Catherine Voison
email: cathoison@wanadoo.fr

So Multiples Multiplications de peaux d'artistes et autres curiosités biologiques reproductibles

Dès les années 1990, les techniques manipulatoires du vivant liées au développement constant des biotechnologies 1 deviennent la source d'un nouveau type de fabrication et de diffusion d'artefacts biologiques et nous assistons aujourd'hui à de singulières procédures de transformation et de reproduction du microcosme organique in ou ex vivo et in vitro dans le champ de l'art. Ces biofacts 2 produits grâce aux nouvelles pratiques de la biologie cellulaire et du génie génétique sont-ils admis dans ce qu'il est convenu d'appeler le « monde de l'art » 3 parce qu'ils sont seulement l'oeuvre d'une innovation technologique destinée à confondre l'activité artistique et celle de l'ingénieur ? Plus encore, ces pratiques de la biologie contemporaine peuvent-elles correspondre aux « normes » d'une esthétique contemporaine en vigueur susceptibles d'orienter nos modes de perception et de représentation du réel vers de nouveaux modèles vivants issus d'une programmation couplant mécanismes naturels et technologie ? L'hégémonie d'une nature primordiale et universelle qui a parcouru le courant naturaliste du XVIII e siècle et l'émerveillement que procure le dynamisme génésique du vivant ne sont plus de mise quand des « artistes », parvenus à franchir le seuil des laboratoires, modifient les mécanismes internes qui gouvernent l'activité des cellules et les contraignent à se multiplier et à adopter un nouveau mode d'existence technique. La notion de reproduction liée à la génération, à l'engendrement naturel est abandonnée au profit de techniques manipulatoires telles que la transgénèse [START_REF]Introduction de gènes étrangers dans le génome de cellules embryonnaires d'animaux, de plantes ou de bactéries[END_REF] ou le clonage 5 et l'équation vivant/naturel n'a plus lieu d'être. Toutes les inventions produites par la reconfiguration technique de la nature en une sorte de sur-création jette un pont entre la culture scientifique et l'art et prend la forme d'un « design biologique » qui n'est autre qu'une substitution accomplie de la nature à la culture. La mise en vue de ces artefacts sur le devant de la scène artistique est contestable tant d'un point de vue esthétique que d'un point de vue éthique. L'exposition de ces nouveaux objets nous incite à réfléchir à ce qui peut encore « faire oeuvre » aujourd'hui et à reconsidérer nos positions à l'égard du concept d'art tel qu'il apparaît à l'heure où la nature du vivant contrôlable et contrôlée devient techniquement reproductible. Il y a quinze ans déjà que Joe Davis résolu à ne faire aucune distinction entre ses activités scientifiques et artistiques a réussi à convaincre les biologistes moléculaires de l'école de médecine d'Harvard et de l'Université de Berkeley en Californie de lui apprendre à synthétiser la molécule d'ADN et à l'insérer dans le génome de bactéries vivantes. L'ADN synthétique issue de segments de la molécule, (les gènes) produite chimiquement en laboratoire) est devenu un matériau et un outil privilégié de ses travaux. Ce substrat physicochimique contenu dans les gènes de toutes les cellules est synthétisé et encodé selon un algorithme que l'« artiste » programme. Joe Davis rêve de transformer les constituants de cette longue chaîne de nucléotides en un moyen de communication élargi à l'univers intra-stellaire et célèbre à sa manière le génie génétique qui lui permet d'introduire de nouveaux messages dans les cellules. Il crée ainsi de nouveaux gènes à partir de la numérisation d'un symbole graphique qu'il encode dans une séquence génétique de la bactérie Escherica Scoli 7 , il les nomme infogènes et les définit ainsi : « un gène qui peut être traduit par la machine humaine en une signification et non par la machine cellulaire en une protéine » 8 . Les bactéries porteuses de ces infogènes, peuvent s'exprimer de façon inchangée sur une longue période de temps et sont assez résistantes, même dans des conditions extrêmes (dans l'espace par exemple) pour se répliquer un très grand nombre de fois [START_REF]À partir d'une cellule bactérienne initiale, on peut obtenir en peu de temps une colonie extraordinairement étendue : au rythme courant de deux divisions à l'heure, une bactérie alimentée suffisamment engendrerait en quarante-huit heures plus de vingt trillions de descendants, et en quelques jours une masse du volume de la terre[END_REF] . Joe Davis s'appuie ainsi sur la singularité du mode de reproduction, une reproduction par scissiparité des organismes unicellulaires (les bactéries), pour répondre à son projet : diffuser à l'infini, dans un temps et un espace non limités les signes tangibles de l'intelligence de l'homme. médiatisé par les « artistes ». Quelle faculté, autre que celle de la raison, si ce n'est une croyance aveugle, nous inciterait à croire que les biotechnologies nous ouvrent la voie vers le meilleur des mondes, si nous ne savons pas en calculer les risques ? Les résultats des travaux Joe Davis réalisés à partir de ce substrat physico-chimique fournissent des résultats dont la portée symbolique l'emporte sur la dimension spectaculaire. En effet, les réalisations de Joe Davis peu visibles à l'oeil nu sont assorties de symboles graphiques et de nombreux commentaires de l'« artiste » 11 . L'artiste fait appel à l'ingénierie génétique pour produire des objets capables de s'autoorganiser ou de se répliquer comme des organismes vivants. Cependant, l'auteur de ces bactéries infogénisées est-il en mesure de contrôler ce qu'il a conçu ? La diffusion ad libitum de tels organismes fait émerger la question de la responsabilité du concepteur face à la prolifération de ses artefacts hybrides. Par ailleurs, la bactérie re-construite par l'« artiste » non seulement pour être reproduite techniquement mais pour se multiplier perd son statut d'objet biologique original pour devenir un matériau de production et un appareil de diffusion à la portée du plus grand nombre. Ce mode de fabrication qui transforme le vivant en objet ordinaire et que nous pourrions comparer à un ready-made « aidé » de Marcel Duchamp, inaugure une nouvelle forme de multiple [START_REF] Artiste | Daniel Spoerri est le premier artiste à développer la notion de multiple qui faisait descendre l'art de son piédestal en insufflant un rôle nouveau aux objets et aux documents ; il fonde en 1959 l'édition[END_REF] . Paradoxalement, le multiple, objet « sauvé » de son aspect fonctionnel grâce à l'art qui le conceptualise plus qu'il ne le fabrique, peut rapidement se transformer en produit de consommation. Il en va ainsi du travail de Chrissy Conant qui fait usage du marketing, une pratique issue des grandes sociétés industrielles pour garantir la qualité et la valeur de ses productions et non des moindres puisqu'il s'agit de ses propres gamètes qu'elle produit en série. Nous avons donc été plus loin comme cobayes […] et maintenant nous offrons nos peaux à leurs expériences de multiplication de cellules d'épiderme. Ce n'est plus la multiplication des pains, mais la multiplication d'un morceau vivant, extrait, développé, entretenu dans sa vie hors de notre corps. Fascinant, ils nous présentent nos petits morceaux de peaux d'épidermes gonflés, grandis, prêts à la greffe…et comme nous sommes attirés par ces petits morceaux de nous hors de nous, ils veulent bien nous en laisser quelques-uns sur des dermes animaux, nous en ferons ce que nous voulons si nous leur abandonnons encore un peu de nos cellules fraîches. Nous sommes repartis avec ces morceaux, ainsi qu'une recette de fabrication plutôt délicate à appliquer hors d'un laboratoire stérile. [START_REF] Huyghe | Art Orienté Objet[END_REF] Les artistes cultivent leurs peaux sur le tissu conjonctif du derme d'un cochon mort. (Ce mammifère est celui qui possède le plus d'analogies physiologiques avec l'homme). Les cellules vivantes de leur épiderme se multiplient ainsi sur le tissu et un nouveau fragment de peau se développe. Les échantillons de leurs peaux tatouées sont ensuite présentés dans des bocaux de formol au centre d'une vitrine.

Il nous faut souligner l'analogie de présentation de ces fragments de peaux mihumaines mi-animales destinées à être achetées par un collectionneur et ceux qui contiennent des fragments anatomiques, destinés à être montrés dans les cabinets d'histoire naturelle au siècle des Lumières [START_REF]Ainsi, pour que rien n'échappe à la curiosité des mondains de ce « siècle des Lumières », toutes les « merveilles » du corps humain, de la nature, font l'objet d'exposition dans les cabinets d'histoire naturelle et les cabinets de cire des particuliers[END_REF] . Toutefois, la mise en vue de ces fragments anatomiques contemporains ne relève pas d'une démarche naturaliste visant à vulgariser les découvertes scientifiques d'une époque mais à dénoncer la valeur marchande de ces fragments humains inédits et la fascination aveugle que les pratiques scientifiques contemporaines exercent sur le public.

Nous avons donc été plus loin comme cobayes […] et maintenant nous offrons nos peaux à leurs expériences de multiplication de cellules d'épiderme. Ce n'est plus la multiplication des pains, mais la multiplication d'un morceau vivant, extrait, développé, entretenu dans sa vie hors de notre corps. Fascinant, ils nous présentent nos petits morceaux de peaux d'épidermes gonflés, grandis, prêts à la greffe… [START_REF] Laval-Jeantet | La grec[END_REF] Les nouvelles fonctionnalités des entités technobiologiques sont soustraites à des données que la science objective. En revanche, elles sont subjectivées par l'art, soit comme l'expression d'une résistance aux impératifs d'une nouvelle forme de consumérisme du vivant, soit pour anticiper les perspectives prometteuses des technosciences.

  http://www.so-multiples.com page 2 Notre propos visera d'autre part à interroger ce qu'il reste de l'incarnation originelle du vivant lorsqu'il est fragmenté, délocalisé et devient l'outil d'une reproduction programmée techniquement. Lorsque certains « bioartistes » considèrent le vivant sous l'aspect d'un support réinscriptible, programmable et donc capable de recevoir de nouvelles données, gamètes, bactéries, cellules tissulaires animales ou humaines, cellules souches 6 , protéines etc. sont à l'épreuve d'un nouveau formatage de la matière organique. Pour approcher le mode de fabrication et de diffusion des biofacts qui « font oeuvre » aujourd'hui et pour illustrer notre propos sur la reproductibilité de ces objets hybrides, intéressons nous plus avant aux travaux de Chrissy Conant qui vit et travaille à New York, à ceux du « sculpteur » Joe Davis, chercheur au centre d'arts visuels du M.I.T. (Massachusetts Institute of Technology) et aux « échantillons » du couple français Art Orienté objet (AOo) formé de Marion Laval-Jeantet et de Benoît Mangin. Les modes de production « bio-artistiques » de Joe Davis ou de Chrissy Conan sont, à ce titre, révélateurs d'une nouvelle pratique d'exposition et de diffusion en série d'artefacts vivants.

Ill 3 .

 3 C'est ainsi que Joe Davis parvient à réaliser Microvenus (ill. 1, 2, 3 et 4), organisme recombinant « messager » d'un nouveau type de langage (contenant selon lui le signe de l'intelligence humaine) à destination d'hypothétiques espèces vivantes intelligentes elles aussi, qui font aujourd'hui l'objet d'une nouvelle science : l'exobiologie. Davis choisit Escherichia coli, une bactérie dont les humains ont besoin pour digérer correctement et qui, selon les expérimentations de la NASA, a survécu plus de cinq ans après une exposition au froid intense et à de fortes radiations. Pour son message, il sélectionna Microvenus, un symbole simple -comme un Y et un I superposé -qui est à la fois la rune germanique représentant la vie et le tracé schématique de la configuration externe des organes génitaux féminins […] Convertie en données numériques et transposée dans un brin d'ADN pourvu de 28 nucléotides, Microvenus se glissa pour la première fois entre les gènes d'Escherichia coli en 1990. La bactérie se multiplia rapidement dans ses boîtes allant jusqu'à contenir un milliard de cellules, chacune d'entre elles portant un exemplaire distinct de l'icône. […]. « Je suis probablement l'éditeur le plus célèbre de l'histoire » dit Davis avec un sourire. « Il y a plus de copies de mon travail que du travail de Salvador Dali, d'Escher et de tous les autres réunis ». 10 Ill. 1. Symbole Microvenus Ill 2. Image matricielle du symbole Double brin d'ADN Microvenus formé de 28 nucléotides Source : http://www.aec.at:festival2000/texte/artistic_molecules_2_e.htm Ill 4. Boîtes de Pétri contenant les Bactéries porteuses de Microvenus MIT Council for The Arts, Ars Electronica 2000 Source : http://www.aec.at:festival2000/texte/artistic_molecules_2_e.htm L'américain idéalise la structure et les fonctions de la molécule d'A.D.N. Or, nos connaissances actuelles sur les conséquences des nouvelles pratiques biotechnologiques sont insuffisantes pour que nous accordions du crédit à un idéal d'existence technique page 4

5 )

 5 Après s'être soumise à un traitement hormonal stimulant chimiquement l'activité ovarienne en vue de produire plusieurs ovocytes lors d'une ovulation, Chrissy Conant récolte et conditionne une douzaine d'ovules en référence aux oeufs de poules qui sont emballés par douze dans le commerce. Placés et scellés dans des pots de verre semblables aux bocaux qui contiennent le caviar Beluga, les douze « oeufs » sont labellisés : "Caucasian : Packed by Private IVF Center, USA. Product of Conant Ovaries, Keep Refrigarated." (ill. 5) Ce label est présent sur le couvercle de chaque pot sous la forme d'une étiquette empruntant les couleurs à celle du caviar Beluga. Une photo séduisante de l'« artiste » est reproduite au centre de cette étiquette[START_REF]Conant espère que sa production « surpasse le caviar Beluga et devienne le fin du fin des objets de consommation de luxe. Dans le contexte des beaux-arts, en utilisant mes gènes comme matériau, je fais de l'art avec mon corps[END_REF] . Chrissy Conant expose et met en vente ses ovules placés sur un présentoir réfrigéré semblable à ceux que l'on rencontre dans les épiceries[START_REF]L'exposition intitulée Chrissy Caviar® a eu lieu à la galerie Saatchi de Londres en[END_REF] . Ces cellules non fécondées et néanmoins indispensables à la reproduction sont exposées et conditionnées comme des marchandises de luxe périssables (au même titre que le caviar). Ill 5. Chrissy CONANT, Chrissy Caviar®, douze oeufs humains (un par pot), fluide tubaire humain, Polyester, nylon, verre, cuivre, équipement de réfrigération, 122 cm w x 131 cm h x 126 cm, Galerie Saatchi, Londres, 2001-2002. Étiquette du label créée par Chrissy CONANT, Caucasian : Packed by Private IVF Center, USA. Product of Conant Ovaries, Keep Refrigarated , 2001-2002, impression à jet d'encre sur papier 100% coton, 69 cm x 89 cm (in The Molecular Gaze, Art in the Genetic Art, Suzanne Anker, Dorothy Nelkin, Cold Spring Harbor Press, New York, 2003) Faire travailler son organisme et le contrôler jusqu'à obtenir la production d'objets consommables, c'est à cette tâche hors normes que se livre Chrissy Conant qui expose et vend sur son site Internet ses ovules produites en nombre grâce aux techniques médicales de fécondation assistée et qui selon l'auteur lui permettront d'améliorer son mode de vie et de mettre fin à son célibat. J'ai passé un contrat avec un embryologiste qui, dans un laboratoire de fécondation in vitro, a conditionné chacun de mes ovules dans les conditions requises de stérilité. Il m'a présenté un endocrinologue qui m'a prescrit un traitement hormonal d'un mois.-et a surveillé mes progrès par ultrasons et dosages sanguins-afin qu'à l'ovulation je produise environ douze oeufs, et non un seul comme dans un cycle normal. […] Dans chaque bocal, au lieu d'oeufs de poissons, il y a mes ovules. Chacun est scellé hermétiquement dans un tube de verre, avec du fluide utérin humain, comme le font les scientifiques avec les ovules et/ou les embryons humains ou murins (les oeufs de souris et de hamster sont physiologiquement et génétiquement comparables aux oeufs humains, et sont utilisés dans les laboratoires de fécondation in vitro pour l'entraînement et la recherche). 15 page 6 Le corps comme outil de production artistique n'est pas nouveau dans le monde de l'art. Toutefois cette nouvelle forme d'art « charnel » qui révèle l'obsolescence de l'unicité du corps et stigmatise sa fragmentation, fait émerger l'idée que la reproductibilité technique du vivant nous permet d'accéder à de nouvelles réalités d'existences que certains artistes biotechnophiles fabriquent à l'envi. Mais jusqu'où la science peut-elle dicter nos désirs ? Consommateurs d'une existence « sur mesure » sommes nous appelés à devenir des usagers des biotechnologies de plus en plus exigeants ? Lorsque la « machine cellulaire » a perdu son système originel et original de production au profit d'une re-programmation externe qui remet en cause le destin des formes vivantes, il y a lieu de considérer l'artefact biologique comme un objet altéré. En exposant des corps biologiques « étrangers » à la nature, produits d'un montage issu des appareils qui pénètrent la matière vivante, les bio-designers transforment l'objet vivant en support de tous les possibles. Désormais « le corps sort de la nécessité pour entrer dans le possible » et l'évolution scientifique qui fait de l'homme l'oeuvre de lui-même nous ouvre la voie de l'hominescence. 16 Le scientifique à l'instar de l'ingénieur soumet le matériau biologique à des opérations techniques afin d'améliorer son fonctionnement et d'accroître son rendement, en revanche, le concepteur d'objets techno-naturels -le designer -qui fait seulement usage des biotechnologies pour reconfigurer le vivant à des fins esthétiques, est-il pour autant un artiste ? Quel crédit accorder à ce nouveau mode de fabrication d'objets baptisés « oeuvres d'art » si nous ne sommes pas en mesure de leur accorder le moindre jugement de goût ? Les « artistes » qui mettent en avant les résultats de la recherche scientifique chercheraientils à nous convaincre que le vivant est devenu un matériau comme les autres ? Si l'art souscrit aux mêmes stratégies que celles de la science : convaincre à tout prix de la fabrication technique du vivant par un langage rationnel et cohérent, il échappe à toute forme d'autonomie et de singularité. Un grand nombre d'artefacts biologiques exposés dans le champ de l'art, ont en commun avec les ready-made d'être réalisés dans l'indifférence à l'égard de l'aspect qui s'en dégage 17 . Toutefois nous ne pouvons placer tous les artefacts biologiques donnés en spectacle au rang de ready-made même s'ils ont une valeur critique qui remet en cause l'art et son mode de réception 18 . Certains auteurs, certes, font basculer l'objet biologique au rang de ready-made vivant lorsque leurs pratiques se jouent de toute détermination sensible liée à sa « fabrication ». En revanche, certains artefacts issus de la vie suscitent des émotions tant de la part des artistes que des « regardeurs » attestant une interprétation sensible des réalités scientifiques. Il en va ainsi du travail du couple AOo. Après avoir été les cobayes pour divers programmes de recherche médicale dans le centre De Framingham près de Boston, Marion Laval-Jeantet et Benoît Mangin deviennent les cobayes de leur propre recherche, en se manipulant eux-mêmes et en réalisant par biopsie Cultures de peaux d'artistes (ill. 6) 19 . Au sein de ce laboratoire expérimental en biotechnologie sont prélevés des fragments de leurs peaux en vue d'une culture et d'une étude sur leur développement. Après accord avec les chercheurs, ils réussissent à se procurer une dizaine de fragments de ces peaux. Ill 6. AOo, Culture de peaux d'artistes, derme de porc, cellules humaines, formol, tatouages de figures animales et Végétales (in Art Orienté objet, Marion Laval-Jeantet, Benoît Mangin, CQFD, Paris)

  Quelle tâche spécifique se donne l'art bitech' en diffusant de multiples exemplaires d'organismes reprogrammés techniquement ? Pour notre part, nous pensons que l'art biotech n'est pas un savoir-faire impliquant l'idée d'habileté de l'artisan ou du technicien 23 mais un faire savoir qui prend la forme d'une pratique spéculative : obtenir du vivant reproductible « sur mesure » pose la question du pire ou du meilleur des mondes à venir ? Cette forme d'art devient une procédure susceptible de nous confronter à l'image d'un nouveau monde de plus en plus objectivé où le savoir prend le pas sur le savoir-faire. Il nous faut encore distinguer la part de création ou de re-création (dans toute l'acceptation de ce terme), qu'il nous faut accorder à ce mode de fabrication qui actualise des possibles et met au monde des sujets plutôt que des objets, des sujets ex-situ dépourvus de toute fonction spontanée de reconnaissance du monde dans lequel ils sont destinés à évoluer. La fabrique d'objets ou de sujets « d'art biotechnologique » ne doit donc ni se résumer à une surenchère de l'hubris humaine sur la nature ni se limiter à la diffusion de pratiques remettant en cause la place de l'humain dans la société. Pour répondre à la dénomination d'artiste, ce « fabriquant » ou ce « démonstrateur » d'objets biologiques doit pouvoir satisfaire à certains critères esthétiques susceptibles de nous permettre de questionner les limites du champ d'application des pratiques biotechnologiques et à faire bon usage du vivant. De manière moins radicale nous pensons que les artefacts biotechnologiques produits par les artistes sont le résultat d'un compromis entre la vision du monde comme accomplissement d'un programme dicté par la raison calculante et la conception du monde comme Création. Hervé Fischer résume en ces termes, ce que sont ces artistes, au regard de la rationalité scientifique : « Au fond, les artistes sont les prêtres et les déviants du mythe de la création, tandis que la science assume le rôle instrumental de la post-création. » 24 Catherine VOISON Catherine VOISON née en 1956, professeure agrégée d'arts plastiques vit et travaille actuellement sur la côte nord du Finistère. Diplômée de l'Ecole Nationale Supérieure des Beaux-Arts de Nancy elle a participé à plusieurs expositions à Paris, en province et à l'étranger durant une quinzaine d'années Formée à l'histoire de l'art à l'Université de Strasbourg puis à l'Université de Paris 1, elle a réalisé le texte du catalogue du « Vent des Forêts » dans la Meuse. Titulaire d'un master en art et sciences de l'art dirigé par Marc Jimenez, elle est actuellement doctorante en esthétique à l'Université de Paris 1.

Nous entendons sous ce terme ce que Joël de Rosnay a défini comme l'ensemble des moyens qui « permettent, grâce à l'application intégrée des connaissances et des techniques de la biochimie, de la microbiologie, de la génétique et du génie chimique, de tirer parti, sur le plan technologique, des propriétés et des capacités des micro-organismes et des cultures cellulaires. » Joël de Rosnay, L'aventure du vivant, Éditions du seuil,Paris, 1966,

ème éd., p. 186.


Salah Khardani

Chafik Samir
email: chafik.samir@uca.fr

Relative Regression on Riemannian Manifolds

Keywords: Riemannian manifolds, Consistency, Asymptotic normality, Relative regression, Mean squared relative error

The considerations of this paper are restricted to random variables with values on Riemannian manifolds M, and hence we propose a geometric framework to estimate their relative regression function. Suppose we are given observations

In this work we define and study a new estimator of the regression function on Riemannian Manifold M. Precisely, we use the mean squared relative error (MSRE) as a loss function to construct a nonparametric estimator of the regression operator on Riemannian Manifolds. Under some standard assumptions in Riemannian Manifolds data analysis, we establish the almost sure consistency, with rates, of the constructed estimator as well as its asymptotic normality. Then, a simulation study, on finite-sized samples, was carried out in order to show the efficiency of our estimation procedure.

Introduction

Regression analysis is a fundamental statistical tool for determining how a measured variable is related to one or more potential explanatory variables. The most widely used regression model is linear regression, due to its simplicity, ease of interpretation, and ability to model many phenomena. However, if the response variable takes values on a nonlinear manifold (non Euclidean space) , a linear model is not applicable. Theoretically, when studying the statistical problem of estimating a regression or density function defined on the non-Euclidean, all techniques used in the Euclidean space cannot be properly used. For solving this problem, the exponential map will be introduced, which was actually one of the main topics in [Park, 2004]. Non-parametric models have been used extensively in last years in fitting data lying on Riemannian manifolds, including kernel-based methods and approaches.

This research is motivated by the necessity for automated methods to analyze such objects. Indeed, for many applications in several branches of science, involving medical imaging and computer vision, (directional data, transformations, tensors, and shape)it is desirable to be able to characterize objects and predict their behavior at unobserved times or in the future. Several works have studied the regression problem on manifolds. [START_REF] Kim | Geometric structures arising from kernel density estimation on riemannian manifolds[END_REF] studied a geometric structures arising from kernel density on Riemannian manifolds. [Park, 2012] studied a density kernel estimator defined by a exponential map on S d . [Hendriks, 2003] developed a generalization of estimation with Fourier series to the case of a compact Riemannian manifold without boundary. [START_REF] Healy | An empirical Bayes approach to directional data and efficient computation on the sphere[END_REF]] studied a nonparametric convolution density estimation on the unit sphere. [Pelletier, 2005] proposed a family of nonparametric estimators for the density function based on kernel weight when the variables are random object valued in a closed Riemannian manifold. The Pelletier's estimators is consistent with the kernel density estimators in the Euclidean, i.e., it coincides to the kernel estimator considered by [Rosenblatt, 1956] and [Parzen, 1962] when the manifold is IR d . [START_REF] Prentice | Spherical regression on matched pairs of orientation statistics[END_REF] introduces an extension of spherical regression using a Stiefel manifold as the sample space. [START_REF] Lee | Nonparametric curve estimation on Stiefel manifolds[END_REF] investigate density and curve estimation on compact smooth submanifolds of a Euclidean space using caps. In a more general setting, [START_REF] Bhattacharya | Nonparametic estimation of location and dispersion on riemannian manifolds[END_REF] studied a nonparametric estimation of location and dispersion on Riemannian manifolds. Under classical assumptions of the kernel and the bandwidth, [Pelletier, 2006] obtained the asymptotic bias and variance and proved the same L 2 rate convergence as kernel regression estimators on Euclidean spaces using a geodesic distance. [Yanovich, 2016] studied the asymptotic properties of local sampling on Riemannian manifolds.

Usually, to estimate the nonparametric regression model, many previous works used the mean squares error (MSE) as a criterion for constructing the predictors, see some details in [START_REF] Calsavara | Defective regression models for cure rate modeling with interval-censored data[END_REF]. This method is very sensitive to outliers, and therefore, the presence of large outliers can lead to inappropriate results. The mean square relative error (MSRE) criterion is more convenient as a measure of performance since the notion of relative regression is more recent than the others, although the results are still limited. [START_REF] Park | Relative-error prediction[END_REF]] studied prediction for situations in which relative prediction error is more important than the usual prediction error and derived the form of the best mean squared relative error predictor. [START_REF] Jones | Relative error prediction via kernel regression smoothers[END_REF] studied nonparametric prediction via relative error regression. [Khardani, 2019] studied nonparametric relative regression under random censorship model. They established the uniform consistency with rate over a compact set and asymptotic normality of the estimator. [START_REF] Mechab | Nonparametric relative regression for associated random variables[END_REF] studied this regression model when the observations are weakly dependent. Recently, [START_REF] Khardani | Nonparametric relative regression under random censorship model[END_REF] studied the asymptotic properties of a consistent estimator of this model by using the kernel method for twice censored data. In this paper, we show that it makes sense to define a relative regression on M as the result of a least squares minimization. The proposed method of calculating the regression estimator will be shown to enjoy a number of nice properties: it is a natural analogue of the Euclidean solution which could be considered as a generalization and the solution exists and is unique in many common situations. In particular, we consider the nonparametric estimation of a relative regression function on a Riemannian manifold by using a generalization of the kernel method, introduced by [Pelletier, 2005, Pelletier, 2006] for classical regression estimation. The paper is organized as follows: in Section 2 we present some geometric background and we define the model and the estimator. In Section 3 we give some assumptions and results for the proposed estimator. Applications (Simulations and real data) are drawn in Section 4. The technical details and the proofs are deferred to Section 5.

Preliminaries and estimators

Geometric background

Let M be a finite dimensional manifold. An important geometrical construct for the statistical analysis is the definition of the tangent space. Since M could be seen a Riemannian submanifold in L 2 (I, R N +1 ) [START_REF] Samir | A gradient-descent method for curve fitting on Riemannian manifolds[END_REF], at any point p ∈ M, we define the tangent space and we denote T p M [Helgason, 1978]. We equip the tangent space of M with a smoothly varying Riemannian metric that measures infinitesimal lengths on M. In many cases, the inner product is first defined on L 2 and then induced on the tangent space of M. More precisely, let v and w be two tangent vectors in T p M, the metric is defined as,

v, w = 1 0 (v(s), w(s)) R N +1 ds.
(1)

We denote by g the metric defined on M with nice physical interpretation and by d g the distance induced by g. Another important step in our analysis is to compute geodesic paths between points with respect to the chosen metric. With respect to the geometric structure, M is considered as submanifold in L 2 (I, R N +1 ) and consequently we can use some standard results from Riemannian geometry, see for example [Siegel, 1964, Lang, 1995], including geodesics and exponential map. Therefore, we assume that (M, g) is geodesically complete, at least locally. So, for any point p ∈ M, there exist a ball B ρ (p) where all geodesics exist and are unique.

For simplicity, we call B ρ (p) the cut locus of p with maximum ρ and inj g (M) the radius of injectivity of M.

For the rest of the paper, we will assume that inj g (M) is strictly positive. There are many examples of manifolds that have this property, we cite for example an Euclidean space with infinite inj g (M), the unit finite-dimensional sphere with inj g (M) = π, the manifold of symmetric positive definite matrices, the Riemannian n-hyperboloid with constant negative curvature, etc. The interested reader can refer to [Jost, 2011, Lang, 1995] for more details. Under these conditions, the geodesic path χ t (p, q), between any two points p and q with q ∈ B ρ (p) on M can be expressed in terms of a tangent direction v ∈ T p M with, t ∈ [0, 1], χ 0 (p, q) = p and χ 1 (p, q) = q. This formulation translates the constant-speed parametrization of the geodesic passing through p with velocity vector v at t = 0. As a result, the exponential map exp : T p M -→ M is given as exp p (v) = q. So, instead of working on M directly, we can use the tangent space T p (M) that has a natural differential structure. This is made possible because exp is a bijective isometry on B ρ (p). It is worth noticing that, when M is Euclidean, exp is simply the identity. The velocity vector along a geodesic path χ t can be obtained as ∂χt dt | t=0 . Thus, given two points p and p, the inverse exponential map (also known as the logarithmic map) allows the recovery of the tangent vector v between them.

To summarize, the exponential map takes points from M in the tangent space M, preserving distance from p; it also preserves the tangential direction from p. Concretely, the exponential map only preserves angles and distances for points in the tangent plane which have the geodesic distance d g (p, .) ρ(p) from p ; however, we shall implicitly assume this condition holds whenever it is needed. Given the above tools for constructing geodesics and inverse exponential maps on M, we will indicate in the next section how these tools may be used to solve the problem of regression from a given set of observations on M. Moreover, we will need to introduce the notion of the volume density function θ p (q) on M as defined by [Besse, 1978], we have

θ p : q -→ θ p (q) = µ exp * p g µ gp (exp -1 p (q)) (2)
as the quotient of the canonical measure of the Riemannian metric exp * p g on T p (M).

Now, let X be a random object on M, i.e. a measurable map from a probability space (Ω, A, P) to (M, B), where B denotes the Borel σ-field of M. Let (X 1 , Y 1 ), . . . , (X n , Y n ) be independent, identically distributed (i.i.d.) pairs of random variables valued in M × IR + with the same distribution as (X, Y ). For regression analysis, our main objective is estimating Y given X = p. As it is largely admitted, the natural solution of this problem in Euclidean spaces is given by the estimators of the regression function r defined by Y = r(X) + ε. Following [Pelletier, 2005, Pelletier, 2006], we present a solution on M and we discuss the estimators in the next section.

Model and the estimator

Let us introduce n pairs of random variables (X i , Y i ) for i = 1 • • • n that we assume drawn from the pair (X, Y ) which is valued in M × IR . Furthermore, we assume that the variables X and Y are connected by the following relation

Y = r(X) + ε
where r is an unknown function and ε is a random error variable independent of X.

Usually, we use the least square error IE [(Y -r(X)) 2 |X = p] as a loss function to estimate the regression function. However, this kind of loss function is very sensitive to outliers. Indeed, this loss function treats all variables as having an equal weight. Thus, the presence of large outliers can lead to irrelevant results. For this reason, in this contribution, we overcome this drawback by using an alternative loss function based on the relative squared error which is defined, for Y > 0, by

E Y -r (X) Y 2 |X , for Y > 0. (3)
Clearly, this criterion is more interesting than the squared error, when the range of predicted values is large. In particular, this kind of obtained regression is less sensitive to the presence of outliers, see [START_REF] Khardani | Nonparametric relative regression under random censorship model[END_REF]. Moreover, the solution of the minimization of Equation ( 3) can be explicitly expressed by

r(p) := E[Y -1 |X = p] E[Y -2 |X = p]
.

Then, our proposal in this paper is to estimate r (p) by:

r n (p) := ψ 1,n (p) ψ 2,n (p) (4)
where

E[Y -|X = p] := ψ l (p) f (p) ψ l,n (p) = 1 n n i=1 Y - i 1 h d n 1 θ Xn (p) K d g (p, X i ) h n for l ∈ {1, 2}.
and

f n (p) = 1 n n k=1 1 h d n 1 θ X k (p) K d g (p, X k ) h n

Assumptions and main results

The closed ball in (M, g)of center p and of radius h will be denoted by B M (p, h), and the closed ball in IR d centered at the origin and of radius h will be denoted by B(h).

Throughout the paper, we denote by r (

•) = E[Y -|X = •] the conditional -inverse moments of T given X and by ψ (•) = E[Y -|X = •]f (•), with f ( = 0) the density of X; ∈ {1, 2, 3, 4}.
When no confusion is possible, we denote by M and/or M any generic positive constant. Define J 0 := { p ∈ M such that ψ 2 (•) > 0} and let S be a compact subset of J 0 . Then we suppose that

inf p∈S ψ 2 (p) > 0 and A = x ∈ S, {ψ 2 (x) -2r(x)ψ 3 (x) + r 2 (x)ψ 4 (x)} ψ 2 2 (x) = 0 .
In this section, we wants to establish the uniform almost complete convergence (a.s.) of r n (x) to r(x) over a compact S. Firstly, we consider the following assumptions : Assumption H1. : f is bounded and two-times continuously differentiable at p such that:

(i) inf p∈S f (p) > M > 0. (ii) inf p,q∈S θ p (q) > M > 0. Assumption H2. h n < h * 2 where h * = min{inj(M), π 2 √ κ }.
where κ is the least upper bound of sectional curvatures of M if this upper bound is positive, and κ = 0 otherwise.

Assumption H3. The smoothing parameter (h n ) satisfies:

(i) lim n→+∞ log n nh d n = 0, (ii) lim n→+∞ nh d+4 n = 0.

Assumption H4.

(i) The kernel K is Continuously differentiable compactly supported density function and is Holder of order γ.

(ii) R v 2 K( v )dv < ∞. (iii) B(0,1) K (||v||) vdv, gradψ (v) = 0 for = 1, 2.
Assumption H5. The inverse conditional moments of the response variable

r (p) = E[Y -|X = p] is two-times continuously differentiable at p and sup p∈M | (∇ 2 ψ (p) i,j ) | < +∞, r 2 (p) > M , ∀ p ∈ M, for ∈ {1, 2}.
Assumption H6. The inverse conditional moments verify:

∀ p ∈ M, and ∀ 1, r (p) = E[Y -|X = p] exists.
Discussion of the assumptions.

We impose the condition (H2) on the bandwidth : h n < inj g (M) where inj g (M) denotes the injectivity radius of M (Chavel, 1993, p108). Since M is compact, inj g (M) is strictly positive by the theorem of Whitehead. The condition h n < inj g (M) guarantees for each p ∈ M the existence of a normal coordinate neighborhood at p containing B M (p, h n ), the ball in M centered at p and of a radius h. Assumption (H4) is a standard assumption when dealing kernel estimators. Furthermore, Assumptions (H5 -H6) are regularity conditions which permit us to evaluate the bias and variance terms of the estimator (4). The fact that θ p (p) = 1 for all p ∈ M guarantees that H1(ii) for a small compact neighborhood of p.

Theorem 3.1 Under Assumptions (H0)-(H5), we have

sup p∈S |r n (p) -r(p)| = O max log n nh d n , h 2 n a.s. (5)
The following corollary is a particular case of the previous theorem. a.s.

Theorem 3.3 Under assumptions (H0)-(H6), then for any p ∈ A, we have

nh d n σ 2 (p) 1/2 (r n (p) -r(p)) D → N (0, 1) as n → ∞,
where

σ 2 (p) = {ψ 2 (p) -2r(p)ψ 3 (p) + r 2 (p)ψ 4 (p)} ψ 2 2 (p) B(1)
K 2 ( z )dz.

and

A = p ∈ M, {ψ 2 (p) -2r(p)ψ 3 (p) + r 2 (p)ψ 4 (p)} ψ 2 2 (p) = 0 , D → denotes convergence in distribution, N the Gaussian-distribution.

Applications

As part of potential applications, we would like to point out that the method developed in this work can be used for more general problems under the assumptions detailed in section 3. As a first example, we want to simulate a regression problem on Riemannian manifolds with positive constant curvature. This setting is very common in different applications where data are longitudinal. We cite for example, wind direction, temperature values and changes, hurricane-force winds from the center, tropical-storm-force winds, bird migration, etc. We now examine in details the geometric structure of finite-dimensional Riemannian manifolds with positive constant curvature k > 0, and provide explicit formulas for computation. In order to avoid trivial cases we assume N ≥ 2. The most used space is the hypersphere for two reasons: i) manifolds with constant k > 0 can be bijectively mapped to the unit sphere of the same dimension, and ii) lot is known about the geometry of the sphere. Here we provide some of the useful definitions from spherical geometry. For more details, we refer the reader to [John M. Lee, 1997]. Since there is no restrictions for the dimension, we denote by S N the unit sphere in

IR N +1 . S N = {p ∈ R N +1 | < p, p > 2 = 1}, (6)
where < ., . > 2 is the standard inner product. Thus, for any p and q we have < p, q >= N +1 i=1 p i q i .Every Riemannian manifold isometrically diffeomorphic to S N will be identified with S N . For p ∈ S N , the tangent space to S N in p is given by

T p (S N ) = {v ∈ R N +1 | < p, v > 2 = 0}.
The restriction of the inner product to closed ball with radius small than π in T p (S N ) is positive definite. This metric is differentiable and therefore S N has a locally complete Riemannian structure. Starting from a point p ∈ S N and a unit vector v ∈ T p (S N ) with ||v|| π, we can construct a unique geodesic γ : R → S N and the arc-length (geodesic distance) between distinct and non antipodal p and q on M is the angle β = arccos (< p, q > 2 ).

The exponential map is a bijective isometry from the tangent space T p (M) to M. For any w ∈ T p (M), we write

Exp p (w) = cos( w 2 )p + sin( w 2 ) w w 2 . ( 7
)
Its inverse, the log map is defined from M to T p (M) as

Log p (q) = β sin(β) (q -cos(β)p). ( 8
)
For any two elements p and q on M the map Γ : T p (M) → T q (M) parallel transports a vector w from p to q and is given by:

Γ p q (w) = w -2 (p + q) ||p + q|| 2 2 < w, q > 2 (9)
Now that we have explicit formulas for geometric tools that can insure conditions in section 3, we can apply our method for regression on S N . An important issue in any kernel-based estimation is the choice of the optimal bandwidth In order to apply our method for the simulation study and the real application considered in next sections, we use a standard method for the choice of the bandwidth. In particular, an optimal choice of h n which minimizes the asymptotic mean squared error (MSE) of the estimator r n at a point x is d+8) . In our experiments, we use the most common method based on least square cross-validation method to search for an asymptotically optimal data-driven bandwidth and thus, adaptive datadriven estimators. Consequently, what we call training data are points used to find optimal bandwidths where the rest is kept for evaluation. For both simulations and real application, the performance of an estimate r of r is measured using the Root Mean Square Error (RMSE).

(h n ) = log n n 1/(d+8) d 8 2/(

Simulations

In this section, we consider two applications, in each investigating the spherical regression models towards different statistical goals. The first involves a simulated data set with a scalar covariate, y ∈ R. We use simple data-generating models to illustrate the flexibility within this regression framework for the mean direction and dispersion to depend on the covariate; to investigate the performance of regression.

For simplicity, we consider an equivalent situation where X is a random position on S N (N = 2) representing the sphere and Y ∈ [0, 1]. Consequently, the regression problem consists in using available observations (X i , Y i ), i = 1, . . . , n in order to predict Y for unknown positions. To better understand this situation, we display two examples of noisy data on S 2 (earth here) Figure 2(a), use some observations to better approximate the model Figure 2(b) and then use the proposed method for regression Figure 2(c). We illustrate this idea with Y-value colors for a better visualization.

Real Application

The second dataset concern the air quality as the covariate and the position (longitude and latitude) as the manifold-valued measurement In this application, we use a dataset consisting of 400 measured at different locations in Europe from January 2015 to July 2019. The dataset is a part of the data archive provided by the European Environment Agency and is available at https://www.eea.europa.eu. The dataset contain observations about the air quality in individual countries, regions and cities. The quality is based on concentration values of different key pollutants to give an idea about the impact on health. Since data do not cover all locations (monitoring stations) and some values are either missing or wrong when data are not formally verified, the relative spherical regression is more convenient. 75% was used for training and the remaining 25% was reserved for testing the performance of the proposed model. For all applications, the proposed model showed a good performance for predicting covariate. In particular, simulations results in simulation study show shows robustness against noise and outliers. Note that the optimal bandwidth is data-dependent but because the geodesic distance on distance on the sphere is defined on wide ball, the impact of h n is moderate. We expect this property to hardly impact the performance if the radius of injictivety is relatively small. Nevertheless, there are many Riemannian manifolds where the geodesics are available with explicit formulas or can be efficiently approximated. Thus, the proposed model can be generalized for other manifolds with different geometric structures but the conclusions

Proofs and auxiliary results

Lemma 1 Under assumptions of Theorem 3.1 and for l ∈ {1, 2}, we have

sup p∈S |E(ψ l,n (p)) -ψ l (p)| = O(h 2 n ).
Lemma 2 Under assumptions of Theorem 3.1 and for l ∈ {1, 2} we obtain

sup p∈S |ψ l,n (p) -E(ψ l,n (p))| = O log n nh d n .
Proof of Theorem 3.1. We consider the following decomposition:

r n (p) -r(p) = 1 ψ 2,n (p) [ψ 1,n (p) -ψ 1 (p)] + [ψ 2 (p) -ψ 2,n (p)] r(p) ψ 2,n (p) (10)
The proof of this theorem is based on the decomposition given by ( 10) and is a direct consequence of Lemmas ( 1) and ( 2), for which the proofs are given in the appendix.

Proof of Theorem 3.3: We write

r n (p) -r(p) = 1 ψ 2,n (p) [D n + W n (ψ 2,n (p) -E [ψ 2,n (p)])] + W n , (11)
where

W n = 1 E [ψ 2,n (p)] ψ 2 (p) {E [ψ 1,n (p)] ψ 2 (p) -[E [ψ 2,n (p)]]ψ 1 (p)} D n = 1 ψ 2 (p) [[ψ 1,n (p) -E [ψ 1,n (p)]]ψ 2 (p) + [E [ψ 2,n (p)] -ψ 2,n (p)]ψ 1 (p)].
Then, it follows from (11), that

r n (p) -r(p) -W n = 1 ψ 2,n (p) [D n + W n (ψ 2,n (p) -E [ψ 2,n (p)])] = : D n + W n J 2n (p) ψ 2,n (p) ,
where

J jn (p) = ψ j,n (p) -E [ψ j,n (p)] , for j = 1, 2.
Therefore, Theorem 3.3 is a consequence of the convergence rate of Lemmas (1) and

(2) and the following intermediate results ( Lemmas 3-4), for which the proofs are also postponed to the appendix.

Lemma 3 Under assumptions of Theorem 3.3, we obtain

nh d n σ 2 (p)ψ 2 2 (p) 1/2 (D n -E[D n ]) D → N (0, 1).
Lemma 4 Under assumptions of Theorem 3.3, we obtain

ψ 2,n (p) P → ψ 2 (p), nh d n σ 2 (p)ψ 2 2 (p) 1/2 W n → 0, and
nh d n σ 2 (p)ψ 2 2 (p) 1/2 W n (ψ 2,n (p) -E [ψ 2,n (p)]) P → 0,
where P → denotes convergence in probability.

Appendix

Proof of Lemma 1 Using a conditional expectation, we have

E ψ ,n (p) -ψ (p) = E Y - i 1 h d i 1 θ X i (p) K d g (p, X i ) h i -ψ (p) = M r (q) 1 h d n 1 θ q (p) K d g (p, q) h n f (q)dν g (q) -ψ (p) = B M (p,hn) 1 h d n 1 θ q (p) K d g (p, q) h n (ψ (q) -ψ (p))dν g (q) = 1 2 ∇ 2 ψ (p) i,j B (1) K( u )u i u j du h 2 n + o(h 2 n ) Assumptions (H4 -H5) conclude the proof. Proof of Lemma 3. It is clear that: E[D n ] = 0, so V ar[D n ] = E[D 2 n ]. Then E[D 2 n ] = E{[ψ 1,n (x) -E(ψ 1,n (x))] 2 } + ψ 1 (x) ψ 2 (x) 2 E{[ψ 2,n (x) -E(ψ 2,n (x))] 2 } -2 ψ 1 (x) ψ 2 (x) cov(ψ 1,n (x), ψ 2,n ( = V ar(ψ 1,n (x)) + r 2 (x)V ar(ψ 2,n (x)) -2r(x)Cov(ψ 1,n (x), ψ 2,n (x)).
We have

V ar[ψ 1,n (x)] = n -1 h -2d n var 1 θ Xn (p) Y -1 n K d g (p, X n ) h n = 1 nh d n E 1 h d n 1 θ 2 Xn (p) Y -2 n K 2 d g (p, X n ) h n - 1 n E 2 1 h d n 1 θ Xn (p) Y -1 n K d g (p, X n ) h n = 1 nh d n E 1 h d n 1 θ 2 Xn (p) Y -2 1 K 2 d g (p, X n ) h n - 1 n (ψ 1 (p) + o(1)) 2 =: K 1n (p) + K 2n (p).
For K 2n (x) it is clear,

K 2n (x) = 1 n (ψ 1 (p) + o(1)) 2 = o(1). (12)
For K 1n (x), under (H4-H6), using again the conditional expectation properties and a Taylor expansion, we get,

K 1n (p) = 1 nh d n E 1 h d n 1 θ 2 Xn (p) Y -2 1 K 2 d g (p, X n ) h n = 1 nh d n M 1 h d n 1 θ 2 q (p)
K 2 d g (p, q) h n ψ 2 (q)dν g (q)

Y -1 i 1 θ X i (p) K d g (p, X i ) h n , 1 nh d n n j=1 Y -2 j 1 θ X j (p) K d g (p, X j ) h n = 1 nh 2d n E Y -3 i 1 θ 2 X i (p) K 2 d g (p, X i ) h n -E Y -1 i θ X i (p) K d g (p, X i ) h n E Y -2 i θ X i (p) K d g (p, X i ) h n
A simple calculation ensures that

1 nh 2d n E Y -1 i 1 θ X i (p) K d g (p, X i ) h n = o(1) and 1 nh 2d n E Y -2 i 1 θ X i (p) K d g (p, X i ) h n = o(1).
A Taylor expansion ensures that Then the combination of ( 14), ( 15) and ( 16) ensures that Under assumptions H1, H4 and H5-H6 , we have :

V ar[ψ 2,n (p)] = n -1 h -2d k var 1 θ X k (p) Y -2 k K d g (p, X k ) h k = 1 nh d n E 1 h d k 1 θ 2 X k (p) Y -4 k K 2 d g (p, X k ) h k - 1 n E 2 1 h d n 1 θ X k (p) Y -2 k K d g (p, X k ) h k = 1 nh d n E 1 h d k 1 θ 2 X k (p) Y -4 1 K 2 d g (p, X k ) h k - 1 n (ψ 2 (p) + o(1)) 2 = O n -1 h -d n = o(1).
Hence ψ 2,n (p) P → ψ 2 (p). Next, it is clear that the second limit of Lemma 4 is a consequence of the above convergence. Then, it suffices to treat the last result. For this, we use the fact that

Corollary 3. 2

 2 Under Assumptions (H0)-(H5) and for (h n ) = log n n 1/(d+4) , we have sup p∈S |r n (p) -r(p)| = O log n n 2/(d+4)

Figure 1 :

 1 Figure 1: Simulation 1 (top) & 2 (bottom) for regression on unit sphere S 2 . For each example: (a) represents original data with noise, (b) examples from training, and (c) the predicted values.

Figure 2 :

 2 Figure 2: Regression on air quality: (a) represents examples from original data, (b) training, (c) test supposed unknown, and (d) original covariate in red and the prediction in green. In this application, the RMSE between test and predicted by the proposed model is 0.0072.

EB

  Y -2 |X = p + r 2 (p)E Y -4 |X = p -2r(p)E Y -3 |X = p -E[D n ]) D → N (0, 1). Proof of Lemma 4. It follows from Lemma 1 that E[ψ 2,n (p) -ψ 2 (p)] → 0

V

  ar[(ψ 2,n (p) -E [ψ 2,n (p)])] = V ar[ψ 2,n (p)] → 0.Then, by the application of Lemma 2 we obtain W n = O(h 2 n ). Then, we deduce that


Isabelle Cattiaux-Huillard

Laura Saini

Characterization and extensive study of cubic and quintic algebraic trigonometric planar PH curves

Keywords: Pythagorean hodograph, trigonometric functions, B-basis, trigonometric polynomial, circle arc Mathematics Subject Classification : 65D17, 51N05

This paper deals with Pythagorean hodograph curves of the spaces of algebraic trigonometric functions and trigonometric polynomials, span {1, t, {cos (kt) , sin (kt)} m k=1 } and span {1, {cos (kt) , sin (kt)} m k=1 } , respectively. First, we propose a general characterization of planar PH curves in these spaces. Next, we consider the particular cases m = 1 and m = 2. For each of them, we give the general form of the control polygon of the PH curves, the implicit relations defining these curves, and their geometrical interpretations. Some examples and particular cases complete this study.

Introduction

The rational parameterization, including polynomial parameterization (NURBS, Bézier curves and surfaces) is one of the most commonly used schemes in CAD and CAGD for the representation of curves and surfaces. However, this scheme, mainly defined with the help of Bézier's model, contains some drawbacks, as detailed by Mainar & All [START_REF] Mainar | Shape preserving alternatives to the rational bézier model[END_REF]. Among them, we may mention in particular the fact that it only allows the representation of a limited number of curves. Even if the trigonometric curves can be parameterized in rational form, this transformation is made at the expense of the quality of the parameterization: for instance, it at been demonstrated in [START_REF] Rida | Real rational curves are not 'unit speed[END_REF] that such a curve can not be parameterized by its arc length. Even in the case of a simple curve as the circle, the rational parameterization has desavantages: difficulty in representing a whole circle or loss of regularity at the junction points (see [START_REF] Fiorot | The circle as a smoothly joined br-curve on [0, 1][END_REF]).

The works of Carnicer and Peña have first shown in [START_REF] Carnicer | Shape preserving representations and optimality of the bernstein basis[END_REF] that among the normalized totally positive bases of polynomial spaces, the Bernstein basis was optimal for shape-preserving properties (decrease of the variation, convex hull property, tangency to the control polygon endpoint). They then established that, in any space having normalized totally positive bases, we can find a unique basis, called normalized B-basis, possessing these shape-preserving properties [START_REF] Carnicer | Totally positive bases for shape preserving curve design and optimality of b-splines[END_REF]. These results thus allow the transfer of the most interesting properties of the rational and polynomial Bézier curves to other spaces. In [START_REF] Mainar | Optimal bases for a class of mixed spaces and their associated spline spaces[END_REF], Mainar and Peña first constructed normalized B-bases in spaces spanned by algebraic polynomials or trigonometric or hyperbolic functions. Then, Mainar, Peña and Sanchez-Reyes determined in [START_REF] Mainar | Shape preserving alternatives to the rational bézier model[END_REF] the normalized B-bases for spaces of curves for which the parameterization consists of trigonometric and polynomial functions, in particular for the sets of trigonometric polynomials F m (K) = span {1, {cos (kt) , sin (kt)} m k=1 } and for the algebraic-trigonometric functions P m (K) = span {1, t, {cos (kt) , sin (kt)} m k=1 } , for m = 1 and 2, where K can be either R or C. Let us notice that such curves could be described as a particular case of generalized splines in their trigonometric space (see [START_REF] Wang | Unified and extended form of three types of splines[END_REF] and [START_REF] Manni | Generalized b-splines as a tool in isogeometric analysis[END_REF]). The spaces P m (K) present numerous advantages: they allow the representation of several classic curves, such as ellipses, cycloids, lemniscates, etc., supplying in particular the arc length parameterization of the circle. Contrary to the rational Bézier curves, the complexity of the representation does not increase with the derivative. Finally, these spaces contain the arc-length parameterization of line segments and arcs of a circle, in the same framework. This point is important in some industrial applications where the curves are still mainly defined by these elements.

However, this will result in having to manipulate the elements derived from these objects, such as the arc length, the curvature, the offset curves or surfaces, etc. Therefore, it is important that these quantities can be computed explicitly. This aim was reached within the framework of polynomial planar curves by Farouki and Sakkalis [START_REF] Rida | Pythagorean hodographs[END_REF] with the Pythagorean hodograph curves, denoted PH curves: a polynomial curve R(t) is said to be a PH curve if and only if the Euclidian norm of its hodograph R (t) is also a polynomial. They have been exhaustively studied by Farouki (see [START_REF] Farouki | Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable[END_REF]) and a survey of the work done on PH curves in the decade 2008-2018 can be consulted in [START_REF] Farouki | New Developments in Theory, Algorithms, and Applications for Pythagorean-Hodograph Curves[END_REF]. These results can be extended to trigonometric polynomials and algebraic-trigonometric curves and could be adapted to various applications (like interpolations, splines, transition between various geometrical objects, etc).

The present paper is thus dedicated to the studies of spaces of algebraic trigonometric PH curves, called AT P H curves, which have been partially studied by Romani & All [START_REF] Romani | Algebraic-trigonometric pythagoreanhodograph curves and their use for hermite interpolation[END_REF] who were particularly interested in the set P 2 (K) and in its application for Hermite interpolation. Here, we propose more general results allowing us to determine all the AT P H curves of a given order, as well as an in-depth study of those of the spaces P 1 (K), P 2 (K), F 1 (K) and F 2 (K). By extending the works of Farouki and Sakkalis on the polynomial case [START_REF] Farouki | Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable[END_REF], [START_REF] Farouki | New Developments in Theory, Algorithms, and Applications for Pythagorean-Hodograph Curves[END_REF] and [START_REF] Rida | Pythagorean hodographs[END_REF], we verify that AT P H curves are characterized by the Pythagorean condition. Then, we show that the complex expression of these curves, introduced by Farouki for Bézier polynomials [START_REF] Farouki | Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable[END_REF], [START_REF] Farouki | New Developments in Theory, Algorithms, and Applications for Pythagorean-Hodograph Curves[END_REF] and [START_REF] Rida | The conformal map z → z 2 of the hodograph plane[END_REF], allows to describe them in a simple and exhaustive way. In section 3, we present the case of P 1 (K) curves, for which we give the formal expression for the control polygons of the PH curves, relatively to the corresponding normalized B-basis, as well as an intrinsic relation which characterizes these control polygons and their geometrical interpretation. We conclude this section with the particular case of the circle arcs for which we give a geometrical characterization. The next section proposes similar results within the framework of the space P 2 (K), completing the works of Romani & All [START_REF] Romani | Algebraic-trigonometric pythagoreanhodograph curves and their use for hermite interpolation[END_REF] and Saini [START_REF] Saini | New tools for animation and design : a haptic-based system for stop motion camera animation and curve design by algebraic-trigonometric Pythagorean Hodograph curves[END_REF]. Finally, in the last section, we present the PH curves of F 1 (K) and F 2 (K) spaces as particular cases of P 1 (K) and P 2 (K) spaces, respectively. However, the results are expressed in the normalized B-bases specific to F 1 (K) and F 2 (K).

2 Characterisation of Algebraic-Trigonometric Pythagorean Hodograph (ATPH) curves

AT function spaces

For an arbitrary m belonging to N * , considering K = R or C, the main Algebraic-Trigonometric (AT) function spaces we study are described as :

P m (K) = span {1, t, {cos (kt) , sin (kt)} m k=1 } .
As we shall see further, to study the properties of the ATPH curves of P m (K), we need to introduce the spaces

F n 2 (K) =    span {cos (kt) , sin (kt)} p k=0 , with p = n 2 if n is even, span cos 2k-1 2 t , sin 2k-1 2 t p k=1 , with p = n+1 2 otherwise.
Note that the derivatives of the functions of P m (K) are in the F m (K) space (i.e. F n 2 (K) for n = 2m).

A function f (t) of F n 2 (K) can be written using the following trigonometric expressions as

f (t) =    f 0 + p k=1 (f 2k-1 cos (kt) + f 2k sin (kt)) , if n is even, p k=1 f 2k-1 cos 2k-1 2 t + f 2k sin 2k-1 2 t
, otherwise, or, using the exponential expressions:

f (t) =    ϕ 0 + p k=1 (ϕ k exp(ikt) + ϕ -k exp(-ikt)) , if n is even, p k=1 ϕ k exp(i 2k-1 2 t) + ϕ -k exp(-i 2k-1 2 t) , otherwise.
However, even if f (t) belongs to F n 2 (R), the coefficients of its exponential form are often complex. So we need conditions which ensure that such a form defines a function f (t) of

F n 2 (R). Proposition 1 Let f (t) ∈ F n 2 (C) be expressed in its exponential form. This function belongs to F n 2 (R) if and only if    ϕ 0 ∈ R and ϕ -k = ϕ k , for k = 1, . . . , p, if n is even, ϕ -k = ϕ k , for k = 1, . . . , p, otherwise. (1)
Proof : We consider f (t) ∈ F n 2 (C) with an even n. If the coefficients ϕ i verify (1), it is easy to verify that

f (t) = ϕ 0 + p k=1 ((ϕ k + ϕ -k ) cos (kt) + i(ϕ k -ϕ -k ) sin (kt))
has real coefficients in trigonometric form. Reciprocally, we can write f (t) as

f (t) = f 0 + p k=1 1 2 (f 2k-1 -if 2k ) exp(ikt) + 1 2 (f 2k-1 + if 2k ) exp(-ikt) , so ϕ k = 1 2 (f 2k-1 -if 2k ) and ϕ -k = 1 2 (f 2k-1 + if 2k
) then, if the coefficients f k are real, for k = 0, . . . 2p, (1) is verified. When n is odd, the proof is similar.

Introducing the notation T = exp(it), we can write

f (t) = 1 T n 2 ϕ(T )
where

ϕ(T ) :=    p k=-p ϕ k T p+k , if n is even, - 1
k=-p ϕ k T p+k + p-1 k=0 ϕ k+1 T p+k , otherwise. Obviously, the map which associates the polynomial ϕ to the AT function f is bijective. This observation permits to extend some definitions from polynomials to AT functions.

Definition 1

The polynomial ϕ(T ) is called the polynomial associated with f (t). Moreover, two functions a(t) and b(t) of the space F n 2 (R) have w(t) for their GCD if its associated polynomial is the GCD of those of a(t) and b(t). They are said to be relatively prime if their associated polynomials are relatively prime.

Remark 1 The degree of ϕ(T ) associated with f (t) of F n 2 (K) is n, whether n is even or odd.

Pythagorean condition for F m (R) functions

As previously noted, the derivative of a function of P m (K) is in the F m (K) space. So the PH curves of P m (R) are those whose hodograph, belonging to F m (R), verifies the Pytagorean condition.

Theorem 1 Let a(t), b(t) and c(t) be non-zero real functions of the space F m (R) such that a(t) and b(t) are relatively prime. The Pythagorean condition

a 2 (t) + b 2 (t) = c 2 (t) (2)
is satisfied if and only if we can find two functions u(t) and v(t) in the space

F m 2 (R) such that a(t) = u 2 (t) -v 2 (t), (3) b
(t) = 2u(t)v(t), (4) c(t) = u 2 (t) + v 2 (t), (5)
where u(t) and v(t) are relatively prime.

Proof : By substituting the conditions (3-5) into (2), we immediately verify they are sufficient.

To prove they are also necessary, we consider α(t), β(t) and γ(t) the respective associated polynomials of a(t), b(t) and c(t) which are supposed to verify [START_REF] Carnicer | Totally positive bases for shape preserving curve design and optimality of b-splines[END_REF]. These polynomials also verify the Pythagorean condition (2). According to the similar result proved in [START_REF] Rida | Pythagorean hodographs[END_REF], two relatively prime polynomials ν(t) and µ(t) can be found such that

α(t) = ν 2 (t) -µ 2 (t), β(t) = 2ν(t)µ(t), γ(t) = ν 2 (t) + µ 2 (t).
We

u(t) = 1 T m 2 ν(T ) and v(t) = 1 T m 2 µ(T ),
two functions in the space F m 2 (R) which verify [START_REF] Cattiaux | Normalized B-Bases for Trigonometric Polynomial Curves[END_REF][START_REF] Farouki | Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable[END_REF][START_REF] Farouki | New Developments in Theory, Algorithms, and Applications for Pythagorean-Hodograph Curves[END_REF]. Without loss of generality we may write

u 2 (t) = 1 2 (a(t) + c(t)) ∈ F m (R), (6) v 2 (t) = 1 2 (c(t) -a(t)) ∈ F m (R), (7) u(t)v(t) = 1 2 b(t) ∈ F m (R). ( 8
)
We observe that u(t) and v(t) are both either real or imaginary. Both of these cases lead to the same functions a(t), b(t) and c(t). For this reason, we consider that u(t) and v(t) only belong to F m 2 (R). In this space, (u(t), v(t)) and (-u(t), -v(t)) are the only two solutions of (6-8). Moreover, ν(t) and µ(t) being supposed relatively prime, u(t) and v(t) are relatively prime too.

Remark 2

The Theorem 1 can be easily extended to the case where a(t) and b(t) are not relatively prime. Denoting by δ(t) the GCD of a(t) and b(t), the Pythagorean condition becomes

a(t) = u 2 (t) -v 2 (t) δ(t), b(t) = 2u(t)v(t)δ(t), c(t) = u 2 (t) + v 2 (t) δ(t),
where u(t) and v(t) are two relatively prime functions of the space F m 2 (R).

Definition 2 A planar parametric curve R(t) = (x(t), y(t)), where x(t) and y(t) are nonzero real functions in the space P m (R), whose first derivative R (t) = (x (t), y (t)) verifies the Pythagorean condition (2) is called an Algebraic Trigonometric PH curve or ATPH curve.

Complex representation

Any parameterization R(t) = (x(t), y(t)), where x and y belong to P m (R), can be bijectively associated with the complex map r(t) = x(t) + iy(t) of P m (C). Farouki in [START_REF] Rida | The conformal map z → z 2 of the hodograph plane[END_REF] proved the practicality of such a representation for the polynomial PH curves. The same approach can be used for ATPH curves. Within this context, they can be characterized by the following result.

Proposition 2 For any m ∈ N * , a P m (R) regular curve R(t) is an ATPH curve if and only if its complex form r(t), belonging to P m (C), can be written as

r(t) = r 0 + t 0 w 2 (τ )dτ, ( 9
)
where w is a F m 2 (C) function and r 0 = r(0).

Proof :

Considering w(t) = u(t) + iv(t) ∈ F m 2 (C), we have

r (t) = w 2 (t) = u 2 (t) -v 2 (t) + 2iu(t)v(t) ∈ F m (C), so x (t) = u 2 (t) -v 2 (t)
and y (t) = 2u(t)v(t). Thus, according to Proposition 1, R(t) is an ATPH parameterization of P m (R).

Reciprocally, as R(t) is a regular ATPH curve, x (t) and y (t) are relatively prime elements of F m (R). Therefore, according to Theorem 1, two relatively prime functions u(t) and v(t) of F m 2 (C) can be found such that

x (t) = u 2 (t) -v 2 (t), y (t) = 2u(t)v(t).
So, for w(t) = u(t) + iv(t), the result is evident.

Remark 3

We can easily see that, for a parameterization R(t) and its complex form r(t), the choice of w(t) that satisfies (9) is unique up to a sign. In the same manner, having chosen the function w(t), changing the complex value r 0 gives the same curve modulo a translation.

In the remainder of this paper, unless specified, we will only represent a curve parameterization R(t) by its complex formulation r(t), so to simplify the notations, P m (C) and F m (C) will be written as P m and F m .

w(t) = z 1 cos t 2 + z 2 sin t 2 . (10)
So, they can be represented by :

r(t) = r 0 + 1 2 z 2 1 + z 2 2 t + z 1 z 2 (1 -cos t) + 1 2 z 2 1 -z 2 2 sin t. (11)
To obtain a regular curve, we need w(t) = 0, for any value of t. This condition is verified when z 1 z 2 is not a real value, i.e. when Im(z 1 z 2 ) = 0.

In [START_REF] Mainar | Shape preserving alternatives to the rational bézier model[END_REF], the authors described the normalized P 1 B-basis, for α belonging to the interval ]0, 2π[ and t ∈ [0, α]. It can be defined as : where

Z k (t) = m 0,k + m 1,k t + m 2,k cos t + m 3,k sin t, for k = 0, 1, 2, 3, with, for α = π m 0,0 = α S α , m 0,1 = - S α + R α S α R α sin α, m 0,2 = sin α R α , m 0,3 = 0, m 1,0 = - 1 S α , m 1,1 = C α S α R α sin α, m 1,2 = -m 1,1 , m 1,3 = -m 1,0 , m 2,0 = - sin α S α , m 2,1 = -m 0,1 , m 2,2 = -m 0,2 , m 2,3 = 0, m 3,0 = cos α S α , m 3,1 = sin α S α R α (C α -α sin α), m 3,2 = m 1,1 , m 3,3 = m 1,0 ,
S α = α -sin α, C α = 1 -cos α, R α = 2 sin α -α -α cos α.
This general expression cannot be used for α = π, in which case it needs a particular description

m 0,0 = 1, m 0,1 = - 1 2 , m 0,2 = 1 2 , m 0,3 = 0, m 1,0 = - 1 π , m 1,1 = 1 π , m 1,2 = - 1 π , m 1,3 = 1 π , m 2,0 = 0, m 2,1 = 1 2 , m 2,2 = - 1 2 , m 2,3 = 0, m 3,0 = - 1 π , m 3,1 = 1 π , m 3,2 = 1 π , m 3,3 = - 1 π .
In both cases, these coefficients define the transformation matrix M α = (m i,j ) 0≤i,j≤3 between the basis {1, t, cos t, sin t} and the normalized B-basis. When we determine its inverse M -1 α , the indetermination for α = π vanishes in the general formulation. For all cases we thus obtain :

M -1 α =          1 0 1 0 1 S α C α 1 S α C α 1 S α + R α C α -1 + α sin α C α S α + R α C α 1 α cos α sin α         
.

Some examples of the normalized B-basis are shown in Figure 1. In this basis, the curve parameterization is described by four control points P 0 , P 1 , P 2 and P 3 , whose complex forms

P 0 Q 0 P 1 P 2 P 3 Q 1 Q 2 Q 3 Figure 2: P 1 PH curve obtained for z 1 = 1 2 + i, z 2 = 3 -3 20
i and α = π 3 (dashed blue line and control points P 0 , P 1 , P 2 , P 3 ) or α = 2π 3 (green line and control points

Q 0 , Q 1 , Q 2 , Q 3 ).
are p 0 , p 1 , p 2 and p 3 , respectively. The parameterization complex expression is

r(t) = 3 k=0 p k Z k (t). ( 12
)
Using M -1 α and (11), we determine the formulation of p 0 , p 1 , p 2 and p 3 when they describe a PH curve :

p 0 = r 0 , (13)
p 1 = r 0 + S α C α z 2 1 , (14)
p 2 = r 0 + z 1 C α (z 1 (S α + R α ) + z 2 (2C α -α sin α)) , (15)
p 3 = r 0 + 1 2 (α + sin α)z 2 1 + z 1 z 2 C α + 1 2 z 2 2 S α . ( 16
)
As we can see in the expression [START_REF] Mainar | Shape preserving alternatives to the rational bézier model[END_REF], the choice of z 1 and z 2 fixes the parameterization of the curve. The α value operates on the length of the arc that is represented and on its control polygon, as shown in Figure 2.

Intrinsic relation

The point p 0 being the curve's initial point r 0 , it can be freely chosen, and the other points can be determined by the relations :

∆p 0 = p 1 -p 0 = S α C α z 2 1 , (17)
∆p 1 = p 2 -p 1 = z 1 C α (z 1 R α + z 2 (2C α -α sin α)) , (18)
∆p 2 = p 3 -p 2 = S α 2C α z 2 1 (1 + cos α) + 2z 1 z 2 sin α + z 2 2 C α . (19)
From these results, we can deduce the following property :

Proposition 3 A curve parameterized in P 1 defined by ( 12) is an ATPH curve if and only if its control points verify

(∆p 1 ) 2 = K α ∆p 0 ∆p 2 with K α = 4 S 2 α 2 sin α 2 -α cos α 2 2 . ( 20
)
Proof : As the points of an ATPH curve control polygon verify relations (17-19), we have

∆p 1 = z 1 C α 4 sin α 2 cos α 2 -2α cos 2 α 2 z 1 + 4 sin 2 α 2 -2α sin α 2 cos α 2 z 2 = 2 z 1 C α 2 sin α 2 -α cos α 2 z 1 cos α 2 + z 2 sin α 2 ,
and ∆p 2 = S α 2C α 2z 2 1 cos 2 α 2 + 4z 1 z 2 sin α 2 cos α 2 + 2z 2 2 sin 2 α 2 = S α C α z 1 cos α 2 + z 2 sin α 2 2 .
The curve is supposed to be regular so ∆p 0 and ∆p 2 are non-zero and

(∆p 1 ) 2 ∆p 0 ∆p 2 = 4 S 2 α 2 sin α 2 -α cos α 2 2 .
Reciprocally, let us consider a curve whose control points written in the complex form p 0 , p 1 , p 2 , p 3 verify (20). Parameters r 0 and α being previously chosen, we can determine z 1 verifying (17) then z 2 verifying (19). The relation (20) ensures that (18) is verified too, so the curve is an ATPH curve.

Geometrical interpretation

Proposition 4 Denoting by l k = |∆p k |, for k = 0, 1, 2 and by θ 1 and θ 2 the interior angles as shown in Figure 3, the implicit condition (20) is equivalent to

l 2 1 = K α l 0 l 2 and θ 1 = θ 2 .
Proof : K α is positive, so, if we denote ∆p k = l k exp(iβ k ) the polar form of ∆p k for k = 0, 1, 2, condition (20) can be written as

l 2 1 = K α l 0 l 2 and 2β 1 = β 0 + β 2 .
Having θ 1 = π + β 1 -β 0 and θ 2 = π + β 2 -β 1 , the conclusion is straightforward.

Arc-length expression

For the parameterization described by [START_REF] Mainar | Shape preserving alternatives to the rational bézier model[END_REF], the instantaneous speed

|r (t)| = |w(t)| 2 is |r (t)| = 1 2 |z 1 | 2 + |z 2 | 2 + 1 2 |z 1 | 2 -|z 2 | 2 cos t + Re(z 1 z 2 ) sin t.
and the arc-length is straighforwardly

s(t) = 1 2 |z 1 | 2 + |z 2 | 2 t + 1 2 |z 1 | 2 -|z 2 | 2 sin t + Re(z 1 z 2 )(1 -cos t).
The complexe values z 1 et z 2 can be deduced from the control polygon by the following relations, obtained from (17-19) :

z 2 1 = C α S α ∆p 0 , z 1 z 2 = C α 2C α -α sin α ∆p 1 - R α S α ∆p 0 , z 2 2 = 1 + cos α S α ∆p 0 - 2 sin α 2C α -α sin α ∆p 1 + 2 S α ∆p 2 .

Application : circle arc control polygon

In P 1 (C), the anticlockwise parameterization of any circle is

r(t) = ω + ρ exp(i(t + γ)), ( 21
)
where ω is the complex form of the circle center Ω, ρ is its radius and γ is the r 0 argument (so, r 0 = ω + ρ exp(iγ)). Such a parameterization has evidently a Pythagorean hodograph but the determination of its B-based control polygon needs some calculations. The expression (21) may be written in the form r(t) = ω + ρ exp(iγ) cos t + ρ exp(i(γ + π 2 )) sin t. Identifying this expression with (11), we have z 1 z 2 = -ρ exp(iγ) and z 2 1 = -z 2 2 = ρ exp i(γ + π 2 ). These expressions are sufficient to determine the control polygon, using relations (13 -16) :

p 0 = ω + ρ exp(iγ), (22)
p 1 = ω + ρ exp(iγ) 1 + i S α C α , (23)
p 2 = ω + ρ exp(iγ) C α (-C α + α sin α + i(αC α -S α )) , (24)
p 3 = ω + ρ exp i(γ + α). (25)
where ω, ρ and γ can be freely chosen. Figure 4 shows some examples of such control polygons and the circles they define.

Remark 4 When t ∈ [0, α], this parameterization describes a circle arc whose central angle is α. This is confirmed by the expression of p 3 . Moreover,

p 1 -p 0 = ρ S α C α exp i(γ + π 2 ) and p 3 -p 2 = ρ S α C α exp i(γ + α + π 2 ), ( 26
)
which ensures that the tangents at the ends of the arc are orthogonal to the corresponding circle radii, as expected.

Proposition 5 A P 1 (R) curve is an anticlockwise parameterized arc of circle whose central angle is α if and only if its control polygon verifies

l 0 = l 2 = 1 √ K α l 1 and θ 1 = θ 2 = α 2 -π, (27)
with Proposition 4 notations. Proof : We denote the control points of the curve by P 0 , P 1 , P 2 , P 3 and their complex form by p 0 , p 1 , p 0 , p 3 . If this curve is a circle arc whose center is Ω and whose central arc is α, its parameterization in P 1 has a Pythagorean hodograph, so according to Proposition 4 and formula (26), we have

l 0 = l 2 = 1 √ K α l 1 .
Moreover, the sum of the interior angles of the pentagon ΩP 0 P 1 P 2 P 3 gives the relation α + π -2θ 1 = 3π, which yields the second constraint of (27).

Reciprocally, let us consider a control polygon verifying (27). We define Ω as the point such that the straight lines (ΩP 0 ) and (ΩP 3 ) are orthogonal to (P 0 P 1 ) and to (P 3 P 2 ) respectively. We denote the distance Ω -P 0 by ρ. We choose Ω as the origin of the coordinate system and the bisector of the segment [P 3 P 0 ] as the y-axis (see Figure 5).

As previously stated, the property of the sum of a pentagon's angles ensures that we have ∠(P 0 -Ω, P 3 -Ω) = α, so the abscissa of P 0 is

x 0 = ρ cos α - π 2 = l 1 2 -l 0 cos π - α 2 , then ρ sin α 2 = 1 2 √ K α + cos α 2
l 0 and S α ρ = C α l 0 . We have thus p 0 = ρ exp(iγ) where

γ = π-α 2 and p 1 -p 0 = l 0 exp i π -α 2 = S α C α ρ exp i π 2 + γ .
Having l 0 = l 2 and θ 1 = θ 2 , the control polygon is symmetrical with respect to the yaxis, therefore we have also

p 3 = ρ exp i(π -γ) = ρ exp i(α + γ) and p 3 -p 2 = l 0 exp(iθ 1 ) = S α C α ρ exp i(γ + α + π 2 )
. So, according to (22,25,26), the represented curve is a circle arc.

B-based representation of the P 2 PH curves 4.1 B-basis of P 2

In [START_REF] Mainar | Shape preserving alternatives to the rational bézier model[END_REF], the authors give the B-basis {B 5 k (t)} 5 k=0 of P 2 which is detailed in [START_REF] Romani | Algebraic-trigonometric pythagoreanhodograph curves and their use for hermite interpolation[END_REF]. It is illustrated by Figure 6. In this basis, the curves are described by

r(t) = 5 k=0 p k B k (t). ( 28
)
The graphics of the basis functions and the P 2 curves can be viewed and manipulated in [START_REF] Cattiaux | Normalized B-Bases for Trigonometric Polynomial Curves[END_REF]. Their expression is

B 5 k (t) = λ k (γ k0 + γ k1 t + γ k2 sin t + γ k3 cos t + γ k4 sin 2t + γ k5 cos 2t) , (29)
with

λ 0 = λ 5 = 1 n 0 , λ 1 = λ 4 = s 1 2n 0 n 1 , λ 2 = λ 3 = s 1 12n 1 n 2 , γ 00 = 6α, γ 10 = -3(2n 4 α -n 0 ), γ 20 = -3(8c 1 n 1 + n 0 -2n 4 α), γ 01 = -6, γ 11 = 6n 4 , γ 21 = -6n 4 , γ 02 = 8c 2 , γ 12 = -4(2n 4 c 2 + s 2 n 0 ), γ 22 = 4(4n 1 (s 1 + c 1 s 2 ) + n 0 s 2 + 2n 4 c 2 ), γ 03 = -8s 2 , γ 13 = 4(2n 4 s 2 -c 2 n 0 ), γ 23 = 4(4c 1 n 1 (1 + c 2 ) + n 0 c 2 -2n 4 s 2 , γ 04 = -c 3 , γ 14 = s 3 n 0 + n 4 c 3 , γ 24 = -(s 3 (8n 1 s 1 + n 0 ) + c 3 (n 4 -8n 1 c 1 )), γ 05 = s 3 , γ 15 = c 3 n 0 -n 4 s 3 , γ 25 = -(s 3 (8n 1 c 1 + n 0 ) + c 3 (n 4 + 8n 1 s 1 )), γ 30 = -3(8c 1 n 1 + n 0 ), γ 40 = 3n 0 , γ 50 = 0, γ 31 = 6n 4 , γ 41 = -6n 4 , γ 51 = 6, γ 32 = 8(2s 1 n 1 -n 4 ), γ 42 = 8n 4 , γ 52 = -8, γ 33 = 4(8c 1 n 1 + n 0 ), γ 43 = -4n 0 , γ 53 = 0, γ 34 = n 4 -8s 1 n 1 , γ 44 = -n 4 , γ 54 = 1, γ 35 = -(8c 1 n 1 + n 0 ), γ 45 = n 0 , γ 55 = 0,
where s q = sin 2 q-2 α , c q = cos 2 q-2 α ,

and

n 0 = 6α -8s 2 + s 3 , n 1 = c 1 (s 2 -3α) + 4s 1 , n 2 = (2 + c 2 )α -3s 2 , n 4 = 8s 4 1 .
These coefficients define the transformation matrix G α = (λ j γ j,i ) 0≤i,j≤5 between the canonical basis {1, t, cos t, sin t, cos 2t, sin 2t} and the normalized B-basis. Determining a control polygon from its canonical parameterization requires the inverse matrix

G -1 α = 1 n 4 γ i,j 0≤i,j≤5
where

γ 00 = n 4 , γ 10 = n 4 , γ 20 = n 4 , γ 01 = 0, γ 11 = 1 2 n 0 , γ 21 = 3 2 (n 0 -4n 2 ), γ 02 = 0, γ 12 = 1 2 n 0 , γ 22 = 3 2 (n 0 -4n 2 ), γ 03 = n 4 , γ 13 = n 4 , γ 23 = 3s 2 (α -s 2 ), γ 04 = 0, γ 14 = n 0 , γ 24 = 3(n 0 -4n 2 ), γ 05 = n 4 , γ 15 = n 4 , γ 25 = (4αs 2 + 4c 2 + c 3 -5), γ 30 = n 4 , γ 40 = n 4 , γ 50 = n 4 , γ 31 = 1 2 (2α(2 + c 3 ) + 4n 2 -3s 3 ), γ 41 = -1 2 (2α(-3 + 4c 2 -c 3 ) + n 0 ), γ 51 = αn 4 , γ 32 = 3 2 (α(3 + 2c 2 + c 3 ) -2s 2 -2s 3 ), γ 42 = 1 4 (c 2 (s 3 -12α) + s 2 (9 + c 3 )), γ 52 = s 2 n 4 , γ 33 = 3s 2 (2s 2 -α(1 + c 2 )), γ 43 = s 1 (6αc 1 -8s 1 -s 3 c 1 ), γ 53 = c 2 n 4 , γ 34 = 3(2α(2c 2 + c 3 ) + 4s 2 -5s 3 ), γ 44 = -6αc 3 -6s 2 + 3s 3 + 2 sin 3α, γ 54 = s 3 n 4 , γ 35 = 3(1 + 4c 2 -5c 3 -2α(2s 2 + s 3 )), γ 45 = 1 + 6αs 3 -6c 2 + 3c 3 + 2 cos 3α, γ 55 = c 3 n 4 .

Remark 5

The above formulae must be used with caution. Indeed, for small values of α (α < 0.005), some tests, with Mathematica, Matlab and Scilab, show numerical instability problems for the functions B 5 2 (t) and B 5 3 (t). L. Saini, in [START_REF] Saini | New tools for animation and design : a haptic-based system for stop motion camera animation and curve design by algebraic-trigonometric Pythagorean Hodograph curves[END_REF] proposed another expression, but it can not be used to determine the classical matrix correspondence between the expressions of a curve in the monomial basis {1, t, sin t, cos t, sin2t, cos 2t} and in the B-basis. The formulation of B 5 2 (t) and B 5 3 (t) we proposed are those which are the less instable among the different attempts we made.

Regularity of a P 2 curve

As in the space P 1 , Proposition 2 shows that the ATPH curves of P 2 are written from the

F 1 functions w(t) = z 0 + z 1 cos t + z 2 sin t, (30)
and have the form :

r(t) = r 0 + 1 2 2z 2 0 + z 2 1 + z 2 2 t + 2z 0 z 2 (1 -cos t) + 2z 0 z 1 sin t + 1 2 z 1 z 2 (1 -cos 2t) + 1 4 z 2 1 -z 2 2 sin 2t. ( 31
)
Remark 6 The parametric speed of such a parameterization is

|r (t)| = |z 0 | 2 + 1 2 |z 1 | 2 + |z 2 | 2 + 2Re(z 0 z 1 ) cos t + 2Re(z 0 z 2 ) sin t + 1 2 |z 1 | 2 -|z 2 | 2 cos 2t + Re(z 1 z 2 ) sin 2t.
Its arc-length is widely described in [START_REF] Romani | Algebraic-trigonometric pythagoreanhodograph curves and their use for hermite interpolation[END_REF]. Such a curve is regular when w(t) = 0 for any value of t, i.e. when the resultant of Re(w(t)) and Im(w(t)), written in a polynomial form via the formulae cos t = 1-u 2 1+u 2 and sin t = 2u 1+u 2 , is non-zero :

x 0 + x 1 2x 2 x 0 -x 1 0 0 x 0 + x 1 2x 2 x 0 -x 1 y 0 + y 1 2y 2 y 0 -y 1 0 0 y 0 + y 1 2y 2 y 0 -y 1 = 0
where z k = x k + iy k for k = 0, 1, 2. After some calculations, this constraint may be written as

Im(z 2 (z 0 + z 1 ))Im(z 2 (z 0 -z 1 )) + Im(z 0 z 1 ) 2 = 0. (32)
In [START_REF] Romani | Algebraic-trigonometric pythagoreanhodograph curves and their use for hermite interpolation[END_REF], w is written in the normalized B-basis { B 2 k (t)} 2 k=0 of the space F 1 as

w(t) = w 0 B 2 0 (t) + w 1 B 2 1 (t) + w 2 B 2 1 (t). ( 33
)
where

B 2 k (t) = 1 C α (n 0,k + n 1,k cos t + n 2,k sin t) , (34)
and n 0,0 = 1, n 0,1 = -1 -cos α, n 0,2 = 1,

n 1,0 = -cos α, n 1,1 = 1 + cos α, n 1,2 = -1, n 2,0 = -sin α, n 2,1 = sin α, n 2,2 = 0. ( 35
)
This basis is illustrated by Figure 7. The coefficients described in (35) define the transformation matrix (cf [START_REF] Sánchez-Reyes | Harmonic rational bézier curves, p-bézier curves and trigonometric polynomials[END_REF])

N α = 1 C α (n i,k ) 0≤i,k≤2
between the basis {1, cos t, sin t} and the B-basis, which allows us to establish that

z 0 = 1 C α (w 0 -(1 + cos α)w 1 + w 2 ) , z 1 = 1 C α (-cos αw 0 + (1 + cos α)w 1 -w 2 ) , z 2 = sin α C α (w 1 -w 0 ).
After some (tedious) calculations, the condition (32) yields

Im(w 0 w 2 ) 2 = 2(1 + cos α)Im(w 0 w 1 )Im(w 1 w 2 ). (36)
In the following, we will assume this criteria satisfied.

Remark 7

The inverse of N α is the matrix

N -1 α = n i,k 0≤i,k≤2
, where

N -1 α =      1 1 0 1 1 sin α 1 + cos α 1 cos α sin α     

Intrinsic relations

The complex forms of the control polygons of the P 2 PH curves can be deduced from (31) with G -1 α . They have already been described in [START_REF] Romani | Algebraic-trigonometric pythagoreanhodograph curves and their use for hermite interpolation[END_REF]. We easily deduce the following slightly different formulation :

∆p 0 = µ 0 w 2 0 , ∆p 1 = (2µ 0 -3µ 1 )w 0 w 1 , ∆p 2 = µ 1 (1 + cos α)w 2 1 + w 0 w 2 , ( 37
) ∆p 3 = (2µ 0 -3µ 1 )w 1 w 2 , ∆p 4 = µ 0 w 2 2 ,
where µ 0 = 1 4C 2 α (6α -8 sin α + sin 2α) and µ 1 = 1 C 2 α (α(2 + cos α) -3 sin α). Figure 8 shows two examples of such control polygons and the curves they defined.

It can be easily verified that the parameters µ 0 and µ 1 are both positive and non null for any α belonging to ]0, 2π[. The curves being assumed regular, ∆p 0 and ∆p 4 are non-zero. Then formulae (37) allow us to establish the following result : Proposition 6 Let us denote by r(t) a P 2 curve defined by its control polygon p 0 , p 1 , p 2 , p 3 , p 4 according to (28). If ∆p 1 = 0 or ∆p 3 = 0, this curve is a PH curve if and only if

∆p 0 (∆p 3 ) 2 = ∆p 4 (∆p 1 ) 2 , ( 38
)
and one of the following relations are verified :

∆p 0 ∆p 1 ∆p 2 - µ 1 µ 0 (∆p 0 ) 2 ∆p 3 -µ 0 µ 1 Γ α (∆p 1 ) 3 = 0, ( 39
) ∆p 2 ∆p 3 ∆p 4 - µ 1 µ 0 (∆p 4 ) 2 ∆p 1 -µ 0 µ 1 Γ α (∆p 3 ) 3 = 0, ( 40
) ∆p 0 ∆p 2 ∆p 3 - µ 1 µ 0 ∆p 0 ∆p 1 ∆p 4 -µ 0 µ 1 Γ α (∆p 1 ) 2 ∆p 3 = 0, ( 41
) ∆p 1 ∆p 2 ∆p 4 - µ 1 µ 0 ∆p 0 ∆p 3 ∆p 4 -µ 0 µ 1 Γ α ∆p 1 (∆p 3 ) 2 = 0, (42)
where

Γ α = (1 + cos α) (2µ 0 -3µ 1 ) 2 .
and anticlockwise if ε = 1. This curve is regular if and only if z 1 = 0, i.e. if the circle radius is non-zero.

Using the F 1 B-basis described by (34) and (35), we determine the control polygon of r(t) as

p 1 = p 0 + tan α 2 z 2 1 , p 2 = p 1 + tan α 2 z 2 1 exp(iα).
for α belonging in (-π, π). So, it is easy to see that an F 1 curve is a PH curve, i.e. a circle arc, if and only if the following intrinsic relation is verified :

∆p 1 = ∆p 0 exp(iα).
Evidently, the geometric form of this condition is

l 0 = l 1 and β 1 = β 0 + α,
where, for k = 0, 1, ∆p k = l k exp(iβ k )) is the polar form of ∆p k . This curve parameterization gives the shortest control polygon for a circle arc among the different families of curves we study in this article.

F 2 PH curves

Similarly to the previous case, F 2 is a subset of P 2 , so the functions w(t) which allow us to describe the F 2 curves belong to F 1 . They can be written in the canonical form (30) or in a Bézier-like form (33). As for the P 2 curves, we can express the regularity of r(t) curve using the condition (32) for the canonical form and (36) for the Bézier-like form. In the following, we assume that the curve is regular. The description (31) of a PH curve of P 2 allows D 1 to be defined as

D 1 = {w(t) = z 0 + z 1 cos t + z 2 sin t with z 0 , z 1 and z 2 verifying 2z 2 0 + z 2 1 + z 2 2 = 0}, If w(t) is written in the normalized B-basis, the condition which define D 1 , 2z 2 0 + z 2 1 + z 2 2 = 0, is rewritten as 3 w 2 0 + w 2 2 + 2 (1 + cos α) w 2 1 (2 + cos α) -3w 1 (w 0 + w 2 ) + 2w 0 w 2 (2 + cos α) = 0. ( 54
)
The normalized B-basis of F 2 is defined (see [START_REF] Romani | Algebraic-trigonometric pythagoreanhodograph curves and their use for hermite interpolation[END_REF] and Figure 11), for k = 0, 1, 2, 3, 4 by cos α) + ∆p 0 ), we obtain

B 4 k (t) = 1 C 2
∆p 1 = K 2 1 + cos α 2 sin α v 0 v 1 -v 0 2 .
According to (64) we have

εv 1 v 2 = ∆p 2 (2 + cos α) -K 1 v 2 2 ,
and

∆p 2 = K 2 1 + cos α 2 sin α εv 1 v 2 -v 2 2 .
So, for w 0 = v 0 , w 2 = εv 2 and w 1 = 1 2 sin α v 1 , the expression of the control polygon agrees with (60-63).

Remark 9 If ∆p 1 (2 + cos α) + ∆p 0 = 0, the curve regularity and the relation (64) ensure that ∆p 2 (2 + cos α) + ∆p 3 = 0, and conversely. In this case, the control polygon shape is particular. Indeed, we have ∆p 1 = -1 2 + cos α ∆p 0 and ∆p 2 = -1 2 + cos α ∆p 3 , with -1 2 + cos α belonging to -1, -

. The polygon is therefore a cross, as shown in Figure 13.

Then ∆p 1 = -K 2 w 2 0 and ∆p 2 = -K 2 w 2 2 . If we identify both of these expressions with (61) and (62), we obtain w1 = 0 or α = π. In both cases, we can write the relation (54) as : 3w 2 0 + 3w 2 2 + 2w 0 w 2 (2 + cos α) = 0 which is verified if the following condition is true : 3∆p 0 + 3∆p 3 + 2ε(∆p 0 ∆p 3 ) 1 2 (2 + cos α) = 0 either for ε = 1 or for ε = -1.

Conclusion

The spaces P m and F m propose efficient alternatives to polynomial or rational Bézier curves for CAGD purpose. Indeed, their normalized B-bases provide the shape-preserving properties of the Bernstein basis. In these spaces, we showed that the PH curves are still characterized by the Pythagorean condition, like for polynomial curves. Using this property, we studied different particular cases, including P 1 and P 2 for which we determined the general control polygon form of PH curves. We noticed its similarity to the polynomial PH curves described by Farouki and Sakkalis in [START_REF] Rida | Pythagorean hodographs[END_REF] and [START_REF] Rida | The conformal map z → z 2 of the hodograph plane[END_REF]. The same is true for the implicit relation characterizing PH curves in both of these spaces.

Such similarities probably have underlying causes, in relation with the B-bases properties, described in [START_REF] Mainar | Shape preserving alternatives to the rational bézier model[END_REF] and [START_REF] Mainar | Optimal bases for a class of mixed spaces and their associated spline spaces[END_REF]. Studying theses causes could provide an interesting generalization of the present paper to any space having a normalized B-basis.

  have 2m = deg α = 2 sup(deg ν, deg µ) and 2m = deg β = deg ν + deg µ then deg ν = deg µ = m. We denote by

  3 B-based representation of the P 1 PH curves 3.1 Cubic ATPH curves in the normalized B-basis of P 1 According to Proposition 2, all the P 1 ATPH curves are obtained from the F 1 2 functions through the relation (9):

1 Figure 1 :

 11 Figure 1: The normalized P 1 B-basis for α = π 4 , 2π 3 , 3π 2 , 11π 6 .

2 Figure 3 :

 23 Figure 3: Geometric elements of a P 1 ATPH curve.

Figure 4 :

 4 Figure 4: Circle arcs and their control polygons for α = π 4 , 2π 3 and 3π 2 .

Figure 5 :

 5 Figure 5: Geometrical elements for Proposition 5.

1 Figure 6 : 4

 164 Figure 6: Normalized P 2 B-bases for α = π 4 , 2π 3 , 3π 2 , 11π 6 .

1 Figure 7 : 4

 174 Figure 7: The normalized F 1 B-basis for α = π 4 , 2π 3 , 3π 4 , 9π 10 .

1 P 2 P 3 P 4 P 5 Figure 8 :

 123458 Figure 8: Two examples of P 2 ATPH curves.

α 1 Figure 11 : 4 1 = 1 , 4 = - 2 , e 2 , 3 = 2 sin α, e 2 , 4 = 0, e 3 , 3 = - 1 - 2 α 0 P 1 P 2 P 3 P 4 Figure 13 :

 11141142232433120123413 Figure 11: The normalized F 2 B-basis for α = π 4 , 2π 3 , 3π 4 , 9π 10 .

Proof : Assuming the curve r(t) is a PH curve, its control polygon has the form described by (37), so (38-42) can be proved with some calculations. Reciprocally, let us suppose that (38) and (39) are true. As the curve is regular, ∆p 0 = 0 and ∆p 4 = 0. Moreover, if ∆p 1 = 0, relation (38) ensures that ∆p 3 = 0 also, and conversely. So we can set a 2 0 = 1 µ 0 ∆p 0 , a 2 2 = 1 µ 0 ∆p 4 and a 1 = 1 a 0 ∆p 1 . According to (38), (∆p 3 ) 2 = a 2 1 a 2 2 , and ∆p 3 = εa 1 a 2 , where ε = ±1. Condition (39) is hence equivalent to µ 0 a 3 0 a 1 ∆p 2 -εµ 1 a 0 a 2 -µ 1 Γ α a 2 1 = 0, and ∆p 2 = εµ 1 a 0 a 2 + µ Γ α a 2 1 . Then the forward differences can be written as

which is coherent with (37) for w 0 , w 1 , w 2 such that

If relation (40), (41) or ( 42) is verified, instead of (39), the approach is similar and yields the same conclusion.

Proposition 7

If ∆p 1 = 0 and ∆p 3 = 0, the curve r(t) is a PH curve if and only if the following relation is verified :

Proof : When r(t) is a PH curve, it is clear, like in the proof of Proposition 6, that (43) is verified. Reciprocally, let us assume (43) is satisfied. Setting

and ∆p 2 = εµ 1 a 0 a 2 where ε = ±1, which agrees with (37) for w 0 = a 0 , w 1 = 0 and w 2 = εa 2 .

Remark 8 In fact, like in the polynomial case (see [START_REF] Rida | Pythagorean hodographs[END_REF]), the curve is a PH curve if one of the two following conditions is verified :

But, if we assume that ∆p 1 = 0 and ∆p 3 = 0, both of these relations can be rewritten as (43).

Geometrical interpretation

As in the cubic case, the relations (38-43) can be reformulated as functions of geometric elements. We denote by ∆p k = l k exp(iβ k ), the polar form of ∆p k , for k = 0, 1, 2, 3, 4. When ∆p 1 = 0 or ∆p 3 = 0, the relation (38) becomes :

Considering the interior angles θ k for k = 1, 2, 3, 4, as shown in Figure 9, we have

and the relation (44) can be geometrically expressed as

Similarly, conditions (39-42) are respectively reformulated as

The real and imaginary parts of each of these equations furnish two geometric conditions. For example, (46) yields

So, relations (45), ( 50) and (51) form a necessary and sufficient condition to obtain an ATPH curve if ∆p 1 = 0 or ∆p 3 = 0. Any of the equalities (47-49) can replace (46) to furnish two equalities (different of ( 50) and ( 51)) which also constitute with (45) a necessary and sufficient condition.

When ∆p 1 = ∆p 3 = 0, the geometrical situation is different (see Figure 10). We denote by As we mentioned earlier, the curves of the spaces F m can be expressed in a rational form, but this comes at the cost of losing some properties of the trigonometric curves. Here we will describe the ATPH of these spaces, in the corresponding bases. For all values of m in N * , F m is a subspace of P m , therefore we can use the Proposition 2 to describe its PH curves with [START_REF] Fiorot | The circle as a smoothly joined br-curve on [0, 1][END_REF]. We denote by D m 2 the set of functions w of F m 2 for which r, defined by ( 9) is in F m .

However, the constant term (which gives the linear term after integration) appears only when w(t) 2 is linearized. So, the definition of D m 2 needs an additional equation to remove this term.

F 1 PH curves

According to [START_REF] Mainar | Shape preserving alternatives to the rational bézier model[END_REF]

2 with z 0 and z 1 verifying z 2 1 + z 2 2 = 0}, and straightforwardly, we obtain

with ε = ±1. Unsurprisingly such a curve is always a circle arc, whose complex form of the center Ω is ω = r 0 + iεz 2 1 and whose radius is

in the latter requires the inverse matrix

where

. This parameterization is thus

where, relatively to z 0 , z 1 et z 2 , the control polygon is defined by

where

. Such curves are shown in Figure 12.

Using the matrix N α (see (35)) and the relation (54), we can rewrite the control polygon as

This family of curves also has an implicit description, although the condition (54) in this form is complicated.

Proof : First, let us prove that (65) is equivalent to (54). Denoting a = 4z 0 z Solving this linear system, we obtain the b, c, d expressions described by (66-68).

Let us now prove the remainder of the proposition. If the curve is a PH curve, its control polygon is described by (60-63) and by the condition (54). It is clear that (64) and (65) are verified.

Reciprocally, let us assume that (64) is satisfied. We write v 0

As the curve is regular, v 0 and v 2 are non-zero complex numbers. Denoting v 1 = 1 v 0 (∆p 1 (2 +


Alessandra Berretta
email: alessandra.berretta@pg.infn.it

Sara Cutini

Giuseppe Greco

Gino Tosti

Pasquale Lubrano

Stefano Germani

Marica Branchesi

Michele Punturo

Pierre Fernique

Thomas Boch

Matthieu Baumann

Francois-Xavier Pineau

Sebastien Derriere

Ada Nebot

Mark Allen

Search for Spatial and Temporal Coincidence Between Fermi-LAT Exposure Maps and Gravitational Wave Sky Localizations

come

Introduction

On August the 17th of 2017, for the first time, we were able to observe a Gravitational Wave (GW) event, produced after the merging of two Neutron Stars (NSs), and its electromagnetic counterpart, in the form of a short Gamma-Ray Burst (sGRB) and a following kilonova (Abbott et al. 2017). This event led the foundations of the Multi-Messenger astronomy. The typical localization area of a GW is of the order of hundreds of square degrees, so localizing an electromagnetic counterpart by gamma-rays telescope such as Fermi, is very challenging. For this reason, more and more advanced tools are continuously developed, in order to support GW's follow-up. The Multi Order Coverage (MOC) method is particularly suitable for this purposes because allows the user to reconstruct and make logical operations between different sky regions, obtained from various experiments, in an intuitive way (Fernique et al.). 1

Multi Order Coverage map method

The Multi Order Coverage (MOC) method is one of the standard algorithms developed within the International Virtual Observatory Alliance context. The MOC procedure can reconstruct a sky region and compute logical operations such as the intersection, the union and the equality between different maps. It is based on HEALPix tesselation [START_REF] Górski | [END_REF]: once you get a spherical description of your region, HEALPix divides the sphere into 12 diamonds, each of them divided into 4 diamonds recursively. So, at each step, an increasing number of indexes (diamonds) characterizes a specific level that, at the end, defines the resolution of the MOC. In this work we used a particular extension of the MOC method called Space Time MOC (STMOC), in which the space and time coverage are connected in order to easily understand where and when the observations took place (Daniel Durand 2019).

LAT exposure map and LIGO/Virgo GW170104

The LAT exposure map can be seen as an Aitoff projection of the sky observed by the telescope in a certain time window after a chosen trigger, in which the total exposure -area multiplied by time -for a given position on the sky can be found. Since the expected number of events is a function of the photons energy, also the exposure map has a dependency on the energy (Fermi LAT Collaboration 2008). We decided to use the event of GW170104, downloaded from the public LIGO/Virgo database (GWO). Then we obtained the LAT exposure map in 10000 s after the trigger time of GW1701004. We used the Aladin interface to visualize and make logical operations between the MOC map 1. The first step consisted on the selection of all the most exposed pixels in the LAT exposure map, in order to compute the Spatial MOC map. We decided to consider only the pixels that have values greater than 10 7 cm 2 s. After that, we read, from the fits header of the LAT file, the starting and the ending time of the observation, to provide a range over which a discretization of the temporal extension of the map can be performed. We then obtained the STMOC map. Since we want to estimate the spatial and temporal intersection between the LAT map and the LIGO/Virgo event, we computed the STMOC map related to GW 170104. We firstly selected a probability threshold of 0.9 to obtain the spatial MOC map. Then, after reading the trigger time from the fits header of the file and copying it in the appropriate Aladin window, we were able to obtain the STMOC for the GW event, as can be seen in the figure 2. The final step consisted on the production of the intersection map. We selected the STMOC maps obtained for the LAT map and the one obtained for the GW's skymap and then computed the intersection map displayed in 3. As can be seen from the time plot, the intersection is not only from the spatial point of view, but also from the temporal one.

STMOC intersection map of LAT exposure map and LIGO/Virgo skymap

Summary

In this work we showed a practical way to search for spatial and temporal coincidence between the LIGO/Virgo event GW170104 and the related Fermi-LAT exposure map, obtaining the intersection map with a technique called Multi Order Coverage method. Using real astrophysical data, this procedure gives a preliminary demonstration on how interesting the MOC method's could be. The capability to construct sky region in a user-friendly way and in small times will certainly represent a strong support for future the electromagnetic follow-up of GWs.

Figure 1 .

 1 Figure1. The visualization on Aladid of, turning clockwise, the sky seen by Fermi, the GW170104 skymap, the LAT total exposure map and the 31st energy bin of the exposure map.

Figure 2 .

 2 Figure 2. STMOC of the LAT exposure map.

Figure 3 .

 3 Figure 3. Intersection map between LAT map and LIGO/Virgo skymap.

Acknowledgments. The research leading to these results has received funding from the European Union's Horizon 2020 Programme under the AHEAD2020 project (grant agreement n. 871158). The authors acknowledgement the ESCAPE H2020 project under grant agreement n. 824064.