Jean-Paul Thibaud

TOWARDS AN AMBIENT ECOLOGY OF THE URBAN

INTRODUCTION

The contemporary urban world challenges our models of intelligibility and frames of reference.

The urban itself tends to become a real enigma that is subjected to unique multifaceted changes, which are more or less expressed and latent, but that destabilize our certainties and most defined ways of living. Do the transformations of the world we inhabit not manifest themselves in our most banal and ordinary sensory experiences? Isn't the sensory domain a particularly powerful analyzer and operator for exploring the changes in our ways of living? As properly stated by Félix Guattari (1995, p. 101), "The aesthetic power of feeling, although equal in principle with the other powers of thinking philosophically, knowing scientifically, acting politically, seems on the verge of occupying a privileged position within the collective Assemblages of enunciation of our era". We must understand that, based on such an aesthetics, it concerns first and foremost sensing -flows, tones, atmospheres, affects and other vectors of the experienced intensity that, from now on, prevails over a pure world of objects and substances. This new aesthetic paradigm that Guattari elaborated and mobilized in order to launch his 'ecosophy' relies on the very core of debates that nowadays cross urban studies.

In fact, the urban milieu works as the condition of possibility, as the very resonator of the contemporary sensible experience. It is set as an excellent laboratory that allows disclosing new forms of sensing, of testing original frames of thinking and of experiencing unique investigation devices.

But then, what about this aesthetic power of feeling? What does this problematic of sensing reveal about the current and future urban world? How does the problematic of ambiances and atmospheres participate in the urban studies renewal?

OPENING THE URBAN TO ITS AMBIENT FEATURE

Ambiances are part of a general movement of openness to the sensitive and participates in the emergence of new frames of sensitivity. Of course, nowadays ambiances are not the ones from yesterday; the atmosphere of Parisian passages from the 19 th century has little to do with the ones from the 21 st -century malls. However, based on an in-depth analysis, it is our way of being sensitive to the spaces we live in that is undergoing changes. A unique landscape of research has to be drawn in order to accurately understand the socio-aesthetic relationship we have with our everyday environment. No doubt, we are living a historical moment concerning the configurations, assemblages and transformations of our sensory conditions of existence. The 'spherology' developed by Peter Sloterdijk (2005a, p. 30) describes in detail this enlightenment moment that contributes to the "elevation of atmosphere to the level of theory". The sensitive is indeed rising to the surface, making itself heard, and the ambiance is one of its most powerful resonators. If this rise of the senses is no longer to be demonstrated, it is possible to ask why ambiances and affective atmospheres prevail in this subject. Two main arguments can be addressed to reinforce the ambiance's ability to call for a strengthened version of the sensorial and to put to test the ordinary feature of the urban.

A strong version of the sensitive

Without getting into details, it is important to understand that an ambiance is not a sensible domain among other ones, but to what the world becomes sensitive. From this viewpoint, the ambiance would be none other than the flesh of the sensitive world. Therefore, it is not an object of perception -such as a spectacle or even a landscape could be, for example -but the very condition for perception. As a medium (air, sound, light, among others), in-between, metaxu, the In other words, we do not perceive the ambiance itself, we perceive based on it. Such an observation is important: it makes the ambiance the place par excellence for the formation of our perceptual habits, the activation of our sensory-motor schemes and for the engagement of our social-aesthetic relationship to the world. In other words, by sharing autonomous percepts and giving impersonal affects to feel, the ambiance permeates our living environments and infuses contemporary sensibility [START_REF] Thibaud | Les puissances d'imprégnation de l'ambiance[END_REF].

The same goes with the shaping power of an ambiance, its ability to open the urban world to its sensitive content and atmospheric nuances. In other words, an ambiance filters what is perceptible and imperceptible, distributes the usual and the unusual in terms of sensation, traces the blind spots of perception, and fundamentally participates in the communing and sharing of the sensory world.

In short, if there is any change in our sensitivity, then we must be able to find traces in the ambiances housing us in daily life. We understand it now -ambiances are rooted in the pathic feature of sensory experience, summon our sense of nuances and our sensitivity to small perceptions, and set the tone for our most ordinary deeds and common gestures. It is similar to what François Bonnet (2015, p. 72) calls infraworld: "the midst of unobjectified sensations, of unconscious impressions, events barely noticed , or not enough to be remembered". We are at the very edge of what is perceptible, at micro-sensation level, in the core of the reign of hyper-sensibility [START_REF] Grossman | Eloge de l'hypersensible[END_REF]. In no way reducible to the register of representation, the ambiance shows a power of immersion, impregnation and contagion that shapes our body states and our very capacities to feel. It derives less from a particular aesthetic appreciation than it leads to restoring a thought of aisthesis and to questioning the aesthetic discipline itself.

The sensitive test of everyday urban situations

But, then, what does the urban do to the ambiances and, conversely, what do they do to it? If the sense of the ambiance allows paying attention to the nuances of the infra-sensible and of its finest inflections, it just effectively exists within concrete situations and specific contexts. Therefore, we must talk about ambiances in the plural, given its diversities and specificities.

We just need to read the numerous narrations reports on the experiences of the cities to find out how their ambiances differ and to understand how the blocks of sensations are formed and fashioned based on diverse urban milieus (THIBAUD; THOMAS, 2004): ambiances in Los Angeles would rather be given as a generalized transport and outburst of urban life (NANCY, 2011), the ones in Las Vegas would be more of a hallucinatory experience [START_REF] Bégout | Zéropolis: l'expérience de Las Vegas[END_REF], while those of Moscow of the last century would rather emerge from destabilization and hesitation [START_REF] Benjamin | [END_REF]. These examples do not aim at essentializing or stereotyping these ambient milieus -it would also be necessary to return more precisely to the concrete scale of situations -but rather to increase their power to inform and to reconfigure ways of feeling, of placing the body in certain sensory-motor and affective dispositions, of shaping and modulating the atmospheric skin of the city (GRIFFERO, 2013).

As occasional visitors in these cities, our three authors are led to shift their focus, to revise the normal forms of their perception, to acclimate to these new atmospheres by becoming significantly different.

In the words of Jean [START_REF] Ladrière | La ville comme inducteur existentiel[END_REF], the city functions as a real affective or existential inducer. The city shows a totalizing power that borders and frames collective affects. If the sensitive aspect of ambiances is fundamentally discreet and diffuse, almost imperceptible, it is, however, channeled. It never expresses itself at the wild state, it is filtered, domesticated and standardized by our forms of urban life. The city therefore puts the sensitive in condition, it gives its consistence by providing it its ordinary and familiar character. It is at this price that our relationship with the daily world is maintain against a background of ontological security, of basic trust ensuring the continuity and reliability of our living environments [START_REF] Courtright | Is trust like an "atmosphere"? Understanding the phenomenon of existential trust[END_REF]. In this regard, working out an urban ecology of ambiances amounts to updating the modalities and processes that participate in this politics of the sensitive.

It would therefore be a mistake to seek to hypostatize the sensitive and to try to think it independently of the conditions which inform it and of the contexts which specify it. If the ambiance turns out to be particularly effective in terms of urban ecology, it is because it postulates a sensitive embedded in territories and materialities, stories and activities, weather and sociabilities, normative expectations and background affects. Paying attention to ambiances thus amounts to showing sensitivity to the complex arrangements that weave the perceptible world, to the indistinct background and to the lines of force that cross a situation. In a way, an ambiance constitutes a basic composition, an elementary format from which the sensitive emerges from the urban and lends itself to experience.

SETTING THE TONE OF URBANITY

This ambient ecology consists in considering the aestheticization processes of urban spaces. Some would see it as a particular case of aestheticization of the world in the age of contemporary capitalism (LIPOVETSKY & SERROY, 2013) while others would recognize an expression of art in a gaseous state [START_REF] Michaud | L'art à l'état gazeux. Essai sur le triomphe de l'esthétique[END_REF]. Cleary, this aestheticization of the urban is in no way reducible to simple operations of embellishment or decoration of inhabited spaces, it proceeds rather from their sensory government, from their setting into ambiances. It is then the place of the body and of sensitive experience that is affirmed, both as a fundamental anthropological value of dwelling and as a privileged instrument of the economy of experience [START_REF] Pine | The experience economy[END_REF]. The very notion of experience is now a socio-economic issue in the making of the city and is essential as one of the key words of urban governance [START_REF] Hasse | Atmospheres as expressions of medial power. Understanding atmospheres in urban gouvernance and under self-guidance[END_REF]. The ambiance -as the privileged operator of the sensitive experience -is thus the object of increasing care, of a deliberate shaping and ordering, even sometimes of a strategic instrumentalization. In other words, it is less and less an unintentional consequence of an action [START_REF] Soubeyran | Pensée aménagiste et improvisation[END_REF].

We can then wonder how the ambiances participate in various figures of the contemporary city: from the historical city turned into patrimony to the new scenes of the creative city, from the functional design of the mobile city to the greening of the sustainable city, from air conditioning of the underground city to the animation of the event city, from the conditioning of the mercantile city to the gentrification of the city center, numerous interventions involve an ambient aestheticization in setting the tone to these spaces. Are we then witnessing the emergence of an "ambiance-manifesto", attentive to becoming the showcase of the urban, exhibiting an attractive brand image, and anxious to display an attractive environment? Has ambiance become an area to show off and exhibit? A plurality of urban worlds unfolds here, and the ambiance plays a specific role in each of them, in its capacity to install them, to qualify them and to bring them to life. The modes of existence of urban ambiances are multiple and varied, playing both on natural-climatic and artificial-technological environments.

Thorough research would have to be initiated on this matter, distinguishing the various forms of setting an ambiance and specifying how the sensitive field is embedded in the processes of transformation of the city. From this perspective, it would not be so much about targeting types of space or city figures, as about focusing on the conditions of the sensitive urban experience: for example, to wonder about examining the sanitization of our environment of life [START_REF] Sennett | La chair et la pierre[END_REF][START_REF] Thomas | L'aseptisation des ambiances piétonnes au XXIème siècle[END_REF], the urban programming of animation (HAJER & REIJNDORP, 2000), the emergence of a guaranteed city (BREVIGLIERI, 2013), the spectacularization of the urban (JEUDY & BERENSTEIN-JACQUES, 2006), the standardization of the living environments [START_REF] Jeudy | Critique de l'esthétique urbaine[END_REF], the reign of weak sensations (KOOLHAAS, 2011) or clean town planning [START_REF] Dolle | Le territoire du rien, ou la contre-révolution patrimonialiste[END_REF]. So many proposals that attempt to explain the shortcomings and forms derived from the aestheticization of the city. Everything happens as if the current city participates in an impoverishment of sensitive experience and a reduction of the field of perceptions.

According to another perspective, we could start from a logic of the sensitive and wonder about the elementary operations that participate in setting in an ambiance. It would be less a question of describing in detail the deterioration of urban qualities and amenities than of accounting for the very power of ambiances to make (and undo) territory with sensitivity. The concept of refrain developed by Gilles Deleuze and Félix Guattari (1980) is a model in this area, making it possible to understand how sensitive traces and expressive marks qualify a territory and operate a territorial arrangement.

We could thus ask ourselves about the capacity of an ambiance to animate (to infuse life), condition (to influence behaviors), monitor (to control environments), atmospherize (to homogenize spaces) and attune (to compose affects) an urban setting. A generative grammar should be developed, which would describe the various methods of raising awareness of inhabited spaces. It is here the critical power of the notion of ambiance that is really at work, making it possible to reveal the underlying processes constituting the installation of a common atmosphere in which everyone bathes. If there is atmospheric politics, it is at the pre-discursive level that it must be sought, in the capacity of a space to immerse and permeate a public, in its aptitude to produce a shareable ambient environment. This psycho-politics then consists in deconstructing the atmospheric conditioning necessary for a public space to exist and last (SLOTERDIJK, 2005b).

It would undoubtedly be a mistake to seek to affiliate excessively the ambient setting of an urban space with a particular type of territory or knowledge. We are indeed dealing with a transverse, diffuse and multifaceted movement that tends to extend to all areas of urban life. An ambiance functions both as an analyzer and an operator of the sensitive transformations of contemporary urbanity. Here lies the heuristic interest of a perspective in terms of ambiance: being able to move between multiple scales (from the urban micro-situation to the globalized space of cities), putting oneself at the service of very diverse questions (safety, health, risk, trade, transport, hospitality, tourism, museography, etc.), articulating technical, aesthetic and social dimensions in the same movement [START_REF] Amphoux | Une expertise "ambiance" est-elle possible? Réserves, propositions et plaidoyer[END_REF]. The proceedings of the three International Congresses on Ambiances (AUGOYARD, 2011;[START_REF] Thibaud | Ambiances en actes[END_REF][START_REF] Remy | demain. Volos[END_REF] bear witness to these very numerous orientations.

If ambiances are often requested in studies dealing with well-being and the quality of urban life, they are also strongly mobilized in a more critical version. It means describing how urban ambiances fit into the framework of the neo-liberal city and of a psychopolitics using emotional design and smart power [START_REF] Han | Psychopolitique: le néolibéralisme et les nouvelles techniques de pouvoir[END_REF]). An ambiance-based approach, for example, allows us to understand how globalization processes are set up and truly take shape [START_REF] Urry | Global utopia? On airport atmospherics[END_REF]. Particularly formatted and conditioned, the ambiances of airports are analyzed as a concrete manifestation of globalization and as a more general phenomenon which is spreading beyond strict airport environments.

Undoubtedly, we have here a completed example of the production of a hermetic and artificial environment, part of an overall movement of capsularization of collective spaces [START_REF] De Cauter | The capsular civilization[END_REF]).

Others will take the shopping mall [START_REF] Crawford | The world in a shopping mall[END_REF][START_REF] Peron | Les boîtes. Les grandes surfaces dans la ville[END_REF] or the theme park [START_REF] Sorkin | See You in Disneyland[END_REF]BEGOUT, 2010) as paradigmatic cases of the ambient future of the world.

Various research works empirically investigate these contemporary urban trends, relying on concrete case studies to highlight a new regime of sensitive experience: the Disneyfication of Times Square shows how an urban landscape is transformed into valuable imagescapes based on a market value and gives rise to a feeling of re-enchantment, of the order of simulation and suspension of disbelief [START_REF] Boyer | Twice-told stories: the double erasure of Times Square[END_REF]; the privatization of public spaces at Potsdamer Platz in Berlin illustrates how the ambiance participates in a new strategy of power playing on inclusion, seduction and a real economy of affect [START_REF] Allen | Ambient power: Berlin's Potsdamer Platz and the seductive logic of public spaces[END_REF]; the arrangement of various ambiances in the gigantic ION Orchard shopping mall in Singapore reveals how each of them addresses a particular segment of society while helping to forge the emotional identity of this City-State [START_REF] Hudson | ION Orchard: atmosphere and consumption in Singapore[END_REF]. In such works, everything happens as if these ambiances were part of a certain form of derealization of the urban experience, of a distended relation to the reality of the order of the floating and the marvelous, arising in a way dream platform [START_REF] Ratouis | La plateforme du rêve. Figures américaines de la fonction de loisir[END_REF]. Did Walter [START_REF] Benjamin | Paris capitale du XIX e siècle[END_REF] not anticipate such an evolution in his time, when he analyzed Parisian passages in terms of urban phantasmagoria consecrating the aestheticization of the commodity? Today's ambiances seem to be part of a massive process of commodification. Therefore, they concerns participating in urban marketing, in implementing new branding, packaging and benchmarking techniques, in order to give an identity to cities and to acclimate the market (GRANDCLEMENT, 2004). In his aesthetic of atmospheres, the German philosopher Gernot Böhme (2017) elaborates on this subject a critique of contemporary aesthetic economy. Based on the thoughts of Walter Benjamin and Jean Baudrillard, he proposes the notion of staging-value, complementary to the more classic notions of use value and exchange value. This critique of the aestheticization of the merchant world thus invites us to take note of the role that ambiances play in their ability to display and enhance consumer products, to encourage purchase by selling an image and arousing desire [START_REF] Michaud | Expériences, arrogance, authenticité[END_REF]. Experienced in the 1970s [START_REF] Kotler | Atmospherics as a marketing tool[END_REF], this commercial strategy intervenes on multiple ambiance factors such as music, colors, smells, temperature, and other tactile sensations. The ambiance here becomes an instrument used for commercial purposes, targets city dwellers as consumers and is designed in such a way as to produce specific effects. In such a context, it loses its critical power to become an instrument at the service of the neoliberal city and of keno-urbanization (DEAR, 2002). In a way, the ambient ecology of the urban meets here the new school of urban ecology of Los Angeles.

As important as they are, such works should not mask the diversity of the urban world and the insistence of the surrounding environments to persevere in their ordinary, plural and informal forms.

If the preceding remarks rightly revealed the incursion of contemporary capitalism into urban spaces, the fact remains that ambiance is deployed most of the time in any banal spaces, more discreet and silent, less spectacular, without a sustainable place to be subscribed of its own [START_REF] Certeau | L'invention du quotidien: arts de faire[END_REF]. This means that an ambiance is not so easily decreed. The production of introverted spaces with formatted ambiances is only one side of contemporary urban reality, even if it testifies to a growing force of influence. The importance of daily social practices, inhabitant memories, local customs and forms of sociability is not for nothing in the ambient existence of an urban situation (ANGIBOUST, DOUSSON, MELEMIS, TIXIER, 2014). In this regard, we could speak of the ambient adherence of a place, of the strength of a local atmosphere to persist and endure, of its resistance to deteriorate in a way that is too voluntarist and sudden. This is how we could also take the measure of ordinary urban practices that contribute to install an ambiance and to configure the ambient environment to make it livable, even hospitable (CHASE, CRAWFORD & KALISKI 2008;[START_REF] Amin | Lively infrastructure[END_REF].

A distinction should perhaps be made about the ambient setting of urban spaces, between establishing and installing an ambiance, between implementing a strategy of seduction and market immediacy and assisting the world on a daily basis by letting operate informal practices and longterm impregnations [START_REF] Thibaud | Installing an atmosphere[END_REF]. This is another issue related to ambiances: their power of immersion and infusion, their ability to affect and attune living environments in a more or less open or restrictive manner.

THE EMERGENCE OF AN AMBIENT SENSITIVITY

Paving the way for a situated, embodied, enacted and shared conception of the sensitive world, the ambiance is located at the interface of research and design. It makes it possible both to carry out the investigation and to design an urban project, to question the sensitive city (heuristic power) and to shape it (operational power). Again, as a sort of two-sided Janus -a dialectical image à la Walter Benjamin -it can function as a powerful tool for criticism and deconstruction of urban environments, as well as an active and effective instrument of the neoliberal city. No doubt, an ambiance is a common asset we can grow and degrade, take care of and be abusive to. In other words, ambiances put to the test the most precarious and fragile character of urban existence.

Finally, what would an ambient ecology reveal regarding our urban condition and what would it bring to urban studies? Let us say, in the first place, that it concerns highlighting a sensitive ecology that gives the city back to labile and fragile entities: flows, atmospheres, envelops, and weather. It means that the urban environment cannot be reduced to a world of objects and well-defined entities, but that it also derives from a fluid, porous and diffuse space, in continual movement, formation and transformation. Therefore, we should learn to recognize such a world of minor existences [START_REF] Lapoujade | Les existences moindres[END_REF], that is most of the time imperceptible, although effective. Such a program could open a path to a new ecology of urban situations, an ecology attentive to the phenomena of envelopment and impregnation common to all situated experience. Placed under the aegis of affect, of our capacity to affect and be affected by the surrounding environment, this ambient ecology would enable to capture and describe the various urban tones that infuse the contemporary world. In this regard, the question would not only be to understand how the urban is produced, but also to what types of affordances, sway and affects it gives rise.

Close to the Chicago School and to the new Los Angeles school, and related to the ecosophy of Felix Guattari as much as to the ecology of perception of James Gibson, this ambient ecology would thus participate in the emergence of a sensitivity to the atmospheric existence of urban life. Sharpening our sensitivity to micro-perceptions and nuances of experience, to vitality affects and powers of action would work as a guide for attention: attention to the diffuse feelings of vulnerability generated by an urban environment, to the sensation of disorientation of a world that we hardly recognize, to low intensity phenomena that make a space habitable and welcoming. It is no less than our sensitive relationship to inhabited spaces which is at stake here, in its vital and existential dimensions. The forms of urban life would then be related to our ways of being sensitive to the spaces we inhabit and to our capacities to pay attention to their most precarious features. Are we not witnessing a movement to sensitize urban studies themselves?

  ambiance is what makes perception possible, what we perceive from, what makes the sensitive exist (COCCIA, 2010).


Charles Barnouin

Benjamin De Witte

Richard Moreau

Arnaud Lelevé

Xavier Martin

Xavier Martin Cost

Cost-Efficient Laparoscopic Haptic Trainer Based on Affine Velocity Analysis

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

 As feedback about skill level is displayed, the simulator should be effective in learning, this needs however to be confirmed by future investigations.

 Learning minimal invasive surgery (MIS) skills is young surgeons' major concern.

 Cognitive load elicited by simulators' use and trainees' spatial abilities seem to impact efficient learning process [START_REF] Van Merriënboer | Cognitive load theory in health professional education: design principles and strategies[END_REF] .

 Objectives : design a basic skill training simulator which objectively evaluate trainees' level.

 Use of Affine velocity [START_REF] Pollick | Constant affine velocity predicts the 13 power law of planar motion perception and generation[END_REF][START_REF] Cifuentes | Automatic gesture analysis using constant affine velocity[END_REF] as assessment variable.

Step 1 -Cognitive conception



  Observation and analyses of surgeons in situ  Inclusion of literatures' recommendations Use of materials enabling haptic feedback and developing a VR environment (Phantom Omni, CHAI3D, laparoscopic devices…) Panel of 77 subjects separated initially into 4 groups : • Expert surgeon : more than 100 interventions • Intermediate : between 5 and 20 • Unexperienced intern (BSS) : witnessed but never performed • Novice Statistical test Kruskal and Wallis on affine velocity (above) can separate every groups but Experts and Intermediates, whereas collision alone could also not separate Novices from Unexperience Interns. and Intermediate fused with variables Collision, Time, and Affine Velocity Relationship between geometry and kinematic, first shown in 2D drawing : 𝒗 = 𝒗 𝒂 𝑲 -𝟏/𝟑 • With the curvature K New power law for 3D movement : 𝒗 = 𝒗 𝒂 𝑲 𝜶 𝝉 𝜷 • With the torsion 𝜏 • 𝛼 and 𝛽 are exponents that depend of the studied movement Logarithmic linearization to find 𝛼 and 𝛽 for laparoscopic gesture : 𝜶 = -𝟎. 𝟎𝟒𝟖, 𝜷 = -𝟎. 𝟎𝟎𝟐𝟔 Step 3 -Simulator validation Define assessment variables to evaluate the trajectory : • Time taken to complete the level • Number of errors (collisions)

HAL Id: hal-01563262 https://hal.science/hal-01563262

	Submitted on 4 Feb 2019
	HAL is a multi-disciplinary open access	L'archive ouverte pluridisciplinaire HAL, est
	archive for the deposit and dissemination of sci-	destinée au dépôt et à la diffusion de documents
	entific research documents, whether they are pub-	scientifiques de niveau recherche, publiés ou non,
	lished or not. The documents may come from	émanant des établissements d'enseignement et de
	teaching and research institutions in France or	recherche français ou étrangers, des laboratoires
	abroad, or from public or private research centers.	publics ou privés.

Cost-Efficient Laparoscopic Haptic Trainer Based on Affine Velocity Analysis Barnouin

  C.[START_REF] Van Merriënboer | Cognitive load theory in health professional education: design principles and strategies[END_REF] , De Witte B. 2 , Moreau R. 1 , Leleve A. 1 , Martin X.[START_REF] Cifuentes | Automatic gesture analysis using constant affine velocity[END_REF] 1 Univ Lyon, INSA lyon, Ampère-UMR CNRS 5005, 25 av. Jean Capelle ouest-F-69621 Villeurbanne Cedex, France.

2 Univ Lyon, University Claude Bernard Lyon 1, Inter-University Laboratory of Human Movement Science. 27-29 Boulevard du 11 Novembre 1918, 69622, Villeurbanne cedex, France.

[START_REF] Cifuentes | Automatic gesture analysis using constant affine velocity[END_REF]

Univ Lyon, University Claude Bernard Lyon 1, Faculty of Medecine, Surgery school, 8 Avenue Rockefeller, 69003 Lyon, France .

.  A cognitive analysis of MIS enables to design a reliable and valid simulator.  Affine velocity is a valid tool and another objective variable to evaluate a trainee skill on his trajectory.  Once a certain level of skill is reached, it becomes harder to differentiate individuals.


Joseph Larmarange

Mapping Demographic and Health Surveys (DHS): a method to estimate regional trends of a proportion

TheCprevRCpackage

  This.approach.has.been.implemented.in.a.free.package.for.RT.. Its.functionalities.areV ...4.bilingual.'English.and.French1 ...4.importation.of.files.provided.on.measuredhsTcom ...4.estimation.of.a.prevalence.andWor.a.risk.ratio.surface ...4.kernel.density.estimation.and.kriging.interpolation ...4.results.exportation.in.Shape.files.and.»SCII.Grids.. ...4.open.Source.Licence.'CeCILL4C1 MoreCInformation prevRConCCRAN http://cran:rNproject:org/web/packages/prevR/ SourceCcodeConCGitHub https://github:com/larmarange/prevR

  For. many. countries8. in. particular. in. sub4Saharan. »frica8. Demographic. and.Health. Surveys. 'DHS1. are. the. main. national. source. of. data. 'depending. on. the.subject1T..Several.DHS.collect.latitude.and.longitude.of.surveyed.clusters. but.the.sampling.method.is.not.appropriate.to.derive.local.estimatesV.sample. size.is.not.large.enough.for.a.direct.spatial.interpolationT Our. objective. is. thus. to. estimate. from. DHS. data8. irrespective. of. administrative.divisions8.a.surface.that.reveals.the.main.spatial.variations.of. the.selected.indicator8.while.retaining.an.infraregional..local.accuracy.for.the. adequately.surveyed.areasT TestingCtheCmethod In. order. to. test. our. approach8. we. devised. a. fictitious. country. for. which. we. simulated. DHSsV. this. makes. it. possible. to. compare. the. prevalence. surface. estimated. from. survey. data. with. the. model's. original. prevalence. surfaceT. DHS.simulations.were.carried.out.to.reproduce.data.comparable.with.actual. Conclusion The.use.of.adaptive.bandwidths.of.equal.number.of.persons.surveyed.makes. it.possible.to.achieve.a.smoothing.effect.that.adapts.to.the.high.irregularity. of. spatial. distribution. among. the. survey. clustersT. . The. surfaces. thus. generated. are. relatively. accurate. for. densely. populated. areas. and. strongly. smoothed.in.sparsely.surveyed.areasT »lthough. local. variations. were. filtered. out. by. this. type. of. technique8. the. regional. component. in. the. spatial. variation. of. prevalence. was. generally. reproduced8. and. the. estimated. prevalence. surfaces. could. be. interpreted. as. regional.trend.surfaces.with.adaptive.bandwidthsT..».surface.of.this.sort8.by. construction8.is.necessarily.spatially.continuous.and.self4correlated.and.in.no. way. implies. any. potential. discontinuities. and. local. variations. in. the. real. surface.of.the.epidemic8.which.remains.inaccessible.in.the.DHS.dataT While.produced.maps.should.be.interpreted.with.caution8.they.do.provide.a. Methods Kernel. estimators. techniques. are. designed. to. construct. a. surface. from. a. scatter.plot8.where.each.point.represents.an.observed.caseT The.use.of.fixed.bandwidths.is.inappropriate.for.DHS8.since.the.clusters.are. very.unevenly.distributedT..».sufficiently.large.radius.needs.to.be.determined. for.estimating.proportions.from.a.sufficient.number.of.individuals8.especially. in.those.areas.where.the.clusters.are.widely.dispersedT..»t.the.same.time8.in. densely. populated. survey. areas8. smaller. circles. could. be. used8. since. the. numbers.are.amply.sufficientTWe. therefore. developed. an. approach. using. adaptive. bandwidth. kernel. estimators. so. that. the. bandwidth. used. for. cases. in. a. single. cluster. would. depend.solely.on.their.location.and.specifically.the.number.of.observations. in. the. neighbourhood. of. that. clusterT. . For. the. estimation. of. the. intensity. surface. of. observed. cases8. the. principle. is. similar. to. the. nearest. neighbour. techniqueT. . ». minimum. number. of. observations. N. is. set. and. the. radius. of. the.smoothing.bandwidth.is.therefore.proportional.to.the.radius.of.the.circle. to.be.drawn.around.the.cluster.in.order.to.capture.this.minimum.numberT.. For. the. positive. cases. we. apply. the. same. bandwidth. as. that. calculated. for. the.control.cases.in.the.same.clusterT

	MappingCDemographicCandCHealthCSurveysCIDHSxCwithCprevRA
	aCmethodCtoCestimateCregionalCtrendsCofCaCproportion
	JosephCLarmarangeCCCCCCentreCPopulationCetCDéveloppementCICEPEDCUMRC-94CParisCDescartesCInedCIRDx
	Context

surveysT Figure. a. represents. the. prevalence. surface. created. for. the. model. 'national. prevalence.of.KRq1T..».DHS.was.simulated.with.a.sample.size.of.L8RRR.people. distributed. in. ERR. clustersT. Figure. b. represents. the. result. of. applying. our. method.on.the.data.obtained.from.this.simulationT Overall8.the.main.variations.in.the.model.prevalence.surface.are.reproducedT.. descriptive. indication. of. the. state. of. a. phenomenon. in. a. country. independent.of.administrative.divisionsT..It.is.a.useful.tool.for.displaying.the. main. spatial. variations. and. identifying. potential. hotspotsT. . »lthough. DHSs.

are. insufficient. for. analysing. spatial. determinants8. they. do. make. it. possible. to. sketch. out. a. preliminary. picture. in. the. absence. of. more. specific. surveys. with.better.geographical.coverageT..Furthermore8.the.method.could.be.easily. applied.using.prevR8.the.dedicated.R.packageT


Jean-Baptiste Hiriart-Urruty

Patrice Lassère

Pierre-Jean Laurent

Pouvez-vous "mesurer" la "forme" d'un triangle ?

Pour un triangle de périmètre p, de cercle inscrit de rayon r et de cercle circonscrit de rayon R, nous présentons une "mesure" définie à partir de ces données, p -(4R + 2r) précisément, qui permet de décider si le triangle en question est rectangle, a un angle obtus, ou a ses trois angles aigus.

Introduction

Le triangle est assurément la figure du plan dont la géométrie est la plus riche. C'est pour cela qu'elle est l'objet d'études depuis des millénaires, qu'on l'utilise encore aujourd'hui comme support d'enseignement des mathématiques dans les collèges et lycées, et qu'elle est toujours source de questions dans les "rubriques de problèmes" de journaux divers et variés.

Notre contribution ici est de caractériser la "forme" d'un triangle (sous-entendu, estil rectangle, obtusangle, acutangle ?) à l'aide d'une "mesure" définie à partir de données comme le périmètre du triangle, le rayon du cercle inscrit et le rayon du cercle circonscrit.

Mais commençons par un mini-quiz comme il en existe de nombreux, et qui pourrait continuer ad libitum.

Considérons trois nombres réels a > 0, b > 0, c > 0. Procédons progressivement. Q1. Existe-t-il un triangle dont les côtés (c'est-à-dire les longueurs des côtés) seraient a, b, c ?

La réponse est clairement : Non, pas toujours. La condition nécessaire et suffisante pour qu'il en soit ainsi est fournie par les trois inégalités, dites triangulaires justement, a < b + c, b < a + c, c < a + b. Disons tout de suite, et ce sera valable pour toute la suite, que nous ne considérons que des "vrais" triangles, pas les aplatis ou dégénérés. D'ailleurs, le degré de platitude ou de non-dégénérescence peut être "mesuré" par le produit des écarts (b + c) -a, (a + c) -b, (a + b) -c, qui est :

(b + c -a)(a + c -b)(a + b -c) = 16A 2 a + b + c , (1)
où A désigne l'aire du triangle de côtés a, b, c. Ceci résulte de la formule de Héron donnant A en fonction de a, b, c, formule que nous aurons l'occasion de revoir plus loin. Notons que la conjonction des trois inégalités triangulaires peut être résumée en une seule :

|b -c| < a < b + c. (2)
Réduisons l'information... On se donne p > 0. On pourrait ainsi continuer la liste des questions... Il existe pour cela des ouvrages fort complets et nous nous contentons de n'en signaler qu'un : [START_REF] René | La géométrie du triangle. Exercices résolus[END_REF] .

La question qui nous préoccupe dans cette note est la suivante : peut-on décider de la "forme" d'un triangle (c'est-à-dire rectangle, obtusangle, acutangle) à partir des seules données p, r, R ? On a vu que, prises séparémént, p ou (r, R), elles ne conduisent à rien... De manière un peu étonnante (pour nous, du moins), la réponse est Oui ; c'est l'objet du paragraphe suivant.

2.

Une "mesure" de la "forme" d'un triangle Le point de départ fut de démontrer que l'on a p = 4R + 2r pour un triangle rectangle, ce qui n'est pas difficile à faire,... puis de démontrer la réciproque, ce qui est un peu plus retors. Et si cet écart p -(4R + 2r) était la "mesure" qui caractérise la "forme" d'un triangle ? Enonçons sans plus attendre le résultat principal.

Théorème. On a les caractérisations suivantes : -Le triangle est rectangle si et seulement si p = 4R + 2r ; -Le triangle est obtusangle si et seulement si p < 4R + 2r ; -Le triangle est acutangle si et seulement si p > 4R + 2r.

Nous obtiendrons les trois caractérisations en un seul coup, en démontrant au préalable une formule qui a son intérêt propre. La voici. Soit donc un triangle ABC ; on note α, β, γ les angles aux sommets A, B, C (que l'on note aussi A, B, C dans la littérature) et a, b, c les (longueurs des) côtés opposés à A, B, C. La clé de la démonstration du théorème énoncé est la relation suivante : Notre parcours pour démontrer le théorème ci-dessus a une histoire. Après avoir démontré le premier cas (caractérisation du triangle rectangle), nos recherches pour démontrer les autres cas nous ont conduit à l'opus de V. Prasolov ([2]). Par une série de résultats se renvoyant les uns aux autres, l'auteur arrive à la formule (3), à une erreur de coefficient près, que nous corrigeons ici.

cos(α) cos(β) cos(γ) = p 2 -(2r + 4R) 2 16R 2 . ( 3
La démonstration proposée ci-dessous est la plus simple et, à notre avis, celle qui présente les enchaînements les plus logiques.

Démonstration de la formule ( 3). Comme souvent en mathématiques, et toujours dans les démonstrations, il faut indiquer les prérequis, bref le matériel dont on dispose. Rappelons donc quelques propriétés classiques (cf. [START_REF] René | La géométrie du triangle. Exercices résolus[END_REF] par exemple).

-Si A désigne l'aire du triangle et s = p/2 son demi-perimètre, on a :

A = pr/2 = sr. (4)
-La dite "règle des sinus" :

sin(α) a = sin(β) b = sin(γ) c = 1 2R . (5)
Comme A = sr (formule (4)) et A = bc sin(α) (ou ac sin(β) ou ab sin(γ)), on déduit :

abc = 4srR. (6)
-La formule de Héron pour l'aire A du triangle :

A 2 = s(s -a)(s -b)(s -c) (7.a) = 1 16 (a + b + c)(b + c -a)(a + c -b)(a + b -c) (7.b) = 1 16 (a 2 + b 2 + c 2 ) 2 -2(a 4 + b 4 + c 4 ) . (7.c)
La variante (7.b) met bien évidence les écarts aux inégalités triangulaires évoqués plus haut. La variante (7.c) peut s'avérer utile dans certains calculs. La formulation (7.a) est la plus connue. Pour la retenir, nous conseillons aux élèves ou étudiants de se rappeler qu'elle est le cas limite de la formule de Brahmagupta donnant l'aire d'un quadrilatère inscriptible de côtés a, b, c, d (et de semi-périmètre s) :

A 2 = (s -d)(s -a)(s -b)(s -c). ( 8
)
Formule très élégante où les quatre côtés interviennent de la même façon. En faisant tendre un côté vers zéro, d par exemple, on arrive à (7.a) sans perdre de vue qu'un triangle est toujours inscriptible ! Nous sommes prêts pour la démonstration, en quatre étapes de complexité croissante. Lemme 1.

a 2 + b 2 + c 2 = 4s 2 -2(ab + bc + ca). (9)
En effet, 4s 2 = 4 a+b+c

2 2 = (a + b + c) 2 = a 2 + b 2 + c 2 + 2(ab + bc + ca). Lemme 2. ab + bc + ca = r 2 + s 2 + 4rR. ( 10
)
Grâce à (4) et (7.a), on a :

A 2 = s 2 r 2 = s(s -a)(s -b)(s -c), d'où : sr 2 = (s -a)(s -b)(s -c) = s 3 -s 2 (a + b + c) + s(ab + bc + ca) -abc = s 3 -2s 3 + s(ab + bc + ca) -abc puisque a + b + c = 2s sr 2 = s 3 -2s 3 + s(ab + bc + ca) -4srR puisque abc = 4srR (cf. (6)).
En divisant par s la dernière égalité, on arrive bien à (10). Des Lemmes 1 et 2, on déduit :

a 2 + b 2 + c 2 = 2s 2 -2r 2 -8rR. (11)
Jusqu'à présent tout était "métrique" (i.e., ne concernait que a, b, c, r, R). Maintenant on va faire intervenir la trigonométrie, avec les angles α, β, γ.

Lemme 3.

sin 2 (α) + sin 2 (β) + sin 2 (γ) = a 2 + b 2 + c 2 4R 2 . ( 12
)
Cela résulte directement de la règle des sinus (cf. (5)). Lemme 4. C'est le plus délicat, ou du moins le plus calculatoire, car il s'agit de relier cos(α) cos(β) cos(γ) et sin 2 (α) + sin 2 (β) + sin 2 (γ). De fait, on a :

cos(α) cos(β) cos(γ) = sin 2 (α) + sin 2 (β) + sin 2 (γ) 2 -1. (13)
Pour démontrer (13), nous allons procéder en deux étapes : d'abord évaluer cos(2α) + cos(2β) (ce qui conduira à 1 -2 sin 2 (α) + 1 -2 sin 2 (β) ), puis évaluer cos(2γ) en tenant compte du fait que α + β + γ = 180 • . Observons d'abord que les deux nombres complexes e i(α+β) e i(α-β) + e i(α+β) e i(β-α) et e i(2α) + e i(2β) sont égaux ; ils ont donc même partie réelle, c'est-à-dire : Il ne reste plus qu'à transformer les cos(2x) en 1 -2 sin 2 (x) pour arriver à :

1 -2 sin 2 (α) + 1 -2 sin 2 (β) + 1 -2 sin 2 (γ) = -4 cos(γ) cos(α) cos(β) -1,
d'où le résultat (13). Des Lemmes 3 et 4, on déduit :

cos(α) cos(β) cos(γ) = a 2 + b 2 + c 2 8R 2 -1. ( 17
)
En réunissant (11) et (17), on obtient :

8R 2 cos(α) cos(β) cos(γ) = 2s 2 -2r 2 -8rR -8R 2 ,
soit, en remplaçant s par p/2 et en multipliant par 2 :

16R 2 cos(α) cos(β) cos(γ) = p 2 -4r 2 -16rR -16R 2 = p 2 -(2r + 4R) 2 .
C'est la formule (3) annoncée. Remarque 1. En utilisant la dite "règle des cosinus", cos

(α) = b 2 +c 2 -a 2 2bc , cos(β) = a 2 +c 2 -b 2 2ac , cos(γ) = a 2 +b 2 -c 2 2ab
, l'égalité (3) devient :

(b 2 + c 2 -a 2 )(a 2 + c 2 -b 2 )(a 2 + b 2 -c 2 ) = (abc) 2 2R 2 p 2 -(2r + 4R) 2 . ( 18
)
Cette formule, que l'on peut qualifier de "pythagoricienne" en raison de son premier membre, est une sorte de mesure du "déficit de rectangularité" d'un triangle.

Dans la référence classique ([1]

), l'auteur dit ceci en page 225 : "En utilisant les formules bien connues ( 4 ), ( 6

2

  cos(α + β) cos(α -β) = cos(2α) + cos(2β). (14) Du fait que α + β = 180 • -γ, il vient cos(γ) = -cos(α + β), puis avec (14) : cos(2α) + cos(2β) = -2 cos(γ) cos(α -β). (15) Toujours pour la même raison, on a : cos(2γ) = 2 cos 2 (γ) -1 = -2 cos(γ) cos(α + β) -1. (16) En rassemblant (15) et (16), on obtient donc : cos(2α) + cos(2β) + cos(2γ) = -2 cos(γ) [cos(α -β) + cos(α + β)] -1 = -4 cos(γ) cos(α) cos(β) -1.

  ), ( 7 .a) et en les injectant dans l'expression p = 2r + 4R, on obtient après un peu d'exercice(b 2 + c 2 -a 2 )(a 2 + c 2 -b 2 )(a 2 + b 2 -c 2 ) = 0". Avouons que nous avons essayé de manière assidue et que nous n'y sommes pas arrivés ; l'expression "avec un peu d'exercice" nous paraît donc an understatement. Notre approche, plus générale car mesurant exactement le produit (b2 + c 2 -a 2 )(a 2 + c 2 -b 2 )(a 2 + b 2 -c 2 ), indique clairement ce que l'auteur demandait de démontrer : le triangle est rectangle si et seulement si p = 2r + 4R.Remarque 2. Les problèmes d'optimisation qui consistent à maximiser ou à minimiser f (α, β, γ) = cos(α) cos(β) cos(γ) parmi tous les angles α, β, γ de trinagles (non aplatis) sont faciles à résoudre. On aboutit à :max f (α, β, γ) = 1 8 ; inf f (α, β, γ) = -1.(19)Toutes valeurs de l'intervalle ]-1, 1/8] sont possibles. La valeur maximale 1/8 est atteinte pour un triangle équilatéral, par exemple avec a = b = c = √ 3, r = 1/2, R = 1. La valeur -1 est une borne inférieure, pas un minimum ; on s'en approche avec un triangle isocèle en A d'angle α tendant vers 180 • .

)

  Ainsi, le produit cos(α) cos(β) cos(γ) est du signe de p -(2r + 4R). Muni de cette clé, le serrurier conclut rapidement : -L'un des cosinus est nul, et donc l'angle correspondant est rectangle, si et seulement si p = 2r + 4R ; Sachant qu'un triangle ne peut avoir qu'un seul angle obtus, il n'y a donc qu'un seul cosinus qui peut être négatif, d'où -Les trois cosinus sont strictement positifs, et donc les trois angles sont aigus, si et seulement si p > 2r + 4R ; -Un seul cosinus est strictement négatif, et donc l'angle correspondant est obtus, si et seulement si p < 2r + 4R.

Nous adoptons ici trois appellations de même assonance : rectangle lorsque le triangle a un angle droit, obtusangle lorsque le triangle a un angle obtus (un seul angle obtus est possible), acutangle lorsque les trois angles du triangle sont aigus.