Amandine Dufaug

Laurent Goujon

Christine Barthod

Impact of a motorised articulated laparoscopic needle holder with ergonomic handle on the gesture smoothness: a pilot study

Keywords: Gesture smoothness, Motorised and articulated instrument, Laparoscopy, Minimal invasive surgery

Purpose:

The objective of this study is to assess the smoothness of a laparoscopic gesture performed with a motorised articulated instrument in comparison with a conventional one. Methods: One right-handed laparoscopic expert surgeon has been recruited for this study. He performed ten vesicourethral anastomoses, in a pelvic stand, using two kinds of laparoscopic instruments (standard and motorised) for different operating room conditions, especially table height adjustments. A 3-axis accelerometer and cameras are used to record information on the dominant upper limb motions during the tasks. Results: Two different methods are applied in order to evaluate the gesture smoothness. Both are using the acceleration data of the elbow collected during the surgical procedure. The first method uses the first-order derivative of the acceleration and is called log dimensionless jerk. The second one is based on the acceleration's Fourier spectrum and is called SPARC. For similar surgical trials of 2.5 stitches, the two methods tend to show that the gesture smoothness is globally improved by using the motorised articulated laparoscopic needle-holder. Conclusions: These preliminary results give a first indication on the benefit of the motorisations and articulations of a surgical instrument on the gesture performance. They have to be validated with more trials.

Introduction

The minimally invasive surgery, especially laparoscopy, represents an important evolution of the surgical practice. It benefits patients by reducing the bleeding, the infection risk and the healing duration compared with open surgery [START_REF] Agha | Does laparoscopic surgery spell the end of the open surgeon?[END_REF]. It also presents well-identified drawbacks for the surgeons. Indeed, the instrument's length, essential to reach the operating site, leads the surgeon to adopt non-ergonomic postures, such as shoulder elevation and abduction, that create increased efforts, pains occurrence and work-related musculoskeletal disorders development [START_REF] Berguer | A comparison of the physical effort required for laparoscopic and open surgical techniques[END_REF] [3] [START_REF] Park | Patients Benefit While Surgeons Suffer: An Impending Epidemic[END_REF]. Moreover, the fixed position of the trocars causes a lack of manoeuvrability increasing these previous risks [START_REF] Choi | A Review of the Ergonomic Issues in the Laparoscopic Operating Room[END_REF] [START_REF] Sánchez-Margallo | Initial experience using a robotic-driven laparoscopic needle holder with ergonomic handle: assessment of surgeons' task performance and ergonomics[END_REF].

 Amandine Dufaug a.dufaug@gmail.com Ergonomic recommendations have been proposed such as the adaptation of the operating room environment, especially the table height, and new instrument handle designs [START_REF] González | Guide design of precision tool handle based on ergonomics criteria using parametric CAD software[END_REF]. Nonetheless, these adjustments have little effect on the instrument manoeuvrability.

Several technical solutions have been proposed to overcome this limitation such as articulations at the tip of the instrument or articulated handles leading to an increased intracorporeal mobility for limited upper limb movements [START_REF] Anderson | Robot-like dexterity without computers and motors: a review of hand-held laparoscopic instruments with wrist-like tip articulation[END_REF] [9] [START_REF] Herman | An articulated handle to improve the ergonomic performance of robotic dextrous instruments for laparoscopic surgery[END_REF]. More recently, motorised articulated instruments have appeared on the market. As described by Sánchez-Margallo, they provide new functionalities to improve the range of movement and surgeon's ergonomics [START_REF] Sánchez-Margallo | Initial experience using a robotic-driven laparoscopic needle holder with ergonomic handle: assessment of surgeons' task performance and ergonomics[END_REF]. The Dex™ (Dex Surgical, France) is one of these devices. Its impact on ergonomics has yet been shown in terms of gesture accuracy, surgeon's posture and muscular solicitation [START_REF] Rosset-Lanchet | Conception et réalisation d'un instrument motorisé à degrés de liberté multiples pour la chirurgie mini-invasive[END_REF] [12] [START_REF] Dufaug | Ergonomic Surgical Practice Analysed Through {sEMG} Monitoring of Muscular Activity[END_REF].

The aim of this study is to compare the laparoscopic gesture smoothness while using the Dex™ instrument in several representative operating room conditions with a standard laparoscopic working configuration using conventional instruments. The smoothness is a good indicator of the gesture performance quality, narrowly linked to the procedure success [START_REF] Fried | Objective Assessment of Technical Performance[END_REF]. In the literature, it is recommended to not apply the smoothness measures to entire rhythmic tasks but to specific patterns of movements [START_REF] Balasubramanian | On the analysis of movement smoothness[END_REF]. To our knowledge, the present preliminary study is the first to evaluate the use of a motorised articulated instrument in terms of smoothness.

The materials and methods used for the comparative study of smoothness are firstly described. Results are then presented and discussed. A conclusion is finally proposed on the contribution of the motorised instrument on the gesture smoothness.

Materials and methods

Participants

One right-handed surgeon was recruited for this study. He is an expert surgeon in the meanwhile of Hiemstra et al. [START_REF] Hiemstra | Intracorporeal Suturing: Economy of Instrument Movements Using a Box Trainer Model[END_REF]. He performed suturing tasks using both standard laparoscopic (Std) and motorised articulated (Dex™) needle holders.

Laparoscopic needle holders

The standard laparoscopic needle holder (Tonglu Kanger Medical Instrument Co., Ltd) presents a non-articulated shaft, a one-mobile jaw tool and a V-type axial handle [START_REF] Matern | Ergonomics: Requirements for Adjusting the Height of Laparoscopic Operating Tables[END_REF]. It can be mobilised among four degrees of freedom. The tool locking is ensured by a rack whilst the opening is allowed by a spring.

The Dex™ motorised articulated needle holder presents seven degrees of freedom whose one is located at the instrument handle [START_REF] Rosset-Lanchet | Motion control of a multi-dof grasper manipulator for laparoscopic surgery[END_REF]. This particular articulation allows the surgeon to both orientate the distal tool of the instrument and reduce the shoulder abduction during the procedure. Three motorisations ensure the opening of the jaws, the unlimited rotation of the tool and the flexion of the instrument's tip.

Experimental setup

For the study requirements, it is essential to create a working environment which is representative of the operating room. In order to be as reproducible as possible between each trial, the patient is replaced by a training box, called pelvitrainer (Fig. 1).

Fig. 1 Vesicourethral anastomosis pelvitrainer

The pelvitrainer contains a simplified representation of the pelvic anatomy after a radical prostatectomy. Two silicon phantoms are used to represent the bladder and the urethra. In order to reproduce several typical working conditions, it is installed on an adjustable height table. The top face of the pelvitrainer (representing the patient abdominal wall) can thus be set from 50 % to 90 % of the surgeon's elbow height [17] [19].

In order to display the intracorporeal gestures on a monitor placed in front of the surgeon, a camera is placed into the box (Fig. 2). Two trocars are inserted through incisions performed in the top face of the box. They create an access to the surgical site for the instruments.

Task

The expert surgeon performed a vesicourethral anastomosis into the pelvitrainer. This surgical gesture ends a radical prostatectomy [START_REF] Rozet | Anastomose urétrovésicale lors de la prostatectomie radicale laparoscopique[END_REF]. It consists in suturing the bladder with the urethra (Fig. 1). A stitch is composed of several phases: jab the semi-circular needle into the phantom, rotate it and pull it out of the soft tissue.

For each kind of working configuration (standard or motorised), the surgeon performed the task for five different working heights. Each trial is followed by a break. For the task realisation, the surgeon used a standard or the Dex™ needle holder in the dominant hand and a standard instrument in the contralateral hand for the ambidexterity conservation. The order of the task realisation was randomised for both used instrument and table height.

The task duration has been fixed equal to 5 min which has been considered long enough to quantify the gesture smoothness on several suturing stitches.

Smoothness definition and implementation

The smoothness is a characteristic feature of a trained movement and increases with motor learning and spatiotemporal coordination [START_REF] Balasubramanian | On the analysis of movement smoothness[END_REF]. A smooth movement presents continuity whatever its duration and amplitude are. It corresponds to a single peaked speed profile [START_REF] Fried | Objective Assessment of Technical Performance[END_REF]. Intermittencies disturb the speed profile by adding zero-speed time lapses, corresponding to successive acceleration/deceleration or null-acceleration [START_REF] Balasubramanian | On the analysis of movement smoothness[END_REF].

A smooth gesture was initially empirically defined as the minimal value of the third-order derivative of the position, called jerk [START_REF] Mazzarino | THE NEED FOR IMPULSIVITY & SMOOTHNESS Improving HCI by qualitatively measuring new high-level human motion features[END_REF]. The more the jerk value is important, the less the gesture is considered as skilled and coordinated. A dependence of the jerk with the signal amplitude and duration is indicated [START_REF] Balasubramanian | On the analysis of movement smoothness[END_REF]. Due to this dependence, most of the jerk-based methods failed to compare smoothness between two equivalent gestures performed by an unique subject or two different ones [22] [23]. A correct smoothness measure is thus defined to be dimensionless, with no dependence to signal duration and amplitude [START_REF] Balasubramanian | On the analysis of movement smoothness[END_REF].

In order to fit with this dimensionless specification, Hogan et al. proposed a formula based on the temporal position signal to quantify the smoothness (Eq. 1) [START_REF] Hogan | Sensitivity of Smoothness Measures to Movement Duration, Amplitude and Arrests[END_REF].

𝐿𝐷𝐿𝐽 = log ( (𝑡 2 -𝑡 1 ) 3 𝑣 𝑝𝑒𝑎𝑘 2 ∫ 𝑥 ⃛(𝑡)𝑑𝑡 𝑡 2 𝑡 1 ) (1)
LDLJ corresponds to the dimensionless jerk, t1, t2 to the temporal interval limits, vpeak to the maximal speed and x(t) to the position.

Balasubramanian et al., in 2015, compared different methods of smoothness quantification, based on jerk analysis, and classified them in terms of technical properties such that: validity, sensitivity, reliability and practicability [START_REF] Balasubramanian | On the analysis of movement smoothness[END_REF]. The LDLJ method is considered as the most relevant jerk based method. Nonetheless it presents a lack of reliability due to noise sensitivity.

These authors proposed a spectral method, called SPARC, which is less sensitive to signal noise than jerk-based methods. The obtained scalar value is directly linked to the number of submovements or intermittencies. The SPARC quantity, computed from the Fourier spectrum of the signal (Eq. 2), is also dimensionless [START_REF] Balasubramanian | On the analysis of movement smoothness[END_REF].

𝑆𝑃𝐴𝑅𝐶 = -∑ √ ( 1 𝐾 𝑐 -1 ) 2 + (∆𝑉 ̂[𝑘])² 𝐾 𝑐 -1 𝑘=1 (2)
SPARC corresponds to the spectral arc length, ΔV ̂[k] to the Fourier spectrum magnitude normalized by the maximal magnitude and Kc to the discrete Fourier transform index which is narrowly linked to the chosen cut-off frequency.

These methods are recommended to be applied on particular gestures of same finality [START_REF] Balasubramanian | On the analysis of movement smoothness[END_REF].

Data acquisition

The LDLJ and SPARC analyses are performed from acceleration acquisition using a three-axis accelerometer (ACL300, Biometrics, Ltd) fixed at the lateral epicondyle of the humerus. It allows following the shoulder's circumduction moves. The orientation of each axis is presented in Fig. 3. Each output is connected to a Datalog (MWX8, Biometrics, Ltd). The corresponding signals (converted in g) are transmitted by a Bluetooth protocol to a laptop through dedicated software at a sampling rate of 1000 Hz.

Data exploitation

From the acceleration signal, the LDLJ method application imposes to compute the acceleration module by a quadratic summation of the 3-axis components before to apply the first-order derivative in order to obtain the jerk.

As the evolutions of SPARC and LDLJ values with smoothness are opposite, we choose to use the absolute value of the SPARC in the following analysis.

During the different exercises, the surgeon does not perform the same number of stitches (Table 1). This leads to a different number of movements (and submovements) which can be associated to a wrong perception of the smoothness. The exercises are thus subdivided in order to compare the smoothness corresponding to the different working conditions for similar suturing sequences in terms of number of performed stiches. The chosen sequence then consists in 2.5 stitches which correspond to the minimal number of stitches practiced during an exercise (Dex at 90% of elbow height).

It is important to notice that the gestures performed by the surgeon to realise the stitches are not perfectly similar between each trial.

Results

The equivalent sequences of 2.5 stitches, acquired for different working heights from 50% to 90% of the surgeon's elbow height, are analysed in order to compare the impact of each kind of instrument on the gesture smoothness.

The LDLJ method is firstly applied on the data. The results are reported in Fig. 4. The analysis of the trials performed with the Dex™ shows that its use globally leads to a decrease of the LDLJ value comparatively with the standard condition for each working height, expected the 60% of the elbow height.

The relative changes computed from the smoothness obtained for the two working configuration are reported in Table 2. The SPARC analysis is then applied on the acceleration data (Fig. 5). The smoothness variations with the working heights are different of the one computed using the LDLJ method. Nonetheless, similarly to the LDLJ analysis, the Dex™ use leads to a weaker value than the standard configuration for all working heights except the 60%. For both LDLJ and SPARC analyses, the weaker value obtained for the standard working configuration corresponds to a working height equals to 60% of the elbow height whilst the maximal smoothness is reported for the Dex™ configuration for a working height of 70% of the elbow height For the standard instrument, the worst value is reported with both methods for the 50% of elbow height condition. The condition leading to the worst value is not the same for the Dex™ instrument: 50% for SPARC and 90% for LDLJ.

Discussion

The results obtained during this preliminary study show a tendency to a gesture smoothness improvement when the Dex™ instrument is used. Nonetheless, the results in terms of smoothness have to be carefully linked with the performed movements.

For a simple movement, with well-controlled components, an increase of LDLJ or SPARC values is unquestionably linked to a decrease of the gesture smoothness. For a more complex movement, such as a suturing, the number of submovements is not exactly the same between two trials. This effect is more important as the number of performed stitches varies that's why the subject has thus to focus on same finality tasks containing specific similar sets of movements.

Moreover, if the stitch phases are well identified (jab, turn and pull the needle into the tissue), the preparation phase is composed of a lot of sub-movement, that can be different each time, which leads to a misestimating of the smoothness.

In conclusion, an increase in the LDLJ or SPARC values can be due to a smoothness decrease, an increased number of motion components or intermittencies (due to static phases) into the motion signal.

Therefore, in order to evaluate the impact of the instrument on the gesture, it is important to correlate the previous results to the surgeon's feeling (Table 4) and the arm motions analysis, available through the external cameras (Table 5). In Fig. 4, concerning the standard instrument, the evolution of the smoothness evaluated by LDLJ as a function of the working height is consistent with the surgeon's feeling, assuming that comfort and smoothness are linked. For the Dex, are globally consistent with the surgeon's feeling, except for the working height corresponding to 60% of the surgeon's elbow height.

For the SPARC analysis, both standard and Dex™ configurations present results close to the surgeon's feeling, except for the highest working condition. However, in this case, the surgeon's arms movements are quasi-static and the number of submovements is strongly decreased. It shows that both SPARC and LDLJ indicators are not reliable enough to quantify the surgeon comfort, and further analysis have to be done.

For both LDLJ and SPARC analyses, the condition which leads to the smoother gesture in standard configuration corresponds to the one which is the less soliciting for the muscle (60% of the elbow height) [START_REF] Dufaug | Ergonomic Surgical Practice Analysed Through {sEMG} Monitoring of Muscular Activity[END_REF]. A possible link between comfort and smoothness can be hypothesised. With both methods, the maximal smoothness is reported for the Dex™ configuration for the working height usually used by the surgeon's in operating room (70% of the elbow height) [START_REF] Dufaug | Ergonomic Surgical Practice Analysed Through {sEMG} Monitoring of Muscular Activity[END_REF]. This observation leads to the question of the training influence on the smoothness for the Dex™ instrument.

The relative changes computed for the LDLJ method are smaller than the one reported when using the SPARC method. It means that the SPARC method allows a better discrimination of the smoothness changes occurring as the working configuration is modified (from the standard to the Dex instrument).

Conclusion

The manoeuvrability reduction introduced by the laparoscopic surgery leads to adoption of awkward static postures. By introducing articulation and motorisation on the surgical instruments, designers aim to improve the laparoscopy ergonomics. The Dex™ needle holder, with its articulated handle, has thus been analysed in terms of gesture smoothness, in order to apprehend its influence on the quality of gesture performance. Two methods, called LDLJ and SPARC, have been identified and applied on motion data (acceleration).

The implementation of both methods tends to show that the use of the Dex™ increases the gesture smoothness. Nonetheless, the SPARC method seems to be more appropriate to highlight the smoothness variation between both working configurations. The SPARC evolution is consistent with the surgeon's feeling, except for the table height equals to 90% of the surgeon's elbow height. Indeed, the SPARC method indicates an increased smoothness for the highest working condition. As the surgeon's movement becomes quasi-static (corresponding to a decrease of the submovements number and to an increase of intermittencies) for this working height, the computed smoothness increased. So we can see that the smoothness by itself is not a sufficient indicator to evaluate the surgeon's comfort. It has to be completed by a complementary EMG analysis [START_REF] Dufaug | Ergonomic Surgical Practice Analysed Through {sEMG} Monitoring of Muscular Activity[END_REF].

Despite the recommendation concerning the table height done at the early 2000 [START_REF] Matern | Ergonomics: Requirements for Adjusting the Height of Laparoscopic Operating Tables[END_REF] [19], 46 % of the surgeons place the patient abdominal wall at a height superior to the elbow level despite the associated risks for them in terms of workrelated musculoskeletal disorders development [START_REF] Modi | Awareness of Ergonomic Guidelines regarding laparoscopic surgeries, its Practice among Surgeons and Comfort level during and after surgery[END_REF]. This paper allows showing that the working height impacts not only the comfort but also the gesture smoothness and, in consequence, its quality of performance, narrowly linked to the success of a surgical procedure.

Although the preliminary status of this analysis, we consider that the obtained results are a first and important step towards the validation of a motorised instrument as the Dex™ to improve suturing performance. Studies with more participants have to be carried out to verify the impact of such an instrument on smoothness as well as its implication in muscular activity and fatigue.

Compliance with Ethical Standards

Fig. 2

 2 Fig. 2 Vesicourethral anastomosis procedure set up

Fig. 3

 3 Fig.3Accelerometer's axes orientation Two cameras are placed in front and in the lateral side of the surgeon in order to record his arms motion, especially the dominant one.

Fig. 4

 4 Fig. 4 LDLJ analysis on a set of 2.5 stitches versus working heights for standard (STD) and Dex instruments

Fig. 5

 5 Fig. 5 SPARC analysis on a set of 2.5 stitches versus working heights for standard (STD) and Dex instruments

Figure 5

 5 Figure 4

Table 1

 1 Number of stitches performed for each task

	Working height	50%	60%	70%	80%	90%
	Standard	3	5	4	4.5	3.5
	Dex™	3.5	3.5	3.5	4	2.5

Table 2

 2 Relative change in LDLJ computation

	Working heights	50%	60%	70%	80%	90%
	εrel	0.09	-0.09	0.07	0.04	0.01

Table 3

 3 Relative change in SPARC computation

	Working heights	50%	60%	70%	80%	90%
	εrel	0.06	-0.10	0.22	0.16	0.14

Table 4

 4 Comfort felt by the surgeon for both working configurations

	Working	50%	60%	70%	80%	90%
	heights
	Comfort	poor		 good 		poor

Table 5

 5 Visual quantity of movement of the right upper limb (elbow) for both working configurations

	Working	50%	60%	70%	80%	90%
	heights
	Posture	static	 dynamic 	static

Acknowledgements

This work has been funded by the Conseil Savoie Mont Blanc and the Université Savoie Mont Blanc.

The authors acknowledge Dr Olivier Skowron for his participation at this study as an expert and Dr Nicolas Forestier for the equipment loan.

We would like to thank Y. Paya, R. Trecani and B. Bourjaillat for their active participation in the project definition (research on preliminary exercises design, data processing) as well as M. Bloch, M. Beauclair, R. Gonnet and D. Paradis for their implication into the bibliography research on the smoothness determination.

Conflict of interest: the authors have no conflict of interest to disclose.

Research involving Human Participants and/or Animal:Yes

Informed consent: Not required


Corentin Fierobe

On the circumcenters of triangular orbits in elliptic billiard

, which proves the same result with the incenters, and [6], which among others introduces the theory of complex reflection in the complex projective plane. The result we present was found at the same time by Ronaldo Garcia in an article to appear in American Mathematical Monthly (no preprint available). His proof uses completely different methods of real differential calculus.

Overview of the problem

The famous great theorem of Poncelet, cf [START_REF] Berger | Géométrie[END_REF] and [START_REF] Poncelet | Propriétés projectives des figures[END_REF], asserts that if there exists an n-sided polygon inscribed in a conic K i and circumscribed about an other one K c , then there are infinitely many such polygons, and you can find such one for each point of K i chosen to be one of its vertices. A classical proof of it can be found in [START_REF] Berger | Géométrie[END_REF]. [START_REF] Ph | Cayley's explicit solution to Poncelet's porism[END_REF] and [START_REF] Schwartz | The Poncelet grid[END_REF] give a way to prove it using complex methods.

It has a lot of consequences (see [START_REF] Berger | Géométrie[END_REF], [START_REF] Dragovic | Bicentennial of the great Poncelet theorem (1813-2013): current advances[END_REF]), especially in billard theory, since it gives a condition to the existence of particular n-periodical orbits in conics. In particular, given an ellipse E, one can find a confocal ellipse γ to E, such that each triangular orbit on E is circumscribed about it; and conversely one can complete each tangent line to γ to a triangular orbit of E.

We study here the set of circumcenters, which are the centers of the circumscribed circles, of all triangular orbits in such an elliptic billard E. We want to prove the following: Theorem 1.1. The set C of the circumcenters of all triangular orbits of the billiard within an ellipse is also an ellipse. Remark 1.2. Theorem 1.1 is obvious in the particular case where the ellipse is a circle, because then the set of circumcenters is reduced to a single point. Thus, from now on we will assume that the ellipse is not a circle.

There are many other results similar to theorem 1.1. Dan Reznik discovered experimentally the same result for the incenters of triangular orbits, see the video [START_REF] Reznik | [END_REF] and the github page [START_REF] Reznik | New Properties of Triangular Orbits in Elliptic Billiards[END_REF] written with Jair Koiller. Romaskevitch (see [START_REF] Romaskevich | On the incenters of triangular orbits in elliptic billiard[END_REF]) confirmed these observations by proving them and her proof widely inspired ours. Tabachnikov and Schwartz, in [START_REF] Schwartz | Centers of mass of Poncelet polygons, 200 years after[END_REF], proved that the loci of the centers of mass (and of an other particular point) of a 1-parameter family of Poncelet n-gons in an ellipse is an ellipse homothetic to the previous one. They also mention that a similar result was proved by Zaslawski, Kosov and Muzafarov for the orthocenters ( [START_REF] Zaslavsky | Trajectories of remarkable points of the Poncelet triangle[END_REF], reference from [START_REF] Schwartz | Centers of mass of Poncelet polygons, 200 years after[END_REF]). And Garcia (see [START_REF] Garcia | Centers of inscribed circles in triangular orbits of an elliptic billiard[END_REF]) uses explicit calculations to prove that the loci of incenters and orthocenters of triangular orbits are ellipses, and describes them precisely.

Before going into details, we give here a brief summary of the proof, which is inspired by [START_REF] Romaskevich | On the incenters of triangular orbits in elliptic billiard[END_REF], and in which we use the same complex methods. We consider a projective complexified version of C, denoted by Ĉ, which turns out to be an algebraic curve as a consequence of Remmert proper mapping theorem, see [START_REF] Ph | Principles of algebraic geometry[END_REF] p. 34. Then we show that the intersection of the complex curve Ĉ with the foci line of the boundary ellipse E is reduced to two points, each one of them corresponding to a single triangular orbit. Further algebraic arguments on the intersection type of Ĉ with the foci line of E allow to conclude that it is a conic, using Bezout theorem. It's then easy to check that Ĉ is an ellipse since its real part is bounded.

As explained, one considers the projective complex Zariski closure of the ellipse E and a complexified version of C, Ĉ. In order to define Ĉ and to prove the first statement concerning the intersection with the foci line, we study an extension of the reflection law and of the triangular orbits to complex domain, as in [START_REF] Romaskevich | On the incenters of triangular orbits in elliptic billiard[END_REF], and we use some of the results contained in the later article such as Proposition 2.14. Complex reflection law and complex planar billiards were introduced and studied by A. Glutsyuk in [START_REF] Glutsyuk | On quadrilateral orbits in complex algebraic planar billiards[END_REF] and [START_REF] Glutsyuk | On Odd-periodic Orbits in complex planar billiards[END_REF]. See also [START_REF] Glutsyuk | On 4-reflective complex analytic billiards[END_REF] where they were applied to solve the two-dimensional Tabachnikov's Commuting Billiard conjecture and a particular case of two-dimensional Plakhov's Invisibility conjecture with four reflections.

In [START_REF] Chang | elliptical billiard systems and the full Poncelet's theorem in n dimensions[END_REF] section III, a billiard reflection law was defined in purely projective terms using polar duality with respect to quadrics. Their definition makes sense in any dimension and over any number field (R, C, . . .). However, the definition of [START_REF] Chang | elliptical billiard systems and the full Poncelet's theorem in n dimensions[END_REF] is given only for billiards within quadrics, but it can be adapted to any smooth hypersurface except at certain points, as we will prove in Subsection 2.4. Later a complex reflection law in two dimensions was introduced in another way by Glutsyuk [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] which also takes into account these exceptional points. In dimension 2 and in the complex case, the two definitions coincide as we will prove in Subsection 2.4. The author follows Glutsyuk's definition in this work.

Section 2 is devoted to the complex reflection law and to complex orbits in a complexified ellipse: in Subsection 2.1, we introduce the complex reflection law; Subsection 2.2 recalls some results about complexified conics; we further define what is a triangular complex orbit in Subsection 2.3; then, in Section 3 we introduce the definition and we study properties of complex circumscribed circles to such orbits: Proposition 3.6 is the main result of this section. Finally, Section 4 is devoted to the proof of Theorem 1.1, using previous results.

Complex triangular obits on an ellipse

Complex reflection law

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 line at infinity. 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 in R 2 can be extended to CP 2 by considering the complexified version of the euclidean squared metric, that is the quadratic form

Q(z, w) = z 2 + w 2
which is a non degenerate quadratic form on C 2 . In a similar way to the euclidean case, it leads to construct a notion of symmetry in C 2 . It has two isotropic subspaces -which by definition are spaces where the quadratic form vanishes identically -of dimension 1 : namely F I := C(1, i) and F J := C(1, -i). Since both spaces F I and F J are contained in their orthogonal spaces, the definition of symmetry of a line with respect to F I or F J cannot be introduced as in the usual euclidean way. A similar notion of symmetry with isotropic spaces is studied in [START_REF] Khesin | Pseudo-Riemannian geodesics and billiards[END_REF] and [START_REF] Dragovic | Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics[END_REF] in pseudo-Euclidean and pseudo-Riemannian cases. Definition 2.1 ([6], definition 1.2). Define the isotropic points at infinity (or cyclic points) to be either I = [1 : i : 0] or J = [1 : -i : 0]. A line in CP 2 is said to be isotropic if it contains either I or J, and non-isotropic if not. Notice that the line at infinity is automatically isotropic. Definition 2.2 ([6], definition 2.1). The symmetry with respect to a line L ⊂ C 2 is defined as follows:

• Case 1: L is non-isotropic. The symmetry acting on C 2 is the unique nontrivial complex isometric involution (for the above quadratic form) fixing the points of the line L. It induces the same symmetry acting on lines.

• Case 2: L is isotropic. We define the symmetry of lines through a finite point x ∈ L: two lines l and l which contain x are called symmetric if there are sequences (L n ) n , (l n ) n , (l n ) n of lines through points x n so that L n is non-isotropic, l n and l n are symmetric with respect to L n , l n → l, l n → l , L n → L and x n → x.

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 2.3 ([6], lemma 2.3). If L is an isotropic line through a finite point x and l, l are two lines which contain x, then l and l are symmetric with respect to L if and only if either l = L, or l = L. This complex reflection law allows to talk about complex billiard orbits on the ellipse, as it will be done in subsection 2.3. Before studying those orbits, it is necessary to introduce some geometric notions about projective conics.

Preliminary results on complexified conics

One needs to present here useful results on confocal conics. They can be found in [START_REF] Berger | Géométrie[END_REF] and [START_REF] Klein | über höhere Geometrie[END_REF]. [START_REF] Glutsyuk | On quadrilateral orbits in complex algebraic planar billiards[END_REF] also cites them in subsection 2.4. This section allows to understand some links between an ellipse and its Poncelet ellipse of triangular orbits, when both are complexified. Thus, by conic (resp. ellipse) we mean here the complex projective closure of a real regular conic (resp. ellipse). This choice of definition is due to the fact that the ellipse in which we study billard orbits is a real ellipse. Later, in order to define circumscribed circle (see section 3), we will understand conics as general complex conics (not just complexified real ones).

Proposition 2.4 ([1] subsection 17.4.2.1).

A conic is a circle if and only if some of the points I or J belong to it. Furthermore, if a conic is a circle, then both I and J belong to it.

In fact, a circle has two isotropic tangent lines intersecting at its center (see the following propositions).

Proposition 2.5 ([1] subsection 17.4.3.1).

A focus f of a conic lies in the intersection of two isotropic tangent lines to the conic.

Proposition 2.6 ([13], p. 179). Two complexified confocal ellipses have the same tangent isotropic lines, which are four isotropic lines taken with multiplicities: one pair intersecting at a focus, and the other one -at the other focus.

This brings us to the following redefinition of the foci: Definition 2.7 ([1] subsection 17.4.3.2). The complex foci of an ellipse are the intersection points of its isotropic tangent lines.

Remark 2.8. The complex projective closure of a real ellipse has four complex foci, including two real ones. Corollary 2.9. A conic has at most four dinstinct finite isotropic tangent lines, each two of them intersecting either at a focus, or at an isotropic point at infinity.

Complex orbits

We have enough material at this stage to study complex triangular orbits. See [START_REF] Glutsyuk | On quadrilateral orbits in complex algebraic planar billiards[END_REF], definition 1.3, for a more general definition of periodic orbits. Definition 2.10 ([6], definition 1.3). A non-degenerate triangular complex orbit on a complex conic E is an ordered triple of points A 1 , A 2 , A 3 on E so that for all i, one has A i = A i+1 , the tangent line T Ai E is non isotropic, and the complex lines A i A i+1 and A i A i-1 are symmetric with respect to T Ai E (with the obvious convention A 0 := A 3 and A 4 := A 1 ). A side of a non-degenerate orbit is a complex line A i A i+1 .

Remark 2.11. The vertices of a non-degenerate orbit are not collinear since a line intersects the ellipse in at most two points.

Remark 2.12. As explained in [START_REF] Glutsyuk | On Odd-periodic Orbits in complex planar billiards[END_REF], the reflection with respect to a non-isotropic line permutes the isotropic directions I and J. This argument implies that a non-degenerate triangular orbit has no isotropic side. We will also study the limit orbits of the above defined orbits, which will be called degenerate orbits.

Definition 2.13 ([6]

). 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 orbits and which is not a non-degenerate orbit. We define the sides of a degenerate orbit as the limits of the sides of the non-degenerate orbits which converge to it. If A j = A j+1 , then it is natural to define the side A j A j+1 as the tangent line T Aj E.

Proposition 2.14 ([18], 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.

During the proof, it will be convenient to distinguish two types of orbits : the ones with no points at infinity, and the others, with at least one point at infinity: Definition 2.15. An infinite triangular complex orbit on a complex conic E is an orbit which has at least one vertex on the line at infinity. The orbits with only finite vertices are called finite orbits.

Proposition 2.16. An infinite orbit is not degenerate, and has exactly one vertex at infinity.

Proof. Suppose two vertices, α, β, of the orbit are at infinity. Then, αβ is the line at infinity. But the tangent T β to the ellipse E in β is not isotropic, and the line at infinity reflects to itself through the reflection by T β . Hence, the orbit is {α, β} = C ∞ ∩ E, which should be a degenerate orbit. But it cannot be a degenerate orbit by Proposition 2.14 since the tangent lines to its vertices α, β are not isotropic. Thus, only one vertex lies at infinity. Therefore, if it is a degenerate orbit, it has two vertices, α, β, corresponding by Proposition 2.14 to two sides, A which is isotropic and tangent to the ellipse in α, and B which is a line containing α and β. Since the tangency points of isotropic tangent lines are finite, α is finite. Thus β is infinite (because the orbit is supposed infinite). Then B and the tangent line T β E to the ellipse in β are collinear (since they have the same intersection point at infinity). But both are stable by the complex reflection by T β , hence T β E = B which is impossible since B is not tangent to the ellipse. ----→ x i x i+1 satisfy the usual reflection law. Then all the sides of the polygon are tangent to a confocal quadric Q. Furthermore, for these quadrics Q 1 , . . . , Q m , Q, there is an (n -1)-parameters family of polygons satifying the same property.

Digression: reflection law on a quadric

----→ x i-1 x i ,
Their definition of reflection has a natural extension to other fields k than R, and we would like to compare it when k = C to the complex reflection law which is described in Subsection 2.1. We are really grateful to the referee who suggested us to read the above-mentionned article.

In the real case one can define locally an inside and an outside of a quadric Q (since a real hypersurface of R n locally separates a small ball centered at its points into two domains). Therefore one can define an orientation on incident and reflected rays, so that the reflected ray is directed to the side where the incident ray arrived from. But in complex spaces this is not the case anymore, and in the following descriptions we will omit any considerations on orientations of rays.

Consider the field k = R or C and let Q be a non-degenerate quadric in P n (k). For any hyperplane u ⊂ P n (k) one can define its pole z with respect to Q (see [START_REF] Chang | elliptical billiard systems and the full Poncelet's theorem in n dimensions[END_REF] for more details). Now suppose Q 1 and Q 2 are two quadrics. If x ∈ Q 1 , one can consider the tangent space u = T x Q 1 to Q 1 at x, and denote by z its pole with respect to Q 2 . Definition 2.18 (section III of [START_REF] Chang | elliptical billiard systems and the full Poncelet's theorem in n dimensions[END_REF] without orientations). Note u = T x Q 1 and suppose that z / ∈ u. Let 1 and 2 be two lines intersecting Q 1 at x transversaly. We say that 1 and 2 satisfy the law of reflection at x on Q 1 about Q 2 if the lines 1 , 2 , xz are contained in a plane P , and if the lines 1 , 2 , xz, P ∩ u form a harmonic set.

As explained in [START_REF] Chang | elliptical billiard systems and the full Poncelet's theorem in n dimensions[END_REF], when k = R and Q 1 and Q 2 are two confocal quadrics (the definition of confocal quadrics is recalled below, see formula (1)), this definition is the same as the usual billiard reflection. We would like to show that this result holds in the case when Q 1 and Q 2 are complex confocal quadrics, namely that genericaly two lines 1 and 2 intersecting at a point x ∈ Q 1 satisfy the law of reflection at x about Q 2 (as in Definition 2.18) if and only if they are symmetric by the complex reflection law at x (as defined in Subsection 2.1).

On CP n (z0:...:zn) , we choose a standard affine chart U = {z n = 1} ⊂ CP n . Consider the complex quadratic form Q = n-1 k=0 dz 2 k on C n which can be seen as a quadratic form on U C n .

Proposition 2.19. Let Q 1 , Q 2 be two distinct confocal quadrics in CP n , x ∈ Q 1 ∩ U , u = T x Q 1 and
z be the pole of u with respect to Q 2 . Then in the chart U , u and xz are orthogonal for Q.

Remark 2.20. We believe this result is well-known by specialists and can be shown using elementary results of polar geometry, but we were unable to find any proof of it. This result is also true in the real case, but we haven't found any proof of it, even in [START_REF] Chang | elliptical billiard systems and the full Poncelet's theorem in n dimensions[END_REF] where it is also mentionned without any proof.

Proof. Any point in CP n can be written as the projection x = (x 0 : . . . : x n ) of a point x ∈ C n+1 . We suppose that Q i = {q i = 0} for i = 1, 2 where q 1 , q 2 are the quadratic forms defined for all x = (x 0 , . . . , x n ) by

q 1 (x) = n-1 k=0 x 2 k a k -x 2 n and q 2 (x) = n-1 k=0 x 2 k a k + λ -x 2 n ( 1
)
with a 1 , . . . , a n , λ being complex numbers such that a k = 0 and a k + λ = 0. Write M 1 (respectively M 2 ) the diagonal (n + 1) × (n + 1) matrix whose diagonal coefficients are the a -1 k (respectively the (a k +λ) -1 ), k = 0 . . . n, and -1. Both matrices M 1 and M 2 are such that q i (x) = x|M i x and the bilinear form associated to q i is b i (x, x ) = x|M i x , where we set x|y = n k=0 x k y k . Note that for any matrix M one has x|M y = M T x|y where M T is the transpose of M . Therefore, for a x ∈ Q 1 ∩ U , the equation of T xQ 1 is given by the set of all ȳ verifying

0 = b 1 (x, y) = M 1 x|y = M 1 x|M -1 2 M 2 y = M -1 2 M 1 x|M 2 y = b 2 (M -1 2 M 1 x, y).
Hence the pole of u = T xQ 1 with respect to Q 2 is the point z where z = M -1 2 M 1 x. Now the diagonal forms of the matrices allow to compute easily that

z = M -1 2 M 1 x = x + λu where u = x 0 a 0 , . . . , x n-1 a n-1 , 0 .
Therefore xz is the line containing x and directed by u. Now the vector subspace of C n U parallel to T xQ 1 ∩ U is defined by the equation Yet if two lines 1 and 2 intersect at x, they are symmetric with respect to u (for the complex reflection law as defined in Subsection 2.1) if and only if the lines 1 , 2 , u and the Q-normal line to C at x form a harmonic set. Note that if u is an isotropic tangent line to C, then u and its Q-normal line through x are the same, and the above mentionned result on harmonic sets makes no sense. But in the other case we have the Corollary 2.22. If u = T x C is not an isotropic tangent line to C, then two lines 1 and 2 through x are symmetric with respect to u for the complex reflection law introduced in Subsection 2.1 if and only if 1 and 2 satisfy the law of reflection at C about any confocal conic C (Definition 2.18). Therefore Definition 2.18 coincide with the complex reflection law introduced in Subsection 2.1 in the case when the reflection takes place on a conic, at a point x satisfying a generic condition of non-isotropy of its tangent line. From Proposition 2.19 we deduce a way to extend Definition 2.18 to any smooth hypersurface Γ which needs not to be a quadric. Definition 2.23. Let Γ ⊂ CP n be a germ of C 1 -smooth hypersurface at a point x ∈ CP n . Consider a quadric Q 1 containing x and such that T x Q 1 = u. We say that two lines 1 and 2 through x are CCS-symmetric with respect to u if and only if 1 and 2 satisfy the law of reflection at Q 1 about any confocal quadric Q 2 given by Definition 2.18. By Proposition 2.19, this definition doesn't depend on the choice of the quadric Q 2 . We still have the Proposition 2.24. Let Γ ⊂ CP 2 be a germ of C 1 -smooth curve and x ∈ Γ such that u = T x Γ is not isotropic. Then two lines 1 and 2 through x are symmetric with respect to u for the complex reflection law introduced in Subsection 2.1 if and only if 1 and 2 are CCS-symetric with respect to u.

We conclude that, in the complex setting, Definition 2.23 extends the law introduced in [START_REF] Chang | elliptical billiard systems and the full Poncelet's theorem in n dimensions[END_REF] to any smooth hypersurface Γ of CP n , and a similar work can be done for hypersurfaces of RP n . In the two-dimensional complex case, this extension coincide with the complex reflection law introduced in [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] except that it is not well-defined at points of isotropic tangency.

Circumcircles and circumcenters of complex orbits

Here we present the last part of the required definitions, which concerns the complex circles circumscribed to triangular orbits. This part is different from the previous one, because here the considered conics are complex and not necessarily complexified versions of real conics.

Definition 3.1.

A complex circle is a regular complex conic passing through both isotropic points at infinity. Its center is the intersection point of its tangent lines at the isotropic points.

Proposition 3.2. For a non-degenerate finite orbit, there is a unique complex circle passing through the vertices of the orbit and both isotropic points at infinity. It is called the circumscribed circle or circumcircle to the non-degenerate orbit.

Proof. Denote by α, β, γ the vertices of the orbit. We have to prove that no three points of α, β, γ, I, J are collinear. Indeed, as no vertices are on the line at infinity, we only need to study two different cases:

1. α, β, γ are not collinear because they are distinct and they lie on the ellipse which has at most two intersection points with any line.

2. α, β, I are not collinear or else the line αβ would be isotropic. But this is impossible for a non-degenerate triangular orbit by Remark 2.12.

We then exclude all other possible combinations of two vertices of the orbit with I or J, using the same arguments.

Let us extend this definition to degenerate orbits.

Definition 3.3.

Let T be a degenerate or infinite orbit. A circumscribed circle of T is the limit (in the space of conics) of a converging sequence of circumscribed circles of non-degenerate finite orbits converging to T . If a sequence of complex circles converges to a conic so that their centers converge to a point c ∈ CP 2 , then c is called a center of the limit conic. A circumcenter of T is a center of its circumscribed circle.

Remark 3.4. A priori, a limit conic K may have several centers in the sense of this definition. Indeed, c depends on the choice of the sequence of circles converging to K. See Case 4 of Proposition 3.5 and its proof for more details.

Even if they are called circles, the circumscribed circles to a degenerate or infinite orbit can degenerate into pairs of lines, as described below.

Proposition 3.5. The limit of a converging sequence of complex circles is one of the following:

1. a regular circle ;

a pair of isotropic non-parallel finite lines ; the corresponding center lies on their intersection ;

3. the infinite line and a finite line d ; the center c lies on the line at infinity and represents a direction which is orthogonal to d ; 4. the line at infinity taken twice : its center can be an arbitrary point in CP 2 .

Proof. The equation of a regular circle D is of the form a(x 2 + y 2 ) + pxz + qyz + rz 2 = 0 where a, p, q, r ∈ C, a = 0 and 4ar = p 2 + q 2 . Both isotropic tangent lines to D have equations 2a(x ± iy) + (p ± iq)z = 0, whose intersection is c = (p : q : -2a), which is the center of D by definition.

If we take a limit of regular circles, the equation of the limit circle is of the same type, that is a(x 2 + y 2 ) + pxz + qyz + rz 2 = 0 but maybe with a = 0 or 4ar = p 2 + q 2 . And the center c is still of coordinates (q : p : -2a).

If a = 0, the limit circle is the union of the line at infinity (z = 0) and the line d of equation px + qy + rz = 0. The line d is finite if and only if (p, q) = 0, and in this case it has direction (q, -p). Since c = (p : q : 0), the direction represented by c is orthogonal to d. If d is infinite, the limit circle is the (double) line at infinity. Note that in this case the center can be an arbitrary point.

If a = 0, but 4ar = p 2 + q 2 , the equation of the limit circle becomes

x + p 2a z 2 + y + q 2a z 2 = 0
which is the equation of two isotropic non collinear lines intersecting at the point (-p 2a : -q 2a : 1) = (p : q : -2a) = c. If a = 0 and 4ar = p 2 + q 2 , the limit circle is regular. Now let us find which triangular orbits have their center on the line of real foci of E. Proposition 3.6. Suppose that T is a complex triangular orbit whose circumcenter lies on the real foci line. Then T is finite, non-degenerate, symmetric with respect to the real foci line of E, and has a vertex on it.

Proof. Let T be a triangular orbit with a circumscribed circle C having a center c on the real foci line of E.

First case : Suppose T is finite and non-degenerate. We follow the arguments of Romaskevich [START_REF] Romaskevich | On the incenters of triangular orbits in elliptic billiard[END_REF] who treated the similar case for incenters. Indeed, at least two vertices should lie outside the foci line. If the line through them is not orthogonal to the foci line, then this pair of vertices together with their symmetric points and the remaining third vertex in T are five distinct points contained in the intersection E ∩C. This is impossible, since E is not a circle. Finally, the remaining vertex has to be on the foci line, or else we could find two distinct orbits sharing a common side, which is impossible by definition of the reflection law with respect to non-isotropic lines.

Second case : Suppose T is infinite. Then the line at infinity cuts C in three distinct points, hence C is degenerate. By Proposition 3.5, C contains the line at infinity. Since T has only one infinite vertex α by Proposition 2.16, and two other finite vertices β, γ, the other line d ⊂ C is not the line at infinity. Again by Proposition 3.5, the center is infinite and represents the orthogonal direction to d. Since it is on the real foci line, the latter is orthogonal to d. Thus d intersects the infinity line at the same point as the line orthogonal to the foci line. This point does not lie in E, and in particular, d does not contain α. Hence, we have d = βγ is a side of T , α / ∈ d and by the same symmetry argument as in the first case α should belong to the real foci line. But this is impossible since the latter intersects E in only two finite points.

Last case : Suppose T is degenerate. Then C cannot be a regular circle, otherwise the latter would be tangent to E in a point of isotropic tangency (by Proposition 2.14): this would imply that this point of isotropic tangency is I or J, which is impossible since they do not belong to E, assumed not to be a circle.

The circumcircle C cannot be the union of the line at infinity and another line d. Otherwise, by the same arguments as in the second case, this line would be othogonal to the real foci line. Since T is finite (Proposition 2.16), d goes through its both vertices, implying that they are symmetric with respect to the foci line. Therefore, both vertices are points of isotropic tangency but this cannot happen for a degenerate triangular orbit.

Finally suppose C is the union of two isotropic lines having different directions. Proof. The tangent line to C n at M n is orthogonal to the line M n c n hence the same is true for their limits. The limit of T Mn C n is obviously the limit of the line M n N n . Since M n and N n are on E, the line M n N n also converges to the tangent line T α E. Hence T α E is orthogonal to αc.

Thus if α is a vertex of isotropic tangency of the orbit, Lemma 3.7 implies that αc is orthogonal to T α E, hence αc = T α E since the latter is isotropic. Recall that α does not lie in the real foci line. Since both isotropic lines constituing the circle go through c, one of them is T α E. Hence, they are both tangent to E by symmetry with respect to the real foci line. Thus the other vertex of T is a point of isotropic tangency of E, which is not possible by the previous arguments (such an orbit is not closed).

Proof of Theorem 1.1

We reall that E is a complexified ellipse, which we will identify with C. Denote by γ the real ellipse inscribed in all triangular real periodic orbits. We use the same notation γ for its complexified version.

Consider the Zariski closure T of the set of real triangular orbits (which are circumscribed about γ). Let T 3 denote the set of triangles with vertices in E that are circumscribed about γ. It is a Zariski closed subset of E 3 CP 1 3 that contains the real orbits and can be identified with the set of pairs (A, L), where A is a point of the complexified ellipse E and L is a line through A that is tangent to γ. The set of the above pairs (A, L) is identified with an elliptic curve, and each pair extends to a circumscribed triangle as above, see the complex Poncelet Theorem and its proof in [START_REF] Ph | Cayley's explicit solution to Poncelet's porism[END_REF] for more details. Hence T 3 is an irreducible algebraic curve. Each triangle in T is circumscribed about γ, by definition and since this is true for the real triangular orbits and the tangency condition of the edges with γ is algebraic. Thus T ⊂ T 3 . Hence T = T 3 , by definition and since the curve of real triangular orbits (which is contained in T ) is Zariski dense in T 3 (irreducibility). Now the set T ⊂ T of complex non-degenerate triangular orbits circumscribed about the Poncelet ellipse γ is a subset of T 3 = T , Zariski open in T (because T \ T is defined by polynomial equations). Note that T \ T is finite (since it is a proper Zariski closed subset of an algebraic curve T ), and T is dense in T for the usual topology. Thus the analytic map φ : T → CP 2 which assigns to a non-degenerate orbit its circumcenter can be extended to a holomorphic map T → CP 2 , being a rational map. And by Remmert proper mapping theorem (see [START_REF] Ph | Principles of algebraic geometry[END_REF]), its image denoted by Ĉ is an irreducible analytic curve of CP 2 , hence it is an irreducible algebraic curve by Chow theorem (see [START_REF] Ph | Principles of algebraic geometry[END_REF]).

Let us show that Ĉ is a conic, using Bezout theorem and studying its intersection with the real foci line of E. In fact, we already know two distinct points lying on this intersection: the circumcenters c 1 and c 2 of both triangular real orbits T 1 and T 2 circumscribed about Poncelet's ellipse γ and having a vertex on the foci line.

Lemma 4.1. The foci line of the ellipse intersects Ĉ in only c 1 and c 2 which are distinct, and for each i the only triangular orbit of T having c i as a circumcenter is T i .

Proof. Take a point c of Ĉ lying on the foci line. Then by Proposition 3.6, an orbit of center c is finite, non-degenerate, and has a vertex on the foci line. If this orbit is in T , it is circumscribed about γ. One of its vertices lies on the foci line, hence coincides with a vertex of some T i . Hence it is T 1 or T 2 , otherwise we could find a number strictly greater than two of tangent lines to γ containing a vertex of E. Furthermore, if c 1 = c 2 , the circumcircle of T 1 would be the same as the one of T 2 by symmetry, and E would share six dictinct points with the former, which is impossible. The result follows.

Theorem 4.2. The set Ĉ ⊂ CP 2 is an ellipse.

Proof. Let us show that c 1 is a regular point of Ĉ, and that the latter intersects the foci line transversally. Fix an order on the vertices of T 1 and consider the germ (T , T 1 ). The latter is irreducible (because parametrized by γ), hence the germ (V, c 1 ) ⊂ ( Ĉ, c 1 ) defined as φ(T , T 1 ) is also irreducible. By Lemma 4.1, any other irreducible component V of ( Ĉ, c 1 ) is parametrized locally by φ and a germ (T , T 1 ), where T 1 is obtained from T 1 by a permutation of its vertices. Thus V = V since φ doesn't change by permutation of the vertices of the orbits: ( Ĉ, c 1 ) is irreducible.

We fix a local biholomorphic parametrization P (t) of the complexified ellipse E, so that P 0 = P (0) is a vertex of the real ellipse E that is also a vertex of the real triangular orbit T 1 . This gives local parametrizations of the orbits T (P ) whose first vertex is P and of their circumcenters c(t) = φ(T (P (t))). We restrict P to the curve P (t) parametrizing the real points of E. We can suppose that P (t) and P (-t) are symmetric with respect to F. Write r(t) = |P (t)c(t)| for the radius of the circumscribed circle to T (t). Thus we have c(0) = φ(T 1 ) = c 1 , and we need to show that c (0) = 0 and that c (0) has not the same direction as the line of real foci of E.

First, we have r(t) = r(-t) by symmetry, and r is smooth around 0 since P (0) = c(0). Thus, r (0) = 0. This implies that the vector c (0) -P (0) is orthogonal to the line c(0)P (0), which is the real foci line by definition. But P (0) is already orthogonal to the foci line (being a vector tangent to E at its vertex P 0 ), hence the same hold for c (0). It's then enough to show that c (0) = 0.

Suppose the contrary, i.e. c (0) = 0. We use again r (0) = 0. If we denote by Q(t) one of the other vertices of T (t) and Q 0 = Q(0), then since also r(t) = |Q(t)c(t)|, the equality r (0) = 0 gives that the line Q 0 c 1 is orthogonal to T Q0 E. It means that the circumscribed circle D to T 1 has the same tangent line in Q 0 as E. Since this is also true in P 0 and in the third point of T 1 (same proof), we get that E and D have three common points with the same tangent lines, which means that E is a circle. But this case was excluded at the beginning (remark 1.2).

Hence c (0) = 0 and c (0) is orthogonal to the line of real foci. The proof is the same for c 2 . Hence by Bezout theorem, Ĉ is an ellipse.

a

  k dx k = 0 which is the equation of the Q-orthogonal space to u. Hence xz and T xQ 1 are Q-orthogonal. Choose n = 2, consider a standard affine chart C 2 ⊂ CP 2 . On this chart define the same quadratic form Q(z, w) = z 2 + w 2 as in subsection 2.1. Let C be a conic and x ∈ C. Proposition 2.19 implies the following Corollary 2.21. The Q-normal line to the tangent line u = T x C to C is the line xz, where z is the pole of u with respect to any confocal conic C to C.

Lemma 3 . 7 .

 37 Let C n be a sequence of circles containing two distinct points M n and N n of E converging to the same finite point α. Suppose C n has a center c n converging to a finite point c = α. Then the line cα is orthogonal to the line T α E.

  Q 1 with respect to another quadric Q 2In[START_REF] Chang | elliptical billiard systems and the full Poncelet's theorem in n dimensions[END_REF] section III, the authors define the reflection law inside a quadric Q 1 with respect to another quadric Q 2 in R n using polar considerations. They use this definition to prove what they call the Full Poncelet's Theorem:

Theorem 2.17 (Chang, Crespi, Shi, see

[START_REF] Chang | elliptical billiard systems and the full Poncelet's theorem in n dimensions[END_REF]

). Let Q 1 , . . . , Q m be confocal quadrics in R n and x 1 x 2 . . . x m be a polygon such that x i lie on Q i and the successive sides

Acknowledgments

This article could not be possible without the precious help, advices and support of Alexey Glutsyuk and Olga Romaskevich, who also suggested me to work on this topic. This article is an adaptation of Olga's proof, and many ideas come from her article, [18]. I am also really grateful to Anastasia Kozyreva for her help.