Corentin Fierobe
email: corentin.fierobe@ens-lyon.fr

Complex caustics of the elliptic billiard

Introduction

It is a well known fact that on a real ellipse, a billiard trajectory remains tangent to the same confocal conic (ellipse or hyperbola) after successive bounces. This is true in particular for periodic orbits. See for exemple [START_REF] Tabachnikov | Geometry and Billiards[END_REF], Chapter 4. In this present paper, we study the caustics of a complex reflection law, which will be defined later (in Section 2). For now, one should just know that it is a natural generalization of the real law and that it was used to prove important results. More precisely, let q be the non-degenerate complex quadratic form q(x, y) = x 2 + y 2 on C 2 . Then to each complex line L passing through a point A, one can associate its orthogonal line (for q) L passing through A. Then in the case when L = L (non-isotropic case) we can define the reflection of lines passing through A as the usual way. In the case when L = L , the reflection of lines is defined as a limit of lines reflected in non-isotropic cases. We give more details in Section 2.

It was introduced by Glutsyuk in order to study Ivrii's conjecture on periodic orbits together with its analogues for pseudo-billiards and complex billiards (cf [START_REF] Glutsyuk | On quadrilateral orbits in complex algebraic planar billiards[END_REF][START_REF] Glutsyuk | On Odd-periodic Orbits in complex planar billiards[END_REF][START_REF] Glutsyuk | On 4-reflective complex analytic billiards[END_REF]). Another use of it was made by Romaskevich in [START_REF] Romaskevich | On the incenters of triangular orbits in elliptic billiard[END_REF], to prove that the set of incenters of triangular orbits in an ellipse is also an ellipse.

The idea to approach a real problem by studying its extension to complex domain is not new. For example a proof of Poncelet theorem was given in [START_REF] Ph | Cayley's explicit solution to Poncelet's porism[END_REF] by generalizing it to complex caustics in CP 2 . Note that Poncelet's theorem was also studied on other fields, see for example [START_REF] Chipalkatti | On the Poncelet triangle condition over finite fields[END_REF] and [START_REF] Hungerbühler | A Poncelet Criterion for special pairs of conics in P G[END_REF]. Since it is a key point in this article, let us recall it. Theorem 1.1 (Poncelet,[START_REF] Ph | Cayley's explicit solution to Poncelet's porism[END_REF]). Let C and D be two conics of CP 2 intersecting transversally. If there is a n-sided polygon inscribed in C and circumscribed about D, then for each point p of C there is such a polygon having p as a vertex.

Billiards with other reflection laws were already studied, see [START_REF] Khesin | Pseudo-Riemannian geodesics and billiards[END_REF] for an introduction to this topic, and [START_REF] Adabrah | Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials[END_REF] for a precise study in the Minkovski settings. Furthermore, new discoveries about the elliptic billiard are made even recently, see for example Reznik's github page [START_REF] Reznik | [END_REF], and [START_REF] Akopyan | Billiards in ellipses revisited[END_REF][START_REF] Bialy | Dan Reznik's identities and more[END_REF] for mathematical proofs of some of its experimental results.

We denote by C λ the (complex) conic given in the affine chart (x, y) by equation

C λ : x 2 a 2 + λ + y 2 b 2 + λ = 1
where x, y ∈ C 2 and λ ∈ C \ {-a 2 , -b 2 } ; let us also define E = C 0 that is E :

x 2 a 2 + y 2 b 2 = 1.
We first prove the following theorem (see Fig. 1):

Theorem 1.2 (Complex caustics). Let T = (M 0 , . . . , M n ) be a (n + 1)-uplet of points of E such that any two consecutive points are dictinct. The following statements are equivalent:

• there is a λ, such that a 2 + λ, b 2 + λ = 0 and for each j ∈ {1, . . . , n -1} the lines M j-1 M j and M j M j+1 realize the two tangent lines to C λ going through M j ;

• T is a non-degenerate non-isotropic piece of complex billiard trajectory whose sides do not contain the real nor the complex foci.

If this λ exists, then it is unique. The definition of non-degenerate and non-isotropic complex billiard orbits will be defined later, in Subsection 2.2. For now, the reader can think about them as "good" orbits. Thus Theorem 1.2 generalizes the theorem of the existence of caustics in the elliptic billiard to its complexification.

For periodic orbits, we improve Theorem 1.2 as follows:

Theorem 1.3. Fix n ≥ 3. There exist N confocal conics γ n 1 , . . . , γ n N satisfying the following properties:

1. any n-sided polygon inscribed in E, circumscribed about a γ n j and having its consecutive vertices distinct, is a periodic billiard orbit (which is nondegenerate non-isotropic and non-flat); 2. any non-degenerate non-isotropic periodic billiard orbit which doesn't pass through a (complex or real) focus of E is circumscribed about one of the γ n j ;

3. for each point p of E and each j, one can find an n-sided polygon inscribed in E and circumscribed about a γ n j , having p as a vertex.

Furthermore, N ≤ n 2 /4 -1 if n is even, and N ≤ (n 2 -1)/4 if n is odd. For n = 3 one has N = 2. For n = 4, N = 3 in the case when a = √ 2b and N = 2 otherwise. Remark 1.4. Theorem 1.3 gives a meaning to another result which can be found in [START_REF] Dragovic | Caustics of Poncelet Polygons and Classical Extremal Polynomials[END_REF]. In this paper, at pages 17-18, the authors use a similar method to ours coming from elliptic functions theory to give a classification of the real caustics of the elliptic billiard. Similarly to us, the caustics are related to the roots of a specific polynomial, and in their case some on these roots do not correspond to any real caustics. For example in the case of triangular orbits, one can find two roots: one of them corresponds to the usual smaller confocal ellipse, the other one defines a confocal ellipse which is bigger than the billiard table and cannot be a caustic for real triangular orbits by convexity. Our result allows to understand this other root also as another caustic, but for the reflection law extended to the complex domain. Remark 1.5. In the proof of Theorem 1.3, for a given n ≥ 3 we exhibit a polynomial B n whose roots are the λ corresponding to all complex caustics of the n-periodic orbits of the elliptic billiard. The coefficients of B n depend on a and b. We show in generic cases that the degree of B n is the same as the upper bound given in Theorem 1.3 (see Proposition 5.8) and that -a 2 and -b 2 are not roots of B n (Proposition 5.11). In order to compute the exact number of complex caustics for the n-periodic orbits (in generic cases), one could try to show that generically B n has simple roots: this fact seems to be true on computer simulations, but we were unable to confirm it.

A rather interesting application of Theorem 1.3 is the classification of 3periodic degenerate orbits (defined as the limits of non-degenerate orbits)which is related to a result in [START_REF] Romaskevich | On the incenters of triangular orbits in elliptic billiard[END_REF]: Corollary 1.6. There are exacty 8 degenerate triangular orbits, given by an isotropic tangency point α of E and the point β of E such that the line αβ is tangent to a conic γ 3 j and non-isotropic.

Structure of the document and main arguments

We exclude the case when E is a circle. The document has the following structure:

1. Section 2 gives more details about the complex reflection law and what complex billiard trajectories are.

2. We present a natural and short proof of Theorem 1.2 in Section 3: it uses Zariski topology and classical results on real billiards.

3.

In Section 4, we present another proof of Theorem 1.2 that simultaneously yields new results on Joachimstal invariant. The latter is a quantity P (M, v) depending on a vertex M ∈ E of a polygon T inscribed in E and a vector v directing a side of this polygon starting at M . We show that this quantity: a) doesn't depend on the vertex M chosen or v if and only if T is a billiard trajectory ; b) is directly related with the existence of a conic inscribed in T .

theorem reduces the proof to the computation of a certain polynomial's degree.

5. The cases of 3-and 4-periodic orbits are studied in Sections 6 and 7, where we compute the exact value of the N appearing in Theorem 1.3, getting similar results as in [START_REF] Dragovic | Caustics of Poncelet Polygons and Classical Extremal Polynomials[END_REF].

Notations and usual properties of conics

In the whole article, we are dealing with the complexification of the real ellipse E and with the complex ellipses C λ . Thus we recall some results about these objects.

The ellipse E has four isotropic tangent lines (which are tangent lines directed by the vectors (1, ±i), cf [START_REF] Berger | [END_REF], Volume 2, Section 17.4.3). The name isotropic is due to the fact that (1, ±i) are isotropic vectors of the complex quadratic form q(x, y) = x 2 + y 2 , i.e., q vanishes on them.

A simple computation shows that their corresponding tangency points have coordinates

± a 2 √ a 2 -b 2 , ± ib 2 √ a 2 -b 2 . ( 1
)
where the signs ± are independent. This allows us to extend the definition of a focus of an ellipse as an intersection point of its non-parallel isotropic tangent lines, cf [START_REF] Berger | [END_REF], Volume 2, Section 17.4.3. In our case, the ellipse E has four foci:

• two real foci of coordinates (±c, 0), where c = √ a 2 -b 2 ;

• two complex foci of coordinates (0, ±ic).

The foci lines of E are defined as two distinct lines : the line joining the complex foci and the line joining the real foci.

Remark 1.7. We see here that E and C λ have the same foci. Hence they have the same isotropic tangent lines !

The following result will be needed, which generalizes a well known result in the real case (concerning the intersection of an ellipse with a confocal hyperbola). To state it, we recall that q(x, y) = x 2 + y 2 is a complex quadratic form, whose associated bilinear form is b defined by

b (u, v) = u x v x + u y v y for all u = (u x , u y ), v = (v x , v y ) ∈ C 2 . Two vector spaces F , G of C 2 are said to be orthogonal if b(u, v) = 0
for all u ∈ F and v ∈ G, and this definition extends naturally to lines in C 2 : two lines are orthogonal if their directions are orthogonal.

Lemma 1.8. Let λ = 0. Then E and C λ have four common points, whose coordinates (x, y) are such that

x 2 = a 2 (a 2 + λ) a 2 -b 2 and y 2 = b 2 (b 2 + λ) b 2 -a 2 .
The tangent lines to C λ and E at these points are orthogonal (for q).

Proof. The coordinates of the common points are obtained by solving the system composed by the equations of C λ and E.

Then, since the equations of the tangent lines of E and C λ in one of these common points can be computed, it is not difficult to check that both tangent lines are orthogonal.

Finally, we will name by S the point of coordinates (-a, 0) of E.

Complex reflection law

Here we introduce the notion of complex billiards, which is somewhat similar to pseudo-euclidean billiards studied by V. Dragovic and M. Radnovic in [START_REF] Dragovic | Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics[END_REF].

Considering an affine chart whose coordinates will be denoted by (x, y), we have the inclusion R 2 ⊂ C 2 ⊂ CP 2 , and CP 2 = C 2 C ∞ , where C ∞ is the infinity line. As introduced and explained in [START_REF] Glutsyuk | On quadrilateral orbits in complex algebraic planar billiards[END_REF], and studied in [START_REF] Romaskevich | On the incenters of triangular orbits in elliptic billiard[END_REF], the reflection law on an algebraic (analytic) curve in R 2 can be extended to CP 2 by considering the complexified version of the canonical euclidean quadratic form, that is the complex-bilinear non-degenerate quadratic form q. It leads to construct a notion of symmetry with respect to lines of C 2 . Just note that q has two isotropic subspaces of dimension 1 (namely C(1, i) and C(1, -i)).

Definition

Definition 2.1. Define the cyclic points of CP 2 as the points I = (1 : i : 0) or J = (1 : -i : 0). Definition 2.2 ([13], definition 1.2). A line in CP 2 is said to be isotropic if it contains either I or J and non-isotropic if not. (Thus, the infinity line is automatically isotropic.) Definition 2.3 ([13], definition 2.1). The symmetry with respect to a line L is defined by the two following points:

• the symmetry acting on C 2 (on points and on lines): it is the unique nontrivial complex-isometric involution (isometric for the form q) fixing the points of the line L, if L is non-isotropic ;

• the symmetry acting on lines: if L is an isotropic line passing through a point x / ∈ C ∞ , two lines and going through x are called symmetric with respect to L if there exist sequences of lines (L n ) n , ( n ) n , ( n ) n through points x n such that L n is non-isotropic, n and n are symmetric with respect to L n , n → , n → , L n → L and x n → x, when n → ∞.

We recall now Lemma 2.3 [START_REF] Glutsyuk | On quadrilateral orbits in complex algebraic planar billiards[END_REF] which gives an idea of this notion of symmetry in the case of an isotropic line through a finite point. Lemma 2.4 ([13], Lemma 2.3). If L is an isotropic line through a finite point x and , are two lines passing through x, then and are symmetric with respect to L if and only if either = L, or = L.

General billiard orbits

Definition 2.5. A piece of non-degenerate complex billiard trajectory is an ordered set of points (M 0 , M 1 , . . . , M n ) on E such that 1. M j = M j+1 for all j ∈ {0, . . . , n -1};

2. for all j ∈ {1, . . . , n -1} the line T Mj E is non-isotropic;

3. for all j ∈ {1, . . . , n -1}, the lines M j-1 M j and M j M j+1 are symmetric with respect to the tangent line T Mj E.

We say that a piece of non-degenerate trajectory is a non-degenerate periodic orbit or just an orbit when M n = M 0 and when the above statements are also true for M 0 . The M j 's are called vertices of the trajectory, and the lines M j M j+1 sides of the trajectory.

Computations will be easier with the following definitions. Definition 2.6. A piece of non-degenerate trajectory is said to be :

1. finite if none of its vertices belong to the infinity line, and infinite if one of them belong to C ∞ ;

2. isotropic if all of its sides are isotropic, and non-isotropic if none of its sides is isotropic.

3. flat if all of its sides coincide with one of both foci lines of E.

Remark 2.7. It is not difficult to see that a piece of trajectory has a side on a foci line if and only if it has all its sides on this foci line.

Then we will use the following proposition, which follows from the fact that the reflection law with respect to a non-isotropic line permutes two isotropic lines, [START_REF] Glutsyuk | On Odd-periodic Orbits in complex planar billiards[END_REF], corollary 2 (there exist exactly two distinct isotropic lines passing through any point x / ∈ C ∞ ).

Proposition 2.8. A non-degenerate periodic orbit with an odd number of sides is non-isotropic and none of its sides lie on a foci line.

When n = 3, periodical orbits are called triangular. The proposition 2.8 implies that there are no non-degenerate isotropic triangular orbits. Definition 2.9 ([13]). A degenerate triangular complex orbit on a complex conic E is an ordered triple of points in E which is the limit of non-degenerate periodic triangular orbits and which is not a non-degenerate triangular orbit. We define the sides of a degenerate orbit as the limits of the sides of the nondegenerate orbits which converge to it. Note that the sides of a degenerate orbit are uniquely defined, see the following more precise proposition.

Proposition 2.10 ([28], lemma 3.4). A degenerate triangular orbit of an ellipse

E has an isotropic side A which is tangent to E, and two coinciding non-isotropic sides B.

Digression: comparison with the projective reflection law

Here we discuss about another generalization of the usual reflection law in the euclidean plane, the so-called projective reflection law, and we compare it to the complex reflection law. This subsection is independant from the rest of the paper and its reading can be skipped without any consequences. If the reader is interested, we refer to [START_REF] Fierobe | On projective billiards with open subsets of triangular orbits[END_REF][START_REF] Fierobe | Examples of reflective projective billiards and outer ghost billiards[END_REF][START_REF] Tabachnikov | Introducing projective billiards[END_REF][START_REF] Tabachnikov | Exact Transverse Line Fields and Projective Billiards in a Ball[END_REF] for more details and recent results about this reflection law.

The projective reflection law in the euclidean plane is defined as follows: a quadruple of lines ( , , L, T ) going through the same point p in the euclidean plane is said to be harmonic if their cross-ratio is equal to -1. We recall that the cross-ratio of four distinct lines going through the same point is the cross-ratio of their intersection points with a fifth line d not containing p (and this doesn't depend on the line d). Remark 2.12. This definition can be naturally extended to the case when L = T p γ, but we will not detail this further. Definition 2.13. A projective billiard is a domain Ω ⊂ R n with boundary endowed with a transverse line field L, called the projective field of lines (see Figure 2). The reflection in planar projective billiard with a boundary curve ∂Ω acts on lines according to the projective reflection law.

∂Ω

Remark 2.14. Notice that the projective law of reflection and projective billiards can also be defined in the complexified euclidean plane, that is in C 2 (in exactly the same way). In this case we write specifically complex projective billiards. Now let us show that this law of reflection generalizes the (real or complex) reflection law. Denote by T a (real or complex) line through a point p and by T ⊥ the (real or complex) line through p which is orthogonal to T (for the standard euclidean metric in the real case, and for q in the complex case). Suppose that T is non-isotropic, so that T = T ⊥ . Two lines and through p are symmetric with respect to T (for the real or complex law of reflection) if and only if the quadruple of lines ( , , T, T ⊥ ) is harmonic, and the projective reflection law is satisfied. In the case when T is isotropic (T = T ⊥ ), we can get the same conclusion. Therefore Proposition 2.15. A real (resp. complex) billiard Ω is a real (resp. complex) projective billiard whose projective field of lines is defined by the normal lines to ∂Ω (resp. for the quadratic form q).

An algebraic proof of Theorem 1.2

In this section we give a short algebraic proof of Theorem 1.2, noticing that the theorem is true for real orbits and using Zariski topology to conclude. Proposition 3.1. For every projective line L passing through neither real, nor complex foci there exists a unique λ / ∈ {-a 2 , -b 2 } such that the conic C λ is tangent to L.

Proof. We prove this result by taking projective duality given by polar duality with respect to the Euclidean metric.

Denote by F 1 and F 2 the real foci of all C λ , and by G 1 and G 2 their complex foci: F 1 , F 2 are on the x-axis, G 1 , G 2 on the y-axis (see Subsection 1.2, or [START_REF] Berger | [END_REF], Volume 2, Section 17.4). Hence the dual line [START_REF] Berger | [END_REF], Volume 2, Section 17.4.3). It induces a point x ij defined as the dual of the line F i G j , which belongs to all dual conics C * λ of C λ . Hence, the family C * λ is a pencil of conics through 4 distinct points x ij . The conics C * λ of the dual pencil are given by the equation (a 2 + λ)x 2 + (b 2 + λ)ỹ 2 = 1. They are smooth for every λ / ∈ {-a 2 , -b 2 }. Now, to any line L passing through neither real, nor complex foci corresponds a dual point y L different from the x ij . By polar duality, Proposition 3.1 is equivalent to say that there is a unique conic passing through y L in the pencil of conics defined by the dual conics of the C λ . And this is a classical result on pencils of conics which is not difficult to prove, see for example [START_REF] Flatto | Poncelet's theorem[END_REF].

F * i of F i is vertical and the dual line G * i of G i is horizontal. Then note that any line F i G j is tangent to any C λ (again, see Subsection 1.2, or
To see that λ / ∈ {-a 2 , -b 2 }, suppose the contrary, for example λ = -b 2 . The conic C * λ is a pair of vertical lines passing through the x ij , hence both lines are the dual lines F * 1 and F * 2 of the real foci. But L do not contain any foci, thus its dual point L * do not belong to F * 1 ∪ F * 2 = C * λ . The same statement holds for complex foci and -b 2 replaced by -a 2 . Remark 3.2. By proof of Proposition 3.1, we see in fact that if L goes through a real focus, its dual point L * belongs to a C * λ only when λ = -b 2 . Hence L is never tangent to a C λ for λ = -b 2 .

Corollary 3.3. The collections of (A, L, λ) such that A lies in the given ellipse E and L is a line through A that is tangent to C λ form an irreducible twodimensional algebraic surface, in which the real part (real lines tangent to real confocal ellipses) is Zariski dense. The image of L under reflection from the tangent line

T A E is again tangent to C λ .
Proof. The first statement of Corollary 3.3 follows immediately from Proposition 3.1. The second statement follows from the fact that it obviously holds on the Zariski dense real part and from algebraicity of the tangency condition for the reflected line.

Proof of Theorem 1.2. If the first statement of Theorem 1.2 is realized, then for each j, by Corollary 3.3 the line M j-1 M j is reflected into the M j+1 M j under reflection from the tangent line T Mj E. Hence T is a billiard orbit.

If the second statement of Theorem 1.2 is true, by Proposition 3.1 there is a unique λ / ∈ {-a 2 , -b 2 } such that the conic C λ is tangent to the line M 0 M 1 . By Corollary 3.3, each M j M j+1 is again tangent to C λ .

Note that if you fix j, M j-1 M j and M j+1 M j are the two tangent lines to C λ passing through M j : indeed, suppose M j-1 M j = M j+1 M j and take any tangent line T to C λ passing through M j . By Corollary 3.3, it is reflected in a tangent line T to C λ . In the case when T = T the line T is orthogonal to T Mj E, and so is the line M j M j+1 for the same reasons. If T = T , we have T or T = M j M j+1 or we would have three distinct tangent lines to E passing through M j which is impossible. Hence

T = T = M j M j-1 = M j M j+1 .

Theorem 1.2 and Joachimsthal invariant

Remark 4.1. This section was inspired by the study of billiards in conics conducted in Chapter 4 -Billiards inside Conics and Quadrics of [START_REF] Tabachnikov | Geometry and Billiards[END_REF]. In this book, Theorem 4.4 shows that for a set of points and directions defined as successive billiard reflections on a real conic E, there is an invariant quantity. Known as Joachimsthal invariant, it is defined by

xv x a 2 + yv y b 2
where (x, y) are the coordinates of a vertex of an orbit, and v a unitary vector having this vertex as starting point and pointing toward the next vertex. The author, Tabachnikov, further explains, without proving it, that one can find such an invariant if and only if one can find a conic tangent to the orbit. Let us mention another reference about Joachimsthal invariant, which was given to us by the referee, which we are grateful to: see [START_REF] Kozlov | Billiards: A Genetic Introduction to the Dynamics of Systems with Impacts[END_REF], chapter IV.

In our case, Joachimsthal invariant doesn't work anymore and we need to change it a little bit: a square power appears, and we have to handle the case of isotropic directions, for which unitary vectors cannot be found (that are vectors v such that q(v) = 1).

A billiard invariant

E M0 M1 M2 v0 v 1 v1 v 2
P (M j , v) = xj vx a 2 + yj vy b 2 2 q(v)
where (x j , y j ) are the coordinates of a finite vertex

M j and v = (v x , v y ) is a directing vector of M j-1 M j or M j M j+1 , is independent on the index j ∈ {0, 1, 2}
of the finite vertex chosen and on v (see Fig. 3).

Remark 4.3. For periodic orbits with an odd number of sides, one can remove the non-isotropic assumption, see Proposition 2.8.

Proof. Since a non-isotropic piece of trajectory has non-isotropic sides by definition, q(v) = 0 for all v taken like in the proposition we want to prove. First case: If M 0 and M 1 are finite, write M 0 = (x 0 , y 0 ), M 1 = (x 1 , y 1 ). Take v 0 a vector such that q(v 0 ) = 1 and directing M 0 M 1 and v 1 vector such that q(v 1 ) = 1 and directing M 1 M 2 . Define the matrix

A = 1/a 2 0 0 1/b 2 .

11

Then since t M j AM j = 1 and since A is symmetric, we get

t (M 1 -M 0 )A(M 1 + M 0 ) = t M 1 AM 0 -t M 0 AM 1 = 0. Since v 0 is collinear to M 1 -M 0 we have further t v 0 A(M 1 + M 0 ) = 0, thus t v 0 AM 1 = -t v 0 AM 0 . ( 2
)
But since M 0 M 1 and M 1 M 2 are symmetric with respect to the tangent line of E in M 1 which is also orthogonal to AM 1 (the gradient in M 1 of the bilinear form defining E), we only have two possibilities : either

v 0 + v 1 or v 0 -v 1 is orthogonal to AM 1 . Hence t (v 0 + v 1 )AM 1 = 0 or t (v 0 -v 1 )AM 1 = 0.
In both cases

t v 0 AM 1 2 = t v 1 AM 1 2
and using equality (2), we get

t v 0 AM 0 2 = t v 1 AM 1 2 (3)
which proves Proposition 4.2 for unitary vectors. For general vectors, it is enough to divide them by a square root of q(v), which explains why there is a 1/q(v) in the invariant formula.

Second case: If M 0 is finite and M 1 infinite (see Fig. 4), then M 2 is finite. Indeed, M 0 M 1 is a finite line and T M1 E is not isotropic. Hence the line symmetric to M 0 M 1 with respect to T M1 E is finite and parallel to M 0 M 1 and to T M1 E. Thus M 2 is finite. And therefore, we need to prove that

P (M 0 , v) = P (M 2 , v) with v a vector directing T M1 E (because v directs the lines M 0 M 1 , M 1 M 2 and T M1 E). But M 2 = -M 0 since T M1 E
goes through the origin O = (0 : 0 : 1) (by property of a tangent line at an point of E on C ∞ ) and the ellipse E is symmetric across O (see Fig. 4). This implies that P (M 0 , v) = P (M 2 , v). Corollary 4.4. Let T = (M 0 , . . . , M n ) be a piece of non-degenerate and nonisotropic trajectory on E. Then the quantity P (M j , v) defined as before doesn't depend on the choice of a finite vertex M j or on v, a directing vector of M j-1 M j or M j M j+1 . Thus we will write P (M j , v) = P (T ). Now we see that the invariant property implies a billiard reflection property. Lemma 4.5. Let M be a finite point on E such that the line T M E is nonisotropic. Let 1 , 2 two non-isotropic lines passing through M and directed by the vectors

v 1 , v 2 . If P (M, v 1 ) = P (M, v 2 ) ( 4
)
then one of the following cases holds:

M0 M1 M2 O E v v v M0M1 M1M2 T M 1 E
(M 0 , v) = P (M 2 , v).
Here E is represented as an hyperbola which allows us to view the tangent line at the infinity point M 1 as the vertical asymptote.

1. 1 = 2 ;
2. 1 and 2 are symmetric with respect to T M E.

Proof. Suppose that case 1 is not true. Let us prove case 2.

We can suppose q(v 1 ) = q(v 2 ) = 1. By the equality (4), we have

t v 1 AM = ± t v 2 AM hence t (v 2 ± v 1 )AM = 0.
Thus we get that

v 1 + v 2 or v 2 -v 1 is orthogonal to AM which is orthogonal to the tangent line of E in M . Hence v 1 + v 2 or v 1 -v 2 is tangent to E in M .
This implies that one of these vectors is fixed by the reflection with respect to T M E. Therefore this means that the components of the v j 's along the direction of T M E ⊥ are the same or have opposite signs. Since the v j 's are unit vectors, their components along the direction of T M E are also the same or have opposite signs.

Hence we have only three possibilities: a) v 1 and v 2 are symmetric with respect to T M E, b) v 1 and v 2 are symmetric with respect to T M E ⊥ , c) v 2 = ±v 1 . Possibility c) cannot happen, otherwise 1 = 2 . Hence case 2 is proven. Lemma 4.6. Let M 0 , M 1 , M 2 be points on E such that M 0 , M 2 are finite and M 1 infinite. Let v j be a vector directing the line M 1 M j . If

P (M 0 , v 0 ) = P (M 2 , v 2 ) ( 5
)
then one of the following cases holds:

1. M 0 = M 2 ;
2. M 0 M 1 and M 1 M 2 are symmetric with respect to T M1 E.

Proof. Suppose that case 1 is not true. Let us prove case 2.

Since

M 1 ∈ E is infinite, M 1 = (a : ±ib : 0). Thus M 0 M 1 and M 1 M 2 are directed by v = (a : ±ib). To simplify, suppose v = (a, ib) = v 1 = v 2 . Thus since P (M 0 , v) = P (M 2 , v
) by equality (5), we have

x 0 a + i y 0 b = ε x 2 a + i y 2 b
with ε ∈ {1, -1}, M j = (x j , y j ). Hence

x 2 -εx 0 a + i y 2 -εy 0 b = 0.
We show that ε = -1. Indeed, if ε = 1, the latter equality means that

----→ M 0 M 2 is orthogonal to v = (1/a, i/b), which is orthogonal to v. Therefore ----→ M 0 M 2 is
colinear to v : but this is impossible, otherwise M 0 , M 1 , M 2 would be three distinct points of E on the same line.

Thus ε = -1. Then, applying the same arguments, we have

M 0 = -M 2 . Hence M 0 M 1 reflects into M 1 M 2 .

Particular values of the invariant

The question we consider here is : for which non-isotropic v do we have P (M, v) = b -2 or P (M, v) = a -2 ? Proposition 4.7. If M is not a point of isotropic tangency of E, we have :

• P (M, v) = a -2 if

and only if v has the same direction as the line going

through M and one of the real foci of E.

• P (M, v) = b -2 if and only if v has the same direction as the line going through M and one of the complex foci of E.

Proof. We just prove the first point, the second one is analogous. First, for a fixed M ∈ E, and a k ∈ C, there are at most two directions v such that P (M, v) = k.

(two collinear vectors have the same direction). Indeed, the equation set v 2 x + v 2 y = 1 in at most two points (weak form of Bezout theorem). And considering the same equation but with -k instead of k , we get at most four unitary vectors such that P (M, v) = k, two of them being the opposite of the others. Hence there are at most two directions such that P (M, v) = a -2 .

We give here the different possibilities for those directions. The real foci of E have coordinates (±c, 0) where c = √ a 2 -b 2 . Hence v ± = (x ± c, y) are directing the lines going through M and the real foci. Then we have

xv +,x a 2 + yv +,y b 2 = x(x + c) a 2 + y 2 b 2 = 1 + xc a 2 = a 2 + xc a 2
and using the fact that M ∈ E, we have

a 2 q(v + ) = a 2 (x + c) 2 + a 2 y 2 = a 2 (x + c) 2 + (ab) 2 -b 2 x 2 = c 2 x 2 + 2a 2 xc + a 4 = (a 2 + cx) 2
Hence, since M is not an isotropic tangency point of E and by (1), we get that

q(v + ) = 0.
Thus,

P (M, v + ) = a -2
and the same is true with v -. There is one case when v -and v + are colinear : when M is one vertex of the ellipse. But this case can be solved easily.

Corollary 4.8 (Forbidden values of P (T ))

. Let T = (M 0 , . . . , M n ), with n ≥ 3, be a non-degenerate and non-isotropic piece of trajectory. If T has none of its sides passing through a real or a complex focus of E, then P (T ) = a -2 and P (T ) = b -2 .

P (T ) = a -2 F1 F2 P (T ) = a -2 , b -2 F1 F2
Figure 6: Trajectories with their respectives P (T )

Proof of Theorem 1.2

Here we prove that the invariant P (T ) charaterizes pieces of trajectories which are tangent to the same conic. We first recall the following elementary fact: Lemma 4.9. let C be a conic in CP 2 given by the equation t XAX = 0 where A is a 3 × 3 invertible matrix, and v = (a , b , c We are now ready to prove the first part of Theorem 1.2: Proposition 4.10. Let T = (M 0 , . . . , M n ) be a non-degenerate non-isotropic piece of billiard trajectory. We suppose that none of its sides pass through a (real or complex) focus. Then there is a unique λ ∈ C such that a 2 + λ = 0, b 2 + λ = 0 and T is circumscribed about the conic C λ .

) ∈ C 3 defining the line v of equation a x + b y + c z = 0. Then v is tangent to C if and only if t vA -1 v = 0.
We have in this case λ = -(ab) 2 P (T ).

Proof. For s ∈ C, let us define the matrix

B s =   a 2 + s 0 0 0 b 2 + s 0 0 0 -1   .
Since the orbit is non-isotropic, two consecutive sides cannot be infinite at the same time. Hence we suppose without loss of generality that M 0 is finite. Then the line M 0 M 1 is defined by the equation in CP 2 v y x -v x y + (v x y 0 -v y x 0 )z = 0.

Here M 0 = (x 0 , y 0 ) and v = (v x , v y ) is a directing vector of M 0 M 1 in C 2 . Hence we have M 0 M 1 = w (in the notations of Lemma 4.9) where

w = (v y , -v x , v x y 0 -v y x 0 ).

It allows us to compute

t wB s w = (a 2 + s)v 2 y + (b 2 + s)v 2 x -(v x y 0 -v y x 0 ) 2 .
Using the fact that M 0 lies on the ellipse, that is, substituting

a 2 = x 2 0 + a 2 y 2 0 b 2 , b 2 = y 2 0 + b 2 x 2 0 a 2
to the above formula, we get

t wB s w = sq(v) + (ab) 2 x 0 v x a 2 + y 0 v y b 2 2 .
Hence the previous quantity P (M 0 , v) defined in Proposition 4.2 appears here again, since t wB s w = q(v) s + (ab) 2 P (M 0 , v) .

Hence if λ = -(ab) 2 P (M 0 , v), it is the only λ for which t wB λ w = 0. And since P (M 0 , v) = P (T ) doesn't depend on the choice of the index j of a M j , the same computations are true for all lines M j M j+1 . Thus, since P (T ) = a

2 + λ = 0. Then each line M 0 M 1 which is tangent to C λ , where M 0 = M 1 ∈ E, is such that 1. M 0 or M 1 is finite; 2. the line M 0 M 1 is non-isotropic; 3. λ = -(ab) 2 P (M j , v)
where v is directing M 0 M 1 and M j is a finite point among M 0 ,M 1 .

Proof. First notice that this line is non-isotropic because otherwise it would be tangent to C λ , hence to E, and we could not have M 0 = M 1 . Furthermore, it is not the infinity line (which is not tangent to C λ ), hence M 0 or M 1 is finite. Therefore, the lemma results from the computation analogous to that of the proof of Proposition 4.10 for the computation of t wB s w. Proposition 4.12. Let λ ∈ C be such that a 2 +λ = 0 and b 2 +λ = 0. Then each n-uplet of points T = (M 0 , . . . , M n ) ∈ E n , two consecutive points being distinct, such that for all j ∈ {1, . . . , n -1} the sides M j-1 M j and M j M j+1 realize the two tangent lines to C λ going through M j , is a non-degenerate and non-isotropic piece of billiard trajectory, with λ = -(ab) 2 P (T ), whose sides avoid the foci of E.

Proof. Lemma 4.11 implies that the sides M j M j+1 of T are non-isotropic and for each j, at least one point among M j or M j+1 is finite.

Furthermore, the quantity P (M j , v) doesn't depend on the finite point M j of T or on the vector v directing M j-1 M j or M j M j+1 .

Therefore for each j, we have two possibilities by Lemmas 4.5 and 4.6: either M j-1 M j = M j M j+1 or both lines are symmetric with respect to T Mj E.

Let us show that the former case is a subcase of the latter case. Indeed, if there is a j such that M j-1 M j = M j M j+1 , then by properties of conics M j-1 = M j+1 . This implies that there exists only one tangent line to C λ going through M j . Hence M j ∈ C λ ∩ E and M j-1 M j is the tangent line T Mj C λ , which is orthogonal to the tangent line T Mj E by Lemma 1.8. Thus M j-1 M j and M j M j+1 are symmetric with respect to T Mj E.

Hence for each j we have a billiard reflection. Finally the sides avoid the foci, since λ = -a 2 , -b 2 and by Corollary 4.8. This concludes the proof.

Proof of theorem 1.3

The finiteness of the number of conics γ n j , which we will call caustics, is not difficult to prove. For a fixed integer n ≥ 3, the set T n of non-degenerate n-periodic orbits is an open set of an algebraic curve of E n CP 1 n (otherwise we could find an open set of inital conditions (M 0 , M 1 ) ∈ E 2 corresponding to n-periodic orbits, contradicting the real case). This curve has then a finite number of irreducible components. Now, for a fixed caustic, the set of n-periodic orbits circumscribed about it is an irreducible algebraic curve included in T n (this follows from the fact that each n-periodic orbit is uniquely defined by its initial condition ( , M 0 ), where = M 0 M 1 is a line through M 0 that is tangent to the given caustic, and the space of initial conditions is an elliptic curve, see [START_REF] Ph | Cayley's explicit solution to Poncelet's porism[END_REF]). But two different caustics cannot have the same set of circumscribed orbits: otherwise their corresponding constant P (T ) would be the same (Proposition 4.10), which is impossible. Hence there is a finite number of caustics:

γ n 1 = C λ1 , . . . , γ n N = C λ N , with pairwise distinct λ j all different from -a 2 and -b 2 .
Our goal now is to estimate the number N of caustics. To do so, we will use Cayley's theorem, proven for example in [START_REF] Ph | Cayley's explicit solution to Poncelet's porism[END_REF]. We will just give an upper bound on N (Propositions 5.4 and 5.8) and explain how we can compute its exact value (Corollary 5.6 and Propositions 5.8 and 5.11).

The following theorem of Cayley is needed. We will say that two conics C and D are in general position if they intersect (transversally) at four points. Note that if λ = 0, C λ and E are in general position (see Lemma 1.8).

A 2 . . . A m+1 . . . . . . . . . A m+1 . . . A 2m = 0, with m = n -1 2 (n odd), or A 3 . . . A m+1 . . . . . . . . . A m+1 . . . A 2m-1 = 0 with m = n 2 (n even).
This condition is reduced to A 2 = 0 when n = 3 and to A 3 = 0 when n = 4.

Remark 5.2. Note that the determinants we are considering in Theorem 5.1 can be rewritten for n odd and n even respectively as det(A i+j ) 1≤i,j≤m and det(A i+j+1 ) 1≤i,j≤m-1 .

Remark 5.3. We recall that an element of CP 2 can be represented by a triple of the form (x : y : z) where x, y, z ∈ C are not all equal to 0. Be careful that this representation is not unique since (x : y : z) = (tx : ty : tz) for any t ∈ C * . Then any polynomial P (x, y) of degree d can be associated to a homogeneous polynomial

P (x, y, z) = z d P ( x z , y z ).
Hence the zeros of P in C 2 can be extended in CP 2 to the set of zeros of P (x, y, z). In our case, the conic C λ can be viewed in CP 2 as the set of (x :

y : z) such that x 2 a 2 + λ + y 2 b 2 + λ -z 2 = 0. Proposition 5.4. Let n ≥ 3.
There is a polynomial B n (λ) such that any of its roots λ / ∈ {-a 2 , -b 2 } has the following property: there exists an n-sided polygon inscribed in E and circumscribed about C λ .

The degree of B n is such that

deg B n ≤ n 2 -1 4 if n is odd n 2 4 -1 if n is even.
Proof. Suppose first that n = 2m + 1 is odd and fix a λ with λ + a 2 , λ + b 2 = 0.

To understand if there is an n-sided polygon inscribed in E and circumscribed about C λ , we apply Cayley's theorem: there is such a polygon if and only if the determinant

A n (λ) = A 2 (λ) . . . A m+1 (λ) . . . . . . . . . A m+1 (λ) . . . A 2m (λ)
vanishes, where the A i (λ) are the coefficients in the analytic expansion of

f : t → det(tC + D λ )
with C and D λ being quadratic forms respectively associated to E and to C λ . Thus, to prove the result we want, we have to show that the determinant A n (λ) vanishes for a finite number of λ. Let us give a more precise fomula of A n (λ). We have by Remark 5.3,

tC + D λ = t a 2 + 1 a 2 + λ x 2 + t b 2 + 1 b 2 + λ y 2 -(t + 1)z 2 hence det(tC + D λ ) = - t a 2 + 1 a 2 + λ t b 2 + 1 b 2 + λ (t + 1)
which we factorize in det(tC

+ D λ ) = - 1 (a 2 + λ)(b 2 + λ) a 2 + λ a 2 t + 1 b 2 + λ b 2 t + 1 (t + 1).
Define the map g : t → a 2 +λ a 2 t + 1 b 2 +λ b 2 t + 1 (t + 1) and write its Taylor expansion as

g(t) = ∞ k=0 B k (λ)t k Since f (t) = ig(t) (a 2 + λ)(b 2 + λ) we have A k (λ) = iB k (λ) (a 2 + λ)(b 2 + λ) .
This shows that A n (λ) is a function of λ which vanishes if and only if the determinant

B n (λ) = B 2 (λ) . . . B m+1 (λ) . . . . . . . . . B m+1 (λ) . . . B 2m (λ)
also vanishes. We thus need to compute the B k 's. Write

√ t + 1 = c 0 + c 1 t + c 2 t 2 + . . .
where

c k = 1 k! 1 2 1 2 -1 . . . 1 2 -k + 1 = (-1) k+1 4 k (2k -1) 2k k . ( 6
)
Therefore for any β,

βt + 1 = c 0 + c 1 βt + c 2 β 2 t 2 + . . . where B k (λ) = u+v+w=k c u c v c w a 2u b 2v (a 2 + λ) u (b 2 + λ) v . ( 7
)
Therefore each B k is a polynomial in λ of degree at least k. Hence B n (λ) is a polynomial in λ verifying: for any λ / ∈ {-a 2 , -b 2 }, B n (λ) = 0 if and only if A n (λ) = 0, which is true if and only if there exists an n-sided polygon inscribed in E and circumscribed about C λ . Now for any permutation σ of {1, . . . , m} we have

deg m j=1 B σ(j)+j = m j=1 deg B σ(j)+j ≤ m j=1 (σ(j) + j) = m(m + 1) and since B n (λ) is a sum of ± m j=1 B σ(j)+j over all σ, we have that deg B n (λ) ≤ m(m + 1) = n 2 -1
4 . Now if n = 2m is even, the existence of B n is treated exactly as in the case when n is odd, but instead, B n (λ) = det(B i+j+1 ) 1≤i,j≤m-1 . Hence for any permutation σ of {1, . . . , m} we have

deg m-1 j=1 B σ(j)+j+1 = m 2 -1 and deg B n ≤ m 2 -1 = n 2
4 -1. Now we are ready for the proof of Theorem 1.3:

Proof of Theorem 1.3 (without its last statement for n = 3, 4). Let B n be the polynomial of Proposition 5.4. By construction, any root λ of B n different from -a 2 and -b 2 corresponds to a C λ inscribed in an n-sided polygon P of E. By Theorem 1.2, P is a n-periodic billiard orbit and C λ is its caustic, therefore λ = λ j for a certain 1 ≤ j ≤ N . Conversely, all λ j are roots of B n since a periodic billiard orbit is a polygon.

The first statement of Theorem 1.3 is obvious by theorem 1.2. The second statement is a consequence of the argument given at the beginning of Section 5.

The third statement comes from Poncelet's theorem: by definition of the γ j , there is at least one n-sided polygon inscribed in E and circumscribed about C λ . Now Poncelet's theorem states that for any p ∈ E there exists such a polygon with p as a vertex.

We can try to know if there are periodic orbits passing through the foci. We recall that in the real case, a billiard trajectory going through the real foci accumulates on the (real) foci line (see [START_REF] Tabachnikov | Geometry and Billiards[END_REF], Exercise 4.3). Proposition 5.5. If an n-periodic orbit goes through the real (resp. complex) foci, then n is even and its edges are on the real (resp. complex) foci line.

Proof. Since a line through a focus is reflected into a line through another focus, n has to be even. Now consider the map f from E to E defined as follows: for A in E, let B be the other intersection point of AF 1 with E and C the other intersection point of BF 2 with E, where F 1 , F 2 are the real foci of E. A nperiodic orbit through the real foci has a vertex M such that f n (M ) = M . Now f is a non trivial automorphism of E CP 1 hence is a Möbius transform, and so is f n . But such tranform has at least two fixed points. Since we already know two of them to be the vertices of E, the latter correspond to the only possible initial points of periodic orbits through the real foci. The same holds with complex foci.

From the proof of Theorem 1.3, we get the if n is odd

n 2 4 -1 if n is even.
Remark 5.9. We show in Proposition 5.10 that p ≥ 1, more precisely that 1 belongs to the collection of {r 1 , . . . , r p }.

Proof. Suppose n = 2m + 1 is odd. By Equation ( 7), B k is of degree ≤ k and the coefficient in front of λ k is

d(B k ) = u+v=k c u c v a 2u b 2v = (-1) k 4 k u+v=k 1 a 2u b 2v (2u -1)(2v -1) 2u u 2v v . ( 8
H m =      Cat 1 Cat 2 • • • Cat m Cat 2 Cat 3 . . . . . . . . . Cat m • • • Cat 2m      .
There are not so easy methods to show that det H m = 1, see for example

deg B n = n-1 2 if n is odd n 2 -1 if n is even. Proof. Suppose n = 2m + 1 is odd. By Equation 7, when a = b = R, for k ≥ 2 B k = k w=0 c k-w 1 + λ a 2 w u+v=w c u c v .
Let us compute u+v=w c u c v : it is the Taylor coefficient at t w of the function

√ 1 + t 2 = 1 + t, therefore we that u+v=w c u c v = 1 if 0 ≤ w ≤ 1 0 if w ≥ 2. Hence B k = c k + c k-1 x where x = 1 + λ/a 2 .
Using the multilinearity of det, it is not hard to see that B n is of degree m if n is odd and m -1 if n is even.

Proposition 5.11.

There exist a at most finite collection of numbers r 1 , . . . , r q ∈ R + such that for all (a, b) with a / ∈ {r 1 b, . . . , r q b}, -a 2 and -b 2 are not roots of B n .

Proof. Suppose n = 2m + 1 is odd. By Equation [START_REF] Dragovic | Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics[END_REF], for k ≥ 2,

B k -a 2 = v+w=k c v c w b 2v (b 2 -a 2 ) v = 1 b 2k v+w=k c v c w b 2w (b 2 -a 2 ) v = 1 b 2k P k (a, b) (9) where P k (a, b) is a homogeneous polynomial in (a, b) of degree 2k. The coefficient in front of a 2k is (-1) k c k = - 1 4 k (2k -1) 2k k = - Cat k-1 2 2k-1 .
As in the proof of Proposition 5.8, for any permutation σ of {1, . . . , m},

m j=1 B σ(j)+j (-a 2 ) = m j=1 1 b 2(σ(j)+j) P σ(j)+j (a, b) = Q σ (a, b) b 2m(m+1) where Q σ (a, b) is a homogeneous polynomial of degree m j=1 deg P j+σ(j) = m j=1 2(σ(j) + j) = 2m(m + 1) whose coefficient in front of a 2m(m+1) is m j=1 - Cat j+σ(j)-1 2 2(j+σ(j))-1 = (-1) m 2 m(2m+1) m j=1 Cat j+σ(j)-1 .
As in proof of Proposition 5.8, B n (-a 2 ) is a sum of products of the form ± m j=1 B σ(j)+j (-a 2 ) hence can be written as

R n (a, b) b 2m(m+1) where R n (a, b) is the sum of ε(σ) m j=1 Q σ(j)+j (a, b) and ε(σ) is the parity of σ. Thus R n (a, b) is a homogeneous polynomial of degree 2m(m + 1) whose coefficient in front of a 2m(m+1) is (-1) m 2 m(2m+1) det H m = (-1) m 2 m(2m+1)
as in the proof of Proposition 5.8. Thus R n (a, b) is a nonzero homogeneous polynomial such that

B n (-a 2 ) = R n (a, b) b 2m(m+1) .
We can do the same with B n (-b 2 ) to obtain the same conclusion, which finishes the proof.

Triangular orbits

As in the proof of Theorem 1.3, any root λ / ∈ {-a 2 , -b 2 } of B 3 (λ) corresponds to a caustic defined by C λ for triangular orbits in the complex domain. Therefore we are going to compute B 3 (λ) and its roots. This section is very similar to [START_REF] Dragovic | Caustics of Poncelet Polygons and Classical Extremal Polynomials[END_REF] p. 17, since we get the same results with a similar method (just notice that the conventions adopted in this paper is different from ours: in [START_REF] Dragovic | Caustics of Poncelet Polygons and Classical Extremal Polynomials[END_REF], a stand for a 2 , b for b 2 and their C λ is the same as our C -λ ). Yet, our conclusion is a little bit more general thanks to our previous study on complex caustics (Theorem 1.2). For the Minkowski plane, similar calculations are given in [START_REF] Adabrah | Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials[END_REF], p. 469, 470. We have

B 3 (λ) = - 1 8a 4 b 4 (a 2 -b 2 ) 2 λ 2 -2(a 4 b 2 + a 2 b 4 )λ -3a 4 b 4
which is a polynomial of second degree. Its roots are

λ ± = a 2 b 2 (a 2 + b 2 ) ± 2a 2 b 2 √ a 4 -a 2 b 2 + b 4 (a 2 -b 2 ) 2
which correspond to the opposite of the solutions found in [START_REF] Dragovic | Caustics of Poncelet Polygons and Classical Extremal Polynomials[END_REF] as predicted. Lemma 6.1. We have λ + > 0 and -b 2 < λ -< 0.

Proof. The inequality λ

-> -b 2 is equivalent to 2a 2 b 2 a 4 -2a 2 b 2 + b 4 -a 2 b 2 (a 2 + b 2 ) < b 2 (a 2 -b 2 ) 2 which can be simplified in 2a 2 b 2 a 4 -2a 2 b 2 + b 4 < 2a 4 + b 4 -a 2 b 2 .
This last inequality is true as we can see by taking the square of both its right and left sides. The same methode can be applied to show that λ -< 0. By Lemma 6.1, the caustics corresponding to λ ± , C λ+ and C λ-, are confocal ellipses which are respectively bigger and smaller than E = C 0 . As noticed in [START_REF] Dragovic | Caustics of Poncelet Polygons and Classical Extremal Polynomials[END_REF], C λ-corresponds to the real motion. And we can add that C λ+ corresponds to no real orbits by convexity. Hence we deduce the following corollary, which is a version of Theorem 1.3 in the case when n = 3: Corollary 6.2. There exist two distinct real ellipses γ 3 1 := C λ-and γ 3 2 := C λ+ which are confocal to E by construction, and such that • all complex triangles inscribed in E and circumscribed about a γ 3 j are billiard orbits;

• any complex triangular billiard orbit of E is circumscribed about either γ 3 1 , or γ 3 2 ;

• any complex orbit inscribed in E and circumscribed about a γ 3 j is a triangular orbit. We can apply Corollary 6.2 to classify the degenerate triangular orbits: Proposition 6.4. There are exacty 8 degenerate triangular orbits, given by an isotropic tangency point α of E and a point β of E such that αβ is tangent to γ 3 j for some j = 1, 2 and non-isotropic.

Proof. Degenerate triangular orbits are limits of non-degenerate (non-isotropic) triangular orbits, hence are circumscribed about a γ 3 j . We apply Proposition 2.10: one side A is isotropic and tangent to a γ 3 j . This gives only 4 possible positions. The other side is non-isotropic, which gives two other possibilities (B should be tangent to the same γ 3 j ), once A is fixed.

E C λ + C λ -

4-periodic orbits

We apply in this section the same ideas as in Section 6 for n = 4. Again these calculations repeat those from [START_REF] Dragovic | Caustics of Poncelet Polygons and Classical Extremal Polynomials[END_REF], p. 18, 19. For the Minkowski plane, similar calculations are given in [START_REF] Adabrah | Periodic Billiards Within Conics in the Minkowski Plane and Akhiezer Polynomials[END_REF], p. 469, 470. In the following, we compute B 4 and its roots We can check that its roots are

λ 1 = - a 2 b 2 a 2 -b 2 , λ 2 = - a 2 b 2 a 2 + b 2 , λ 3 = a 2 b 2 a 2 -b 2 .
They satisfy

λ 1 < -b 2 < λ 2 < 0 < λ 3 . ( 10
)
Note also that

λ 1    < -a 2 when a < √ 2b = -a 2 when a = √ 2b > -a 2 when a > √ 2b (11)
Denote by C i the caustic C λi where i = 1, 2, 3, with C i not defined when a = √ 2b. By Inequations [START_REF] Fierobe | Examples of reflective projective billiards and outer ghost billiards[END_REF], C 2 and C 3 are confocal ellipses, respectively smaller and bigger than E = C 0 . By Inequation [START_REF] Flatto | Poncelet's theorem[END_REF], the conic C 1 is a hyperbola if and only if a > √ 2b. • all complex quadrilaterals inscribed in E and circumscribed about a C j are billiard orbits;

• any complex quadrilateral billiard orbit of E, which do not have its edges on a foci line, is circumscribed about a C j ;

• any complex orbit inscribed in E and circumscribed about a C j is a quadrilateral orbit.

Example 7.2. We consider the case when a = 2 and = 1, see Figure 9. We compute that λ 3 = -λ 1 = 4 3 ≈ 1, 333 and λ 2 = 4 5 = 0, 8.

Let us investigate to which type of orbits corresponds each of the C i . To do so, we fix a point S to be the vertex (-a, 0) of the initial ellipse E, and we want to determine the 4-periodic orbits having S as a vertex. One is already known: it is the flat orbit whose vertices are on the foci-line and denoted by T 4 0 , see Figure 10. i the 4-periodic orbit tangent to C i , where i = 1, 2, 3. Let v i = (v x,i , v y,i ) be a vector directing a side of T 4 i starting from S, and such that q(v i ) = 1. By Proposition 4.12, one has

λ i = -(ab) 2 P (T 4 i ) = -b 2 v 2 x,i . ( 12
)
Thus we can determine v i for each i.

Case i = 1: We compute that

v 1 = 1 √ a 2 -b 2
(±a, ±ib).

A line passing through S and directed by one of the solution for v 2 intersects the ellipse in S and a point at infinity M ± = (a : ±ib : 0). We get the orbit (S, M + , S , M -) where S = -S, see Figure 11. It corresponds to the well-known real orbit (S, P, S , P ) where P = (0, b), P = -P and S = -S, see Figure 12. Case i = 3: Here we have

v 3 = 1 √ a 2 -b 2 (±a, ±i 2a 2 -b 2 ).

Figure 1 :

 1 Figure 1: The confocal caustic C λ inscribed in a piece of billiard trajectory.

Figure 2 :

 2 Figure 2: A projective billiard Ω

Figure 3 :Proposition 4 . 2 .

 342 Figure 3: In Proposition 4.2, we consider all quantities P (M 0 , v 0 ), P (M 1 , v 1 ), P (M 1 , v 1 ) and P (M 2 , v 2 ).

Figure 4 :

 4 Figure 4: A piece of billiard trajectory (M 0 , M 1 , M 2 ) on E with M 1 infinite as in the proof of Proposition 4.2. The points M 0 and M 2 are symmetric across O, hence M 2 = -M 0 and P (M 0 , v) = P (M 2 , v). Here E is represented as an hyperbola which allows us to view the tangent line at the infinity point M 1 as the vertical asymptote.

xvx a 2 + yvy b 2 =Figure 5 :

 25 Figure 5: The non-isotropic line passing through M and directed by v goes through a real focus if and only if P (M, v) = a -2 , see Proposition 4.7.

Figure 7 :

 7 Figure 7: The confocal caustic C λ inscribed in a piece of billiard trajectory.

1 λ

 1 -2 , b -2 by corollary 4.8, we have a 2 + λ = 0 and b 2 + λ = 0, B λ is invertible and B -defines the conic C λ . The above equality t wB λ w = 0 implies that all M j M j+1 are tangent to C λ . Now let us prove the second part of Theorem 1.2. It is a consequence of Proposition 4.12 which comes later. But first we will need the following Lemma 4.11. Let λ ∈ C be such that a 2 + λ = 0 and b

Theorem 5 . 1 (

 51 Cayley). Fix n ≥ 3. Let C and D be quadratic forms defining two regular conics in CP 2 in general position. Write det(tC + D) = A 0 + A 1 t + A 2 t 2 . . . the analytic expansion in 0 of the holomorphic function t → det(tC + D). Then there is an n-sided polygon inscribed in C and circumscribed about D if and only if

Corollary 5 . 6 .Proposition 5 . 8 .

 5658 If the roots of B n are simple and different from -a 2 and -b 2 , then N = deg B n . Remark 5.7. In order to compute the exact value of N , we can first try to compute the exact value of deg B n (see Proposition 5.8). Then we can understand when the assumptions of Corollary 5.6 are satisfied (see Proposition 5.11). Unfortunately we still do not know in which cases B n has simple roots. There exist r 1 , . . . , r p ∈ R + such that for all (a, b) with a / ∈ {r 1 b, . . . , r p b}, we have deg B n = n 2 -1 4

) 4 k 2 (- 1 )

 421 Fix a permutation σ of {1, . . . , m}. j) + j) = m(m + 1) and the coefficient in front of λ m(m+1) is m j=1 d(B σ(j)+j ). Since B n (λ) is a sum of ± m j=1 B σ(j)+j over all σ, we have that deg B n (λ) ≤ m(m + 1), and the coefficient in front of λ m(m+1) is d n (a, b) = d(B 2 ) . . . d(B m+1 ) . . . . . . . . . d(B m+1 ) . . . d(B 2m ) . Let us show that d n (a, b) = 0 except for specific (a, b) as described in Proposition 5.8. Note first that each d(B k ) is a homogeneous polynomial in (a -1 , b -1 ) of degree 2k, and by Equation (9) the coefficient in front of a -2k is (-1) k+1 4 k (2k -Cat k-1 where Cat k = 1 k+1 2k k is the k-th Catalan number. Now by linearity of the determinant, d n (a, b) is also a homogeneous polynomial in (a -1 , b -1 ), and we apply the same procedure as before: for any permutation σ of {1, . . . , m}, we have deg m j=1 d(B σ(j)+j ) = m j=1 deg d(B σ(j)+j ) = m j=1 2 (σ(j) + j) = 2m(m + 1) and the coefficient in front of a -2k is m j=1 j+σ(j)+1 4 j+σ(j) Cat j+σ(j)-1 = (-1) m(m+2) 2 m 4 m(m+1) m j=1 Cat j+σ(j)-1 . Since d n (a, b) is a sum of ± m j=1 d(B σ(j)+j ) over all σ, we have that deg d n (a, b) ≤ 2m(m + 1), and the coefficient in front of a -2m(m+1) is (-1) m 2 m(2m+1) det H m where H m is a Hankel matrix of the sequence (Cat k+1 ) k defined as

Proposition 5 . 10 .

 510 [START_REF] Krattenthaler | Advanced determinant calculus: a complement[END_REF], Theorem 33. Hence d n (a, b) is a nonzero homogeneous polynomial in (a -1 , b -1 ) and therefore there exists a at most finite collection of numbers r 1 , . . . , r p ∈ R + such that for all a, b > 0, we have d n (a, b) = 0 if and only if a ∈ {r 1 b, . . . , r p b}.We can ask the question if deg B n has always the maximal value (described in Proposition 5.8), or if we can find a, b > 0 such that deg B n is less than the value in Proposition 5.8. Proposition 5.10 asserts that we can find indeed such a, b by looking at the case of the circle (a = b). When a = b (in the case of the circle),

Example 6 . 3 .

 63 We consider the case when a = 2 and b = 1, see Figure8. We compute that

Figure 8 :

 8 Figure 8: The initial ellipse with its two caustics C λ-and C λ+ when n = 3, a = 2 and b = 1.

B 4 (

 4 λ) = 1 16a 6 b 6 (a 2 -b 2 ) 2 (a 2 + b 2 )λ 3 + (a 3 b -ab 3 ) 2 λ 2 -(a 6 b 4 + a 4 b 6 )λ -a 6 b 6 .

Corollary 7 . 1 .Figure 9 :

 719 Figure 9: The initial ellipse with its three caustics in the case when n = 4, a = 2 and b = 1.

Figure 10 : 4 0

 104 Figure 10: The flat 4-periodic orbit T 4 0 By Theorem 1.3, each one of the other orbits is tangent to a C i : denote by T 4i the 4-periodic orbit tangent to C i , where i = 1, 2, 3. Let v i = (v x,i , v y,i ) be a vector directing a side of T 4 i starting from S, and such that q(v i ) = 1. By Proposition 4.12, one has

Figure 11 :

 11 Figure 11: The infinite 4-periodic orbit T 4 1 . A change of coordinates y → iy to represent the the orbit properly ; this operation changes E into a hyperbola. The infinite points of the orbit, M + and M -, lie "at the end" of each branch of the hyperbola. Case i = 2: By Equation (12) and q(v 2 ) = 1 we get v 2 = 1 √ a 2 + b 2 (±a, ±b).

Figure 12 :

 12 Figure 12: The 4-periodic orbit T 4 2 .

In Section 5 we prove Theorem 1.3. We use a theorem of Cayley to establish the existence of confocal conics inscribed in an n-sided polygon which is itself inscribed in E, and also to bound their number. This

A line passing through S and directed by one of the solutions for v 3 intersects the ellipse in S and in one of the points

depending on the signs we choose for v 3 's coordinates.

The question is: why are the points N ± not on the x-or y-axis by symmetry of the 4-periodic orbit ? The reason is that the line passing through S and directed by v 3 reflected in the same line at one of the points N ± . Indeed, the system of equations in v P (N + , v) = P (T 4 3 ) and q(v) = 1

(see Lemma 4.6) has a unique solution (up to multiplication by -1), which is in fact v = v 3 . Thus the corresponding orbit is (S, N + , S, N -), see Figure 13. . Its edges SN + and SN -are reflected into themselves in N + and N -respectively. Here the drawing is biased, because we cannot represent T 4 3 which has complex vertices.


Magali S Grison
email: magali.grison@u-bordeaux.fr

Jules D Petit

Marie Glavier

Emmanuelle M Bayer

Quantification of Protein Enrichment at Plasmodesmata

Keywords: Plasmodesmata, PD Index, Protein enrichment, Confocal data analysis, Confocal microscopy

Intercellular communication plays a crucial role in the establishment of multicellular organisms by organizing and coordinating growth, development and defence responses. In plants, cellto-cell communication takes place through nanometric membrane channels called plasmodesmata (PD).

Understanding how PD dictate cellular connectivity greatly depends on a comprehensive knowledge of the molecular composition and the functional characterization of PD components. While proteomic and genetic approaches have been crucial to identify PD-associated proteins, in vivo fluorescence microscopy combined with fluorescent protein tagging is equally crucial to visualise the subcellular localisation of a protein of interest and gain knowledge about their dynamic behaviour. In this protocol we describe in detail a robust method for quantifying the degree of association of a given protein with PD, through ratiometric fluorescent intensity using confocal microscopy. Although developed for N. benthamiana and Arabidopsis, this protocol can be adapted to other plant species.

[Background] Currently, confirmation of protein localization to PD by confocal imaging is based primarily on two different approaches. On the one hand, the molecular composition of the cell wall surrounding PD differs. While the cell wall is highly enriched in cellulose, the environment near the PD is enriched in callose, a beta (1-3) glucan polymer that can easily and specifically be stained with fluorophore Aniline Blue. This staining presents the considerable advantage of being used as a PD marker without crossed lines. On the other hand, proteomic studies have identified specific PD proteins and led to their characterization [START_REF] Faulkner | Proteomic identification of putative plasmodesmatal proteins from Chara corallina[END_REF][START_REF] Fernandez-Calvino | Arabidopsis plasmodesmal proteome[END_REF][START_REF] Grison | Specific membrane lipid composition is important for plasmodesmata function in Arabidopsis[END_REF][START_REF] Brault | Multiple C2 domains and transmembrane region proteins (MCTPs) tether membranes at plasmodesmata[END_REF]. These proteins can be used as PD markers in subsequent studies when tagged with fluorescent proteins in transient or stable expression in plants [START_REF] Thomas | Specific targeting of a plasmodesmal protein affecting cell-to-cell communication[END_REF][START_REF] Simpson | An Arabidopsis GPIanchor plasmodesmal neck protein with callose binding activity and potential to regulate cell-tocell trafficking[END_REF]. Note that PD proteins can be exclusively associated with PD but can also present a dual localization within different cellular compartments as for Synaptotagmin1 (SYT1), Multiple C2 domains and Transmembrane regions Protein 4 (MCTP4) which associate with both the Endoplasmic Reticulum and PD [START_REF] Levy | Synaptotagmin SYTA forms ER-plasma membrane junctions that are recruited to plasmodesmata for plant virus movement[END_REF][START_REF] Brault | Multiple C2 domains and transmembrane region proteins (MCTPs) tether membranes at plasmodesmata[END_REF]. Since PD are dynamic structures responding to developmental and environmental cues, their molecular constituents may vary conditionally (Benitez-

Note:

The Aniline Blue stock solution is 1 mg/ml in water. The Aniline Blue working solution is 0.025 mg/ml. Do not use higher concentration of aniline bleu when the protein of interest is GFP tagged otherwise the Aniline Blue signal will crosstalk with the GFP signal. This method allows PD staining in the cotyledon and the hypocotyl, the aniline blue staining in roots gives a resolution and staining efficiency that we find difficult to combine with PD Index calculation.

Note: For roots, we advise the users to do immunolocalization in whole mount roots using monoclonal antibody: Biosupplies Australia, (1-3)-beta-glucan-directed monoclonal, catalogue number 400-2 and the protocol described in [START_REF] Boutté | Immunocytochemical fluorescent in situ visualization of proteins in Arabidopsis[END_REF] a. Grow Arabidopsis seedlings during 4 to 6 days on ½ MS 1% sucrose agar plate under 16/8 h day/night photoperiodic condition (150 μE/m 2 /s, 22 °C).

b. Take 0.2 ml of Aniline blue solution at a concentration of 0.025 mg/ml in water with the 1 ml syringe.

Note: Abiotic or biotic stress conditions can be tested to compare the PD Index of the protein of interest, in that case the aniline blue can be diluted in water supplemented with different molecules (such as NaCl, Mannitol, …) or can be apply before the aniline blue staining (such as viral infection where the plant is infected few days before aniline blue infiltration).

Note that leaves or seedlings cannot be infiltrated twice. c. Remove the air from the syringe. d. Carefully take the seedling with tweezers. e. Gently push the seedling into the syringe, so that the aerial parts of the seedling (i.e., the cotyledons and hypocotyl) are immersed in the solution (Figures 1A-1D). The seedling's root should protrude from the syringe in order to be able to extract the seedling from the syringe after infiltration. f. Position your finger at the extremity of the syringe and slowly pull the piston out of the syringe and reach the 0.21 graduation, bubbles should appear at the surface of the cotyledons (Figures 1E-1H).

g. Count to 5 and then release the piston very slowly (Figures 1I).

Note: We recommended the users to be extremely delicate and gentle at both steps (f and g) otherwise the cells will explode under the pressure and it will lead to a general blurry blue coloration.

h. Carefully remove the seedling from the syringe by grasping the root of it with tweezers (Figures 1J-1K).

i. Place the seedling on a slide in a drop of water, and gently mount the coverslip. b. Take a small volume, around 0.2 ml, of aniline blue at a concentration of 0.025 mg/ml in water in a 1 ml syringe. c. Gently return the leaf to show its abaxial side.

d. Apply the syringe on the leaf and position your finger at the same location on the other side in order to block the syringe (Figures 2A-2B).

Note: Avoid infiltration in veins area of the leaf, the area to infiltrate should be as flat as possible. Also avoid the puncture site from the previous Agrobacterium infiltration.

e. Apply a small pressure on the leaf with the syringe and push slowly the piston in order to infiltrate (Figures 2C-2F). Nicotiana benthamiana leaf infiltration using a 1 ml syringe.

2. Using PD proteins as PD marker a. Grow Agrobacterium previously electroporated with the relevant binary plasmids of the protein of interest and of the PD marker (Table 1) in liquid Luria and Bertani medium with appropriate antibiotics, at 28 °C and 250 rpm, for 1 day. b. Perform a 1/10 dilution of each culture and grow again at 28 °C and 250 rpm until the culture reach an OD600 of about 0.8 c. Centrifuge the culture at 3,500 x g, discard carefully the supernatant and resuspend in water for a final OD600 of 0.3. d. Mix 1:1 volume of both the agrobacteria cultures transformed with the protein of interest and with the PD marker. e. Use 5 to 6 leaves stage plant and make a very small puncture on the abaxial side of each selected leaf. Avoid the leaf veins. f. Apply a 1 ml syringe containing the agrobacteria mix in water against the leaf at the puncture site g. Position your finger on the other side to block the syringe and gently push the piston while applying a small pressure on the leaf so the liquid can infiltrate. Here, it is advantageous to infiltrate a large portion of the leaf. h. Place the plant in the appropriate culture room for another 3 days for protein expression. 2. The pinhole value needs to be kept at airy 1 to ensure a focal plane as accurate as possible.

3. The excitation wavelength and the spectral acquisition windows should be adjusted according to the fluorescent proteins chosen in your experiment (Table 1).

4. Laser power, photomultiplier (PMT) and photomultiplier offset should be set such that the acquired signals are not saturated. Between experiments and in a same experiment the setting parameters of the PD marker can be modified to obtain the better signal as possible without bleaching. In a same experiment the fluorescent tagged protein of interest channel setting must be kept identical while comparing different mutants or conditions. Between experiments we also recommend keeping the same settings for the fluorescently-tagged protein of interest. However, the PD Index is a ratiometric calculation between Region Of Interest (ROI) in a same picture so changes on the fluorescent protein acquisition setting may be acceptable if really needed. 5. Line average can be applied during image acquisition, the same line average should be kept in all experiments.

6. During image acquisition, the sequential scanning is preferable compared to simultaneous scanning. It is important to control that the excitation wavelength used for one fluorescent protein does not excite the other one, i.e., ensure there is no crosstalk between the PD marker/Aniline blue and the tag of the protein of interest (Table 1).

Notes:

a. The Aniline Blue stock solution is 1 mg/ml in water. The Aniline Blue working solution is 0.025 mg/ml. Do not use higher concentration of aniline bleu when the protein of interest is GFP tagged otherwise the Aniline Blue signal will crosstalk with the GFP signal. b. To avoid crosstalk of fluorescence between channels, it is necessary to perform

Data analysis

A. Data collection using Fiji 1. Open the confocal image with Fiji. Do not split channels.

2. Open ROI Manager (Analyze > Tools > ROI Manager).

3. Use the circle selection tool from main panel and draw a circle on the PD marker channel that is slightly smaller than the size of the PD marker signal (Figure 3A).

Note: Only sharp callose signal should be selected as PD ROI. Do not select ROI either for PD ROI and for PM ROI when the "spot" in the aniline blue channel is blurry.

4.

Click "add" on the ROI Manager once the circle is well positioned and selected. A new line should appear in the ROI Manager.

5. Move the circle, without modifying it, to another signal and click "add" again.

Repeat

Step A4 until you have enough ROI (ROI; best between 5 and 10). Index value below or equal to 1 means that there is no specific enrichment nor accumulation of the protein of interest at PD, whereas a PD Index value above 1 is significant from enrichment of a protein at PD.

Note: For example, a PD resident protein like At-PDCB1, which is also located at the plasma membrane, present a PD Index of 1.45 when transiently expressed in N. benthamiana. QSK1 and IMK2, two PM associated LRR-RLKs which conditionally relocalize to PD, display a PD Index rising from 0.9 to 1 in control condition to 1.5 to 2 upon abiotic stress [START_REF] Grison | Plasma membrane-associated receptor-like kinases relocalize to plasmodesmata in response to osmotic stress[END_REF]. It is also important to note that the system of expression may sorely influence the PD Index value. Illustrating this, At-MCTP4 display a PD Index of 1.85 when transiently expressed in N. benthamiana, whereas the PD Index raised to 5 when stably expressed in Arabidopsis [START_REF] Brault | Multiple C2 domains and transmembrane region proteins (MCTPs) tether membranes at plasmodesmata[END_REF].

For statistical analysis the use of R software and the Rcmdr package is recommended.

Parametrical tests can be use only when n ≥ 20 and when the sample distribution respect the normal law. Non-parametrical tests are systematically used when n < 20.

www.bio-protocol.org/e3545

Bio-protocol 10(05): e3545. DOI:10.21769/BioProtoc.3545

8. Distilled water 9 .

 9 Luria and Bertani medium (LB broth Miller, Sigma-Aldrich, catalog number: L3152) 10. Sucrose (Sigma-Aldrich, catalog number: S7903) Equipment 1. Confocal Microscope: plant imaging was performed using a ZEISS microscope (ZEISS, model: LSM880) 2. 28 °C Shaking Incubator (Dutscher, MaxQ 4000, catalog number: 078381) 3. Centrifuge (Dutscher, Spectrafuge 6 C for 10 ml tubes, catalog number: 096610) Software 1. FIJI (https://imagej.net/Fiji) (Schindelin et al., 2012) 2. R (https://www.r-project.org) (R Core Development Team, 2015) www.bio-protocol.org/e3545 Bio-protocol 10(05): e3545. DOI:10.21769/BioProtoc.3545 Procedure A. Plant material preparation In Arabidopsis seedlings 1. Aniline Blue staining

  j. Immediately proceed to the acquisition under the confocal microscope. Note: Aniline Blue bleaches rapidly. Direct observation should be done with a low power of the mercury lamp to avoid the bleaching of the sample. In confocal mode, please use a www.bio-protocol.org/e3545 Bio-protocol 10(05): e3545. DOI:10.21769/BioProtoc.3545 405 nm laser power as low as possible for the same reason. With the ZEISS 880 confocal device, the laser power is generally set around 0.2 to 5%, but of course this may vary depending on the microscope used. Aniline blue staining can be visualised immediately and is stable during 10 to 15 min.

Figure 1 .

 1 Figure 1. Arabidopsis seedling infiltration. Illustration of the different steps of an Arabidopsis seedling infiltration using a 1 ml syringe.

  www.bio-protocol.org/e3545 Bio-protocol 10(05): e3545. DOI:10.21769/BioProtoc.3545 Note: When the liquid penetrates and migrates in the leaf, a darker area should appear and expend around the syringe. If the pressure applied is correct, the infiltration should be smooth, without resistance nor loss of liquid, and without wounding the surface of the leaf. f. Do not infiltrate the whole leaf, a small area of 1 cm 2 is sufficient. g. With a sharp razor blade, cut the infiltrated area of the leaf. h. Place the sample on a mounting slide, abaxial side of the leaf facing up. i. Add a drop of water on the sample. j. Cover it with the coverslip. k. Immediately proceed to the acquisition under the confocal microscope.

Figure 2 .

 2 Figure 2. Nicotiana benthamiana leave infiltration. Illustration of the different steps (A-F) of

  i. Using a razor blade, cut a square of approximately 1 cm 2 in the infiltrated area of the leaf. j. Place the sample on a mounting slide, abaxial side of the leaf facing up. k. Add a drop of water on the sample. l. Cover it with the coverslip. www.bio-protocol.org/e3545 Bio-protocol 10(05): e3545. DOI:10.21769/BioProtoc.3545 m. Immediately proceed to the acquisition under the confocal microscope. B. Confocal acquisition 1. The use of a water immersion objective 63x (NA ≥1.4) for observation is recommended to have the same refraction index between immersion and mounting. If water immersion objective is not available, the use of an oil-immersion objective is still possible.

  independent and combined excitation and detection at the excitation and emission wavelength of the individual fluorochrome or fluorescent protein, respectively. Control samples labeled only with a single fluorescent protein/PD marker should be prepared in order to verify that the laser excitation wavelength used to excite one fluorochrome does not excite the other. www.bio-protocol.org/e3545 Bio-protocol 10(05): e3545. DOI:10.21769/BioProtoc.3545

7 . 8 .

 78 Change the view to the other channel (the protein of interest fluorescence channel). Using the same circle resume from step A5 and add regions that do not overlap with the AB signals but are at the cell periphery (Figure3B).

9 .Figure 3 .

 93 Figure 3. ROI selection for PD Index calculation. A. The PD ROI are selected in the PD marker channel (white arrows). B. The PM ROI are selected in the protein of interest fluorescence channel (red arrows). Verify that the ROI selected in the protein of interest fluorescence channel do not correspond to a PD in the PD marker channel.

Table 1 . Fluorescence excitation and emission maximum of Aniline Blue and commonly used fluorescent proteins

 1 (FP: Fluorescent Protein or Dye; Ex: Excitation wavelength in nm;

	Em: Emission wavelength in nm)
	FP or Dye	Ex Maximum	Em Maximum
	Aniline Blue	405	450
	CFP	433	475
	mTurquoise	433	475
	mCerulean	433	475
	EGFP	488	507
	EYFP	514	527
	mVenus	515	528
	mCitrine	516	529
	mRFP	584	607
	mCherry	587	610

Acknowledgments J.D.P. is funded by a PhD fellowship from the Belgian "Formation à la Recherche dans l'Industrie et l'Agriculture" (FRIA grant no. 1.E.096.18). This work was supported by the National Agency for Research (Grant ANR-14-CE19-0006-01 to E.M.B), "Osez l'interdisciplinarité" OSEZ-2017-BBRIDGING CNRS program to E.M.B., the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 772103-BRIDGING to E.M.B).

Competing interests

The authors declare no competing financial interests.