communauté Soutien ambivalent, membre l'exploitant agricole Région Financement Ademe Financement Projet -> Projet 1 Projet 2 Projet 3 (dans sa version 2) Catégorie d'acteurs Nom acteur Rôle Nom acteur Rôle Nom acteur Rôle Développeur privé industriel PME régionale Connaissances et compétences projets EnR participatif, Expertise technique EnR, Développement, Financement, Actionnaire Entreprise du CAC 40 Expertise technique EnR, Développement, Financement, Actionnaire PME régionale (développeur 2) Connaissances et compétences projets EnR participatif, Expertise technique EnR, Développement, Financement, Actionnaire, Gouvernance Habitants du territoire Financement participatif, gouvernance, actionnaire territoire Financement participatif, gouvernance, actionnaire exploitant & sa société familiale Prend part aux risques, Actionnaire, Gouvernance Ademe Financement Pôle d'Equilibre Territorial et Rural Participe au comité éolien Parc naturel régional générales  Soutien, membre association Remarques Habitants du Lancement projet alternatif, Agriculteur Initiateur du projet, Investisseur, Projet -> Projet 1 Projet 2  Sur le plan des modes de coopération Au niveau de la gouvernance Au niveau territorial Sur le plan des modes de coopération Au niveau de la gouvernance Projet Au niveau territorial Projet Projet 3 (dans sa version 2) Catégorie d'acteurs Nom acteur Rôle Nom acteur Rôle Nom acteur Rôle Acteur public Commune Propriétaire foncier Commune Validation, soutien et diffusion du projet dual Société d'aménagement communauté d'agglomération Participation au financement, actionnaire SPV, gouvernance Région Occitanie Financement Communes limitrophes Une commune soutient projet dual et s'implique dans société portée par la société civile versus oppositions d'une majorité de communes Syndicat d'énergie Participation au financement, actionnaire SPV, gouvernance Projet -> Projet 1 Projet 2 Projet 3 (dans sa version 2) Catégorie d'acteurs Nom acteur Rôle Nom acteur Rôle Nom acteur Rôle Acteurs caractéristiques d'un codév3 ne sont pas directement associés en tant que collectif mais personnellement ; Postures ambivalentes des      1 1. Une commune dynamique & des habitants sensibles aux 1. Intention, expertise et dynamisme du développeur 1. Des personnes impliquées à titre individuel plutôt que les questions environnementales, en plus d'être engagés dans la vie associative locale ; 2. Implication du développeur privé qui a fait bénéficier aux projets de ces ressources pour faciliter le processus de codéveloppement ; 3. Présence de ressources : exposition solaire, terrains communaux disponibles, dynamiques associatives locales, PLU. concertation & coconstruction avec les habitants. bénévoles experts, projets d'habitat participatif à proximité, privé pour participer à un projet participatif impliquant une collectivité territoriale et des habitants ; 2. Proximité des partenaires et qualité de leurs relations, expériences de coopération préalables ; 3. Reconnaissance et notoriété du maire auprès des différentes catégories d'habitants ; 4. Information, consultation individuelle et collective, 1. Capacité à établir un projet commun avec une ambition de développer d'autres projets EnR. 1 1. Réécriture du PLU ajoutant un délai supplémentaire ; 2. Une majorité d'investisseurs vivent en dehors de la commune et créent des intérêts/attentes et des implications différentes à l'origine de tensions sous-jacentes. organisations ; 1. Coût de raccordement initial demandé par Enedis trop 2. Absence de collèges de sociétaires selon leur qualité, et de onéreux : difficulté de négociation du fait de rapports au sein du conseil de gestion ; inégaux ; 3. Processus de décisions pas clairement ni formellement 2. Communication relative au projet dans la phase établis = marge aux processus informels sources de d'exploitation s'avère moins présente que pendant la phase contestations potentielles ; de développement ; 4. Tensions

acteurs/personnes qui facilitent codév mais troublent répartition des rôles et interrogent une forme de centralisation des décisions ; Rôle du développeur essentiel car facilite aboutissement du projet sur le plan participatif ; Appui financier des acteurs publics déterminant ; Tensions développeurs/structures d'accompagnement car concurrence sur l'expertise participative. Mais tension majeure pour le raccordement avec Enedis.

Importance du comité éolien déterminant comme espace de discussion entre l'ensemble des acteurs ; Création de deux projets portés par deux sociétés de projet différentes : une portée par société civile, une autre par le développeur => le projet n'est pas tout à fait commun ; Négociation importante entre société civile et développeur pour s'accorder sur ce qui est partagé, y compris le temps bénévole des habitants ; Tensions sur validation du projet par certains acteurs publics.

Projet impliquant un grand nombre d'acteurs, dont la composition évolue au fil du temps ; Rapports en tension du fait d'intérêts hétérogènes mais capacité à construire un compromis dans une pacte associés ; Projet particulièrement intégré sur le plan de la gouvernance qui en fait sa force ; Pb actuel de recours liés aux opposants qui bloquent le projet. 2 1. Ouverture & sensibilité des élus communaux à la transition écologique citoyenne qui remportent par deux fois les élections municipales ; 2. Présence d'une dynamique associative locale ; 3. Dimension territoriale d'un projet à but de développement local ; 3. Choix politique et stratégique de la société citoyenne révélant un projet de nature sociétal avec une gestion contrôlée des retombées économiques. 1. Information, consultation individuelle et collective et concertation avec le comité éolien ; 2. Importance de la désignation d'un animateur « neutre », à savoir l'AREC, qui bénéficie d'une légitimité reconnue par les acteurs participants ; 3. Coconstuction d'un cadre commun de fonctionnement : un règlement intérieur ; 4. Réunion des acteurs clefs du territoire aux rôles, fonctions, intérêts et légitimités complémentaires et contrastés ; 5. Implication des habitants intéressés ou motivés pour participer au projet. 1. Composition du collectif de la société civile qui porte la société de projet « citoyenne » ; 2. Connaissances et compétences de certains habitants pour négocier avec le développeur ; 3. Signature du contrat de partenariat entre le développeur et la société de projet citoyenne ; 4. Recherche de répartition des rôles dans une logique de complémentarité : connaissance du territoire de la société citoyenne, techniques et procédures administratives du développeur. 1. ; 2. Présence d'un terrain de surface importante, dans une région ensoleillée, venteuse et avec des terres agricoles fertiles ; 3. Projet de valorisation de l'espace en harmonie avec son environnement, en intégrant davantage l'usine d'enrichissement d'uranium située à proximité ; 4. Dispositifs de financement (Ademe et Région Occitanie). 1. L'agriculteur s'associe à des acteurs clefs soutenant le projet. ; 2. Présence de personnes facilitant la concertation et les discussions entre les partenaires ; 3. Importance des organisations spécialisées dans l'accompagnement de projets participatifs ou EnR ; 4. Médiation du développeur par le biais de son chef de projet en 2019/2020 pour construire un compromis dans le pacte associés. 1. Travail de collaboration et de coopération instauré dès le début pour penser le projet ; 2. Ouverture du projet à de nouveaux acteurs au cours du codéveloppement ; 3. Diversité complémentaire des acteurs composant le projet ; 4. La rentabilité du projet favorisant l'implication continue des acteurs partenaires ; 5. Capacité du développeur à faire émerger un compromis ; 6. Engagement des organisations membres dans le capital de la société de projet.

au sein du conseil de gestion et divergences sur répartition

		et usages des bénéfices.
	1. Aucune collectivité territoriale n'est membre, en tant que commune, de la société de projet portée par la société civile ; 2. Présence d'opposants actifs => l'éolien suscite des oppositions récurrentes ; 3. Dimension territoriale du projet n'apparaît pas complète et source de tensions dans le codéveloppement : une majorité de communes alentours n'est pas favorables au projet.	1. Présence d'opposants qui réalisent des actions fortes contre le projet ; 2. Canaux de communication entre les deux sociétés de projet pas évidents ; 3. Transparence de chaque société de projet étroite du fait d'un manque de confiance ; 4. Processus de coopération « grinçant » avec des « problèmes réguliers de mécanique » ; 5. Négociation

entre deux partenaires de tailles inégales et aux connaissances inégales.

	1. Dualité du projet peu évidente à gérer : complexité du
	montage du projet en deux ;
	2. Longueur du processus de négociation ;
	3. Difficulté à faire émerger un compromis du fait d'intérêts
	divergents importants ;
	4. Tensions entre le processus de décision descendant du
	développeur et celui horizontale de la SAS de la société civile ;
	5. Conséquence : existence de deux gouvernances puisque le
	projet éolien est porté distinctement par deux sociétés
	différentes ;
	6. Adaptabilité

du cadre réglementaire et juridique à ce type de structure.

			1.
	3	1. Réticence des collectivités ou plutôt des élus locaux ; 2. Contraintes liées au terrain Seveso et inondable ; 3. Actions en justice des opposants locaux pour des raisons plutôt pécuniaires.	1. Agendas et intérêts divergents des partenaires ; 2. Poids des opposants avec les recours juridiques ; 3. Places des citoyens encore limitée et reléguée à ce jour.

Présence d'organisations aux intérêts divergents complique le compromis ; 2. Problèmes de transparence de

certains acteurs : jeux politiques territoriaux forts ; 3. Collectif s'est ouvert à nouveaux acteurs aux intentions de coopération variables ; 4. Exclusion d'un acteur pourtant historiquement important dans le collectif ; 5. Place limitée des citoyens dans les décisions du codéveloppement.

  .................................................. 12 Tableau 2 Les enjeux des acteurs des projets d'EnR en codév 3 ......................................................... 20 Tableau 3 Tableau synthétique des différentes étapes de recherche CITENR .................................... 24 Tableau 4 Catégories d'acteurs identifiés pour la phase d'entretiens experts ...................................... 28 Tableau 5 Catégories d'acteurs identifiés pour la phase entretiens projets cibles ............................... 29 Tableau 6 Déroulé et descriptif de tous les ateliers .............................................................................. 40 Tableau 7 Une comparaison entre trois générations de projets ........................................................... 50 Tableau 8 : La matrice des construits codév3 ....................................................................................... 53 Tableau 9 Catégorie des membres des instances de gouvernance ..................................................... 54 Tableau 10 : Les désaccords potentiels entre les acteurs selon la grille de Boltanski et Thévenot ..... 58 Tableau 11 : Analyse comparée du rôle des acteurs dans trois projets EnR codév3 occitans ............ 64 Tableau 11 : Analyse comparée du rôle des acteurs dans trois projets EnR codév3 occitans ............ 65 Tableau 11 : Analyse comparée du rôle des acteurs dans trois projets EnR codév3 occitans ............ 66 Tableau 12 : Analyse comparée des éléments favorables au codéveloppement. ................................ 71 Tableau 13 : Analyse comparée des éléments freinant le codéveloppement ....................................... 73 Tableau 14 Faisceau des droits de propriété selon Ostrom (2010) ...................................................... 75 Tableau 15 Faisceau des droits de propriété spécifiques aux projets d'EnR en codév 3 selon notre idéal-type ............................................................................................................................................... 76 Tableau 16 -. Spécificités des projets en codév3 par régions. Issues des deux Champs de force produits par les différentes régions de l'atelier 3B.

  .............................................. 84 Figure 22 Les dispositifs publics favorables aux projets codév3 ......................................................... 84 Figure 23 : Les relations réciproques entre projets codév3 et dispositifs publics ................................. 85 Figure 24 Freins et leviers au déploiement du codév3 qui ont émergé des ateliers de la région Occitanie et regroupés en familles de facteurs. ................................................................................... 86 Figure 25 : Les freins/leviers au déploiement du codév3 dans une vision interrégionale. .................... 88 Figure 26 : Principaux freins et leviers interrégionaux sur lesquels les acteurs indiquent avoir un degré de contrôle élevé.. ................................................................................................................................. 89 Figure 27 : Thématiques prioritaires à traiter en plan d'action (vote des participants) ......................... 90 Figure 28 : Plan d'actions pour la région Occitanie -par groupes thématiques ................................. 91 Figure 29 -Plan d'Actions produit en atelier Interrégional pour développer le codév 3 et complété à partir du plan d'action Occitanie ............................................................................................................ 92 Figure 30 Plan d'action .......................................................................................................................... 93 Figure 31 -Synthèse des enjeux territoriaux (spécificités régionales, conditions d'émergence et déploiement, freins et leviers du codév3) .............................................................................................. 94 Figure 32 Kit de familiarisation aux outils co-produits en atelier et permettant de se scorer en ligne.. 95 Figure 33 Les 4 outils co-construits dans les ateliers et mis à disposition sur le site pour « scoring » et enrichissement. ..................................................................................................................................... 96 Figure 34 : Fiche d'explication de la roue socratique issue du tool kit de Jacques Chevalier. ........... 96 Figure 35 : La roue d'autoévaluation finale du Codév3. ........................................................................ 97 Figure 36 : Présentation des critères et de leur déclinaison en sous critères constituant la roue du codév3 ................................................................................................................................................... 98 Figure 37 : Construction de la roue du Codév3 (à gauche). Du remue-méninge à la formalisation d'une roue socratique. (à droite). .................................................................................................................... 99


Exploring the Cerebellum with functional Near-Infrared Imaging: a preliminary study

INTRODUCTION `Cerebellum

CONCLUSIONS

• fNIRS is able to discriminate between baseline condition and experimental session, detecting hemodynamic changes due to cerebellar activity (BOLD paradigm).

• The hemodynamic changes in fNIRS signal appear correlated to the subject's engagement in the task, thus stimulus-induced.

• This study pioneers a novel way of exploring the cerebellum and suggests differences in its hemodynamic behavior compared to the brain.

• Extended experimental sessions are planned to gain deeper insight.

• Coupling with other techniques, like EEG, is available and will be implemented in future analyses.

RESULTS

METHOD

Experimental protocol

One subject was sitting on a chair in a dark room performing a finger tapping task.

ACKNOWLEDGEMENTS

This project has received funding from Horizon 2020 COFUND programme "BoostUrCareer" under the Marie Sklodowska-Curie grant agreement no. 847581. We would like to thank LAHMESS -STAPS for supporting our experiments.

AIM

To examine the possibility to use a novel technique to study the cerebellar role in motor and cognitive processes as an alternative to functional Magnetic Resonance Imaging (fMRI), namely functional Near-Infrared Spectroscopy (fNIRS): ✓ Higher time resolution ✓ Low-cost and flexible

Data quality assessment

The neurovascular coupling between the optodes and the scalp was verified by a frequency domain analysis. The Power Spectral Density of the signal was computed, where some components are visible as a proof of adequate coupling:

Data acquisition

The Artinis Oxymon III system was used, by setting the wavelengths at 763 and 858 nm with sampling rate at 50 Hz. Two channels were acquired with the optodes placement inspired by [START_REF] Todd | The human electrocerebellogram (ECeG) recorded non-invasively using scalp electrodes[END_REF], where CB1 and CB2 positions were added to the 10/20 EEG system.

The added optodes, CB1 and CB2, are shown in the red circles. The proportion distances are with respect to the 10/20 system and used as reference for the placement.

The experimental settings: light emitters (in yellow) and receivers ( in blue) held by a headband.

Data analysis

The measured light intensities were converted to local changes in oxygenated (ΔHbO2) and deoxygenated (ΔHbR) hemoglobin concentrations A band-pass FIR filter with order 1000 and cut-off frequencies 0.01 -0.07 Hz was applied [START_REF] Pinti | The present and future use of functional near-infrared spectroscopy (fNIRS) for cognitive neuroscience[END_REF].

• Decrease of ΔHbO2 and ΔHbR in baseline • Linear increase of ΔHbO2 during session with two peaks in both hemispheres after the left and the right finger tapping phases.