Isabel Repiso

Talking about counterfactual worlds A comparative study of French and Spanish

Keywords: conditionality, counterfactuality, modal verbs

Counterfactual thinking is a cognitive process in which reality is compared to an imagined view of what might have been. Previous studies in Linguistics have focused on the description of the conditional construction if P (then) Q. Our study aims to compare the grammatical devices used by thirty French native speakers and thirty Spanish native speakers. Participants were presented with a story that led to a particular outcome and were asked to provide alternative scenarios that might have prevented such an outcome from happening. The results showed (i) that counterfactuality is not expressed most frequently by conditional constructions and (ii) that the use of evaluative modal markers within the mutation core is a salient difference between French and Spanish. The minor role of conditional constructions suggests that the boundaries of conditionality as a conceptualization process need to be reconsidered within the realm of irreality. In addition, the frequency of evaluative modals in French suggests a preference for counterfactual scenarios that integrate an overt critical judgement from the speaker.

Introduction

Counterfactual thinking is the cognitive process in which reality is compared to an imagined view of what might have been [START_REF] Kahneman | The simulation heuristic[END_REF]. Previous studies dealing with the description of counterfactuality have focused on the analysis of conditional sentences [START_REF] Bates | Language and Context: The Acquisition of Pragmatics[END_REF][START_REF] Bloom | Th e Linguistic Shaping of Th ought: A Study in the Impact of Language on Th inking in China and the West[END_REF][START_REF] Reilly | The Acquisition of Conditionals in English[END_REF][START_REF] Au | Chinese and English counterfactuals: the Sapir-Whorf hypothesis revisited[END_REF][START_REF] Bernini | Le frasi ipotetiche nell'italiano di stranieri[END_REF][START_REF] Chini | Meno male che non elo un topo, se no mi mangia un gatto, pe fi nta. Precursori e genesi dei costruitti ipotetici in una bambina italofona[END_REF][START_REF] Harris | Children's use of counterfactual thinking in causal reasoning[END_REF][START_REF] Katis | Th e emergence of conditionals in child language: are they really so late?[END_REF][START_REF] Schouten | The Role of the Native Language in the Non-Native Acquisition of Hypothetical Conditional Structures[END_REF][START_REF] Yeh | Reasoning counterfactually in Chinese: picking up the pieces[END_REF][START_REF] Grevisse | Le bon usage (Paris-Gembloux: Duculot)[END_REF][START_REF] Renzi | Grande grammatica italiana di consultazione[END_REF][START_REF] Campos | De la oración simple a la oración compuesta: curso superior de gramática española[END_REF][START_REF] Alarcos-Llorach | Gramática de la lengua española[END_REF]. However, evidence of the marking of counterfactuality in ways other than using conditional sentences has been found in simple clauses in natural languages [START_REF] Hellberg | Le Système hypothétique dans le français écrit et parlé[END_REF][START_REF] Van Linden | Th e nature and origins of counterfactuality in simple clauses: cross-linguistic evidence[END_REF]. This article provides original insights into the expression of counterfactuality in French and Spanish. Eventual differences in the grammatical devices preferred in French and Spanish might underlie different semantic patterns in the conceptualization of counterfactual states of affairs. The ultimate objective of the present study is to analyse whether the grammatical categories by which the native speakers talk about irreality are salient in the same way in French and Spanish. This article is organized in seven sections. In the next Section, we provide a formal description of conditional constructions and we present the conceptual foundations of counterfactuality. In Section 3, we offer a typological description of counterfactual conditional clauses in French and Spanish. Subsequently, we introduce the materials and instructions used to elicit counterfactual responses. Statistical results are presented in Section 5. The main findings are discussed and related to previous studies, and the conclusions summarized in the final sections.

State of the art

The notion of counterfactuality has typically been linked to a particular linguistic construction, the conditional construction If P (then) Q:

(1) If kangaroos had no tails, they would topple over. This kind of construction generally consists of two clauses, called the protasis and apodosis, respectively. The protasis is a subordinate clause introduced by a conjunction (if) and characterized by a particular tense form: in the English example [START_REF]SBJ: bon déjà il aurait pu choisir les coquilles Saint-Jacques. He could have chosen the Coquilles Saint-Jacques[END_REF], a form that looks like a simple past, but is considered to be a past subjunctive. The apodosis is a main clause typically characterized by a particular verb form, again in [START_REF]SBJ: bon déjà il aurait pu choisir les coquilles Saint-Jacques. He could have chosen the Coquilles Saint-Jacques[END_REF], the combination of would + infinitive (the present conditional). In fact, it is this construction in English, and its counterparts in other languages, that are generally described in traditional grammar books as pertaining to counterfactuality (Section 3) and most of the work on the description of counterfactuality deals with this type of construction (Sections 2.2 and 2.3). However, this is not the only way in which speakers can talk about worlds that are explicitly contrasted to what is really the case. In this study, our goal is to see how speakers proceed when they are attempting to speak about counterfactual worlds in French and Spanish. Do they employ conditional constructions, or do they use different strategies?

Conceptual foundations

It is not our aim to present a complete theoretical picture of the contributions of philosophy, psychology and linguistics to counterfactuality but rather to introduce three frameworks from which counterfactuality has been studied. We consider counterfactual thinking to be a cognitive process in which reality is compared to an imagined view of what might have been [START_REF] Kahneman | The simulation heuristic[END_REF]. This process is based on the mental construction of alternatives to factual states of affairs [START_REF] Wells | Mental simulation of causality[END_REF]. Counterfactuality has been described as a semantic domain related to three main topics:

 causality, because of the causal relation that may hold between the main clause (Q) and the subordinate clause (P) within conditional constructions (Hume 1739;Lewis 1973;[START_REF] Mackie | Causes and conditions[END_REF];

 epistemic modality, because of the speaker's degree of uncertainty involving the content of their statement [START_REF] Lyons | Semantics[END_REF][START_REF] Palmer | Mood and Modality[END_REF][START_REF] Fauconnier | Espaces mentaux[END_REF][START_REF] Dancygier | Conditionals, distancing, and alternative spaces[END_REF];

 irreality, because of the definition of counterfactuality as one semantic subdomain within irreality, alongside the non-exclusion-of-factuality and non-referentiality [START_REF] Pietrandrea | The conceptual structure of irreality. A focus on nonexclusion-of-factuality as a conceptual and linguistic category[END_REF].

Along a semantic scale, counterfactual conditionals express a low degree of probability and a high degree of hypotheticality [START_REF] Comrie | Conditionals: a typology[END_REF]. In counterfactual conditional constructions, the relationship between P and Q holds in the past. This is because counterfactuality is deeply related to the speaker's state-of-knowledge and in particular to their certainty about the non-actualization of a given process.

The past is a better known domain compared to the present and the future and thus counterfactuality represents a well delimited frame from which people can conceive other possible worlds (W₁, W₂, W₃) which differ from the world as it is (W₀). This means that any antecedent (P) introduced by the speaker is false at the time when something is said (the time of utterance). Consider the example below:

(2) If Karen hadn't eaten seafood, she would still be alive. This is regularly understood to mean that the speaker asserts that Karen had in fact eaten seafood, i.e. the sentence Karen has not eaten seafood is false. The certainty of the speaker about the nonactualization of the process is a major feature that differentiates counterfactuality from potentiality [START_REF] Verstraete | The semantics and pragmatics of composite mood marking: the non-Pama-Nyungan languages of northern Australia[END_REF]. Beyond conditional constructions, the combination of a past tense and a modal particle has been described as a frequent marker of counterfactuality (Van Linden and Verstraete 2008). For example:

(3) The police should have done something to prevent the killing.

These authors posit that counterfactual statements are semantically double-layered, since they combine potentiality, i.e. the speaker presents a given state of affairs as possible, desirable, imminent or intended, and pastness, i.e. the speaker is aware of the non-actualization of what is being said. In English, the interpretation of a counterfactual statement comes from the compositional semantics of the modal element should, which frames it in a virtual locus, and the past marker have done, which marks a non-actualized state of affairs.

Rationale

A detailed reading of the literature on counterfactuality leads us to affirm that (i) in general this concept has failed to be analysed beyond conditional constructions and that (ii) a terminological problem remains in respect to its meaning. A major contribution to the linguistic properties of counterfactuality was made by the publication of the collective volumes On Conditionals and On Conditionals Again [START_REF] Ferguson | Overview[END_REF][START_REF] Athanasiadou | Conditionality, hypotheticality, counterfactuality[END_REF], in which conditionality was introduced as a semantic category covering both hypotheticality and counterfactuality. These volumes, together with earlier studies [START_REF] Bates | Language and Context: The Acquisition of Pragmatics[END_REF][START_REF] Bloom | Th e Linguistic Shaping of Th ought: A Study in the Impact of Language on Th inking in China and the West[END_REF][START_REF] Reilly | The Acquisition of Conditionals in English[END_REF][START_REF] Au | Chinese and English counterfactuals: the Sapir-Whorf hypothesis revisited[END_REF][START_REF] Liu | Reasoning counterfactually in Chinese: are there any obstacles?[END_REF] and later studies [START_REF] Bernini | Le frasi ipotetiche nell'italiano di stranieri[END_REF][START_REF] Chini | Meno male che non elo un topo, se no mi mangia un gatto, pe fi nta. Precursori e genesi dei costruitti ipotetici in una bambina italofona[END_REF][START_REF] Schouten | The Role of the Native Language in the Non-Native Acquisition of Hypothetical Conditional Structures[END_REF][START_REF] Yeh | Reasoning counterfactually in Chinese: picking up the pieces[END_REF], contributed to placing counterfactuality under the umbrella of conditional constructions. To date, terminological incoherence subsists across grammatical traditions and some authors claim that counterfactuality strictly covers the semantic domain of irreality [START_REF] Renzi | Grande grammatica italiana di consultazione[END_REF][START_REF] Riegel | Grammaire méthodique du français[END_REF]Real Academia Española 2009), whereas other authors consider it as a broader notion covering both irrealis and imagined conditions (Grevisse and Goosse 2008). Within the realm of irreality, it is generally assumed that conditional constructions can express values related to a past irrealis or a present irrealis [START_REF] Harris | Th e historical development of si-clauses in romance[END_REF]Riegel et al. 2001;Grevisse and Goosse 2008).1 We adhere to the theoretical framework that places counterfactuality in the semantic subdomain of irreality, alongside subdomains of non-exclusion-of-factuality and non-referentiality [START_REF] Pietrandrea | The conceptual structure of irreality. A focus on nonexclusion-of-factuality as a conceptual and linguistic category[END_REF].

We consider counterfactuality to be a well delimited field that, unlike potentiality, implies the certainty of the speaker about the non-actualization of the process (Verstraete 2005).

Cross-linguistic description of counterfactual conditionals

The typological distance between French and Spanish in counterfactual conditionals resides in the protasis: French requires the indicative mood whereas Spanish requires the subjunct ive mood.

Concerning the apodosis or main clause, the two languages converge in the canonical use of the perfect conditional.

Counterfactual conditionals in French

In spoken French the canonical mood in the protasis is the pluperfect indicative (cf. Table 1).2 This means that the indicative mood in French is associated with two different readings: factual and hypothetical, although the presence of the morpheme si (if) usually disambiguates this dichotomy. More problematic is the disambiguation of the pluperfect indicative in the protasis, which can encode either potential or irrealis semantic values. In this case, it is generally assumed that the disambiguation depends on the context [START_REF] Riegel | Grammaire méthodique du français[END_REF][START_REF] Vetters | À propos de l'imparfait après "si[END_REF][START_REF] Patard | L'un et le multiple. L'imparfait de l'indicatif en francais: valeur en langue et usages en discours[END_REF]; LeBlanc 2009).3 Double conditional marking in both the protasis and the apodosis ( 4) is considered a violation of the French standard. For example:

(4) Si vous m'auriez ennuyé, je vous l'aurais dit. [START_REF] Grevisse | Le bon usage (Paris-Gembloux: Duculot)[END_REF](Grevisse : 1686) ) If you 'would have annoyed' me, I would have told you.

Example ( 4) is generally attributed to familiar uses [START_REF] Grevisse | Le bon usage (Paris-Gembloux: Duculot)[END_REF] and to verbal symmetry between the protasis and the apodosis in some hypothetical and concessive constructions [START_REF] Riegel | Grammaire méthodique du français[END_REF]).4

Nevertheless, when the protasis is introduced by the constructions au cas où or dans le cas où (in case) the use of verbal forms of the conditional or the subjunctive is considered canonical, instead [START_REF] Grevisse | Le bon usage (Paris-Gembloux: Duculot)[END_REF]). Among the less frequently described grammatical markers in French, we find the modal verbs pouvoir and devoir appearing in the past participle verbal forms of the perfect conditional:

(5) Jean aurait pu être quelqu'un d'autre. (Fauconnier 1984: 144) Jean could have been someone else.

Table 1. Counterfactual canonical conditionals in French and Spanish

If P (then) Q Example FR Indica- tive Condi- tional
Si tu avais admis cette opinion, tu aurais eu tort [START_REF] Grevisse | Le bon usage (Paris-Gembloux: Duculot)[END_REF] If you had agreed with that opinion you would have been wrong

SP Subjunc -tive
Si te hubieses quedado, habrías visto algo bueno (Alarcos-Llorach 1999)

If you had remained you would have seen something good

Counterfactual conditionals in Spanish

Disambiguation between factual and counterfactual conditionals in Spanish is achieved by using the indicative mood or the subjunctive mood in the protasis (Alarcos-Llorach 1999). The main clause or apodosis is generally in the conditional, whereas the perfect subjunctive in the protasis ( 6) frames the content of the proposition in the past. ( 6) Si te hubieses quedado, habrías visto algo bueno. (Alarcos-Llorach 1999: 474) If you had stayed, you would have seen something good.

In practice, a wide range of verbal combinations are observed in Spanish counterfactual conditionals, including symmetrical verb forms in the conditional tense in both the protasis and the apodosis [START_REF]SBJ: elle aurait pu ne pas être allergique aussi. She could have not been allergic for instance[END_REF] or in the subjunctive mood (7). For example:

(7) Si hubiera tenido dinero, hubiera ido a España. (Campos 1993: 163) If I had money, I would have gone to Spain.

(8) Si habría tenido dinero, habría ido a España. (Campos 1993: 163) If I had money, I would have gone to Spain.

The use of (7) has been described as a frequent structure in the Mexican variety of Spanish [START_REF] Wald | On the evolution of "would" and other modals in the English spoken in East Los Angeles[END_REF] while the use of ( 8) has been observed in general in some dialectal varieties of Spanish (Alarcos-Llorach 1999) and in particular in Buenos Aires [START_REF] Lavandera | Linguistic structure in sociolinguistic analysis[END_REF]) and in the variety of Spanish spoken in the Basque Country (Silva-Corvalán 1982).

Method

The study is based on a task that requires participants to discuss alternative scenarios based on a story describing a chain of related events. We henceforth refer to this task as the 'mutation task'. Note that previous studies dealing with conditional constructions have elicited their data from spontaneous speech [START_REF] Bowerman | First steps in acquiring conditionals[END_REF][START_REF] Chini | Meno male che non elo un topo, se no mi mangia un gatto, pe fi nta. Precursori e genesi dei costruitti ipotetici in una bambina italofona[END_REF][START_REF] Katis | Th e emergence of conditionals in child language: are they really so late?[END_REF][START_REF] Schouten | The Role of the Native Language in the Non-Native Acquisition of Hypothetical Conditional Structures[END_REF], what if-questions [START_REF] Bates | Language and Context: The Acquisition of Pragmatics[END_REF][START_REF] Reilly | The Acquisition of Conditionals in English[END_REF][START_REF] Harris | Children's use of counterfactual thinking in causal reasoning[END_REF] or written questionnaires [START_REF] Bloom | Th e Linguistic Shaping of Th ought: A Study in the Impact of Language on Th inking in China and the West[END_REF][START_REF] Au | Chinese and English counterfactuals: the Sapir-Whorf hypothesis revisited[END_REF][START_REF] Liu | Reasoning counterfactually in Chinese: are there any obstacles?[END_REF][START_REF] Schouten | The Role of the Native Language in the Non-Native Acquisition of Hypothetical Conditional Structures[END_REF]. The mutation task has been previously used by psychological studies as a way to obtain critical judgements related to causality [START_REF] Wells | Mental simulation of causality[END_REF][START_REF] Mandel | Counterfactual thinking and ascriptions of cause and preventability[END_REF]. The data we discuss come from two groups of adults: thirty French native speakers and thirty Spanish native speakers; thirty-five women and twenty-five men, the majority of them recruited at the Aix-Marseille University (cf. Appendix). Guided interviews were recorded and stored as WAV files, then transcribed using the CLAN editor for CHILDES. Data collection was done mainly in the Aix-Marseille area but also in Nijmegen (the Netherlands). The stimulus was the following 200-word text, which participants had to read.

Karen was an assistant editor for a small publishing firm. She had a rare hereditary disease called Karpinson's hemotrysoma, characterized by the lack of an enzyme that normally breaks down certain proteins in the digestive system. Because of this, fermented drinks such as wine and liqueurs can cause a severe allergic reaction in someone with the disease.

Karen had just received a promotion so her boss, Mr Carlson, took her to an expensive French restaurant to celebrate. Mr Carlson had been to this restaurant several times, so he ordered for both of them. As he looked over the menu, Mr Carlson considered what to order for Karen. He first thought about ordering the Coquilles Saint-Jacques, but at the last moment decided on the Moules Mariniere instead. Although Mr Carlson did not know this, the Moules Mariniere was made in a wine sauce whereas the Coquilles Saint-Jacques did not contain any wine.

Karen enjoyed her meal greatly, but began to feel ill shortly after finishing. Within minutes, she went into convulsions and was rushed away in an ambulance. She died on the way to the hospital.

The original text was in English (Wells and Gavanski 1986). For the experiment, it was translated in French and Spanish and then edited by two native reviewers in each language (cf. Appendix). After reading the text, the participants were asked to comply with the following instruction:

Imagine three modifications that could have been different in the story to avoid Karen's death.

We assume that every text answers an implicit question [START_REF] Klein | How to solve a complex verbal task: text structure, referential movement and the quaestio[END_REF]. 5 The identification of the mutation cores, i.e. the segments of the information responsible for a modification, was possible after a qualitative analysis that allowed us to distinguish foreground structures and background structures. Consider the following example:

9. *SBJ: et du coup elle aurait pu les manger. And thus, she could have eaten the Moules.

10. *SBJ: sans être malade et sans mourir.

Without getting sick and without dying.

In the example above, lines 1, 5 and 8 correspond to the text's foreground structure containing the core information that really answers the quaestio, whereas lines 3, 4, 9 and 10 can be considered background structures. The mutation cores were classified according to their formal composition in an Excel file. Statistical analyses were run using the software R.

Results

Among the different elements encoding counterfactuality we can distinguish (i) syntactic elements, e.g. conditional constructions, (ii) lexical elements, e.g. modal verbs, and (iii) inflectional elements denoting a particular verbal tense and mood. (16) Que ella hubiera informado en su trabajo que sufría esa enfermedad. She might have informed people at her job that she had that disease.

Table 2 summarizes the frequency in the use of the four constructions listed above in French. Statistical analyses using the chi-squared test revealed a significant difference between the frequency in the use of Type B and Type C (p-value = 0.0004). This suggests an overt preference to produce mutation cores combining a conditional tense with a modal marker over coordinate clauses in verbal forms of the indicative. No significant differences were found between Type B, i.e. the combination of a conditional tense and a modal marker, and Type A and Type D, i.e. conditional constructions and simple clauses in the subjunctive, respectively. The lack of significant differences suggests that these three grammatical devices play a prominent role in the expression of counterfactuality in French. Table 3 summarizes the frequency in the use of the four constructions in Spanish. The frequency in the use of Type D was found to be significant compared to any of the other three constructions. In other words, simple clauses introduced by que in a subjunctive verbal form are the preferred way to conceptualize counterfactual scenarios in Spanish. The use of conditional constructions (Type A) in the mutation task in Spanish was significantly lower compared to clauses in the subjunctive mood (p-value = 8.566e-15).

Discussion

Conditional constructions are not the most common way of referring to counterfactual worlds, neither for the French native speakers nor for the Spanish native speakers. The attention paid to conditional constructions in previous studies devoted to hypotheticality and counterfactuality has been explained because of its iconicity within the cognitive process of thinking about alternative situations [START_REF] Ferguson | Overview[END_REF]. Our study provides empirical evidence that conditionality is one possible way among others in which one can refer to counterfactual worlds. However, counterfactuality is frequently expressed using grammatical devices other than conditional constructions. In French, evidence has been given about the prominent role of clauses combining the conditional tense and a modal marker (Type B), simple clauses in the subjunctive mood (Type D) and conditional constructions (Type A). In Spanish, we have shown that conditional constructions are rare compared to clauses in the subjunctive introduced by que.

Our results support the conclusion by [START_REF] Hellberg | Le Système hypothétique dans le français écrit et parlé[END_REF] about the predominance of simple sentences with a conditional tense as the most frequent device to express hypotheticality in French. Our study suggests that the role assigned to conditional constructions as a prominent category to express hypotheticality and counterfactuality [START_REF] Grevisse | Le bon usage (Paris-Gembloux: Duculot)[END_REF][START_REF] Renzi | Grande grammatica italiana di consultazione[END_REF][START_REF] Campos | De la oración simple a la oración compuesta: curso superior de gramática española[END_REF][START_REF] Alarcos-Llorach | Gramática de la lengua española[END_REF][START_REF] Ferguson | Overview[END_REF][START_REF] Athanasiadou | Conditionality, hypotheticality, counterfactuality[END_REF] is not empirically based on the observation of natural languages. Previous studies have portrayed conditionality as a way 'to reason about situations, to make inferences based on incomplete information, to imagine possible correlations between situations and to understand how the world would change if certain correlations were different' (Ferguson et al. 1986: 19). The interest paid to conditionality as a resource for conceptualizing irrealis or imagined scenarios has neglected the description of other ways to express these semantic domains and thus has provided an incomplete description of the conceptualization of counterfactuality. The nonprominent role of the conditional constructions to speak about what might have been but was not in Spanish and French suggests that the status of conditionality as a conceptualization process needs to be reconsidered. This idea finds support in the balanced distribution of the conditional constructions, the subjunctive mood and the combination of the conditional tense and modal markers across the mutation cores in French. Each group's preferences in respect to marking counterfactuality have implications in terms of typological distance. The compositional semantics of counterfactuality, i.e. the features of virtuality and non-actualization, was satisfied by the use of the subjunctive in Spanish and by the combination of a past marker and a modal marker in French. The presence of a modal marker in the mutation core was the most salient difference when comparing these Romance languages. The frequent use of evaluative particles in the mutation cores in French was implemented by means of the modal verbs pouvoir and devoir, i.e. pu and dû (could and should in English), in the past participle form within the past conditional structure. This is coherent with the description of the most frequent counterfactual marker in natural languages, which is not one single dedicated marker but rather the combination of a past tense and a modal marker (Van Linden and Verstraete 2008). Modalizers are well known for their wide range of forms, meanings and uses, and by the absence of univocal relationships between these three dimensions [START_REF] Roulet | Des formes et des emplois des modalisateurs de proposition dans l'interaction verbale[END_REF]. However, this is not the case for the modal markers elicited in the French mutation cores. As we explained supra, the French modal verbs pouvoir and devoir were systematically used in the past participle form in a perfect conditional framework. For instance, no pu / dû were found in the passé compose (e.g., Elle a pu choisir toute seule son plat). To us, this fact reveals the univocal relationship between the 'modalized conditional' and its use for altering a factual past scenario in French. A further question concerns the type of modality expressed by this form. When saying Elle aurait pu choisir toute seule son plat ('She could have chosen her own dish'), the speaker is not expressing any epistemic or deontic modal value, since she is not attaching beliefs or nuances of obligation. We posit that the modal values expressed when using the modalized conditional for mutation purposes are evaluative. By saying Elle aurait pu choisir toute seule son plat ('She could have chosen her own dish'), the speaker attaches the pertinence of a specific action within a set of particular conditions. This coincides with the notion of evaluative modality (Palmer 1986: 119), which expresses the speaker's attitude over known factual events. This is why the expression of regret is often linked to evaluative modals (Palmer 1986: 115). The frequency in the use of this 'modalized conditional' in the mutation task is coherent with the French pattern of retelling a linear story from a narrator's point of view by the use of predicates relating to intentions, attitudes and perceptions [START_REF] Carroll | Subordination in narratives and macrostructural planning: a comparative point of view[END_REF]). These authors have shown that the predicates of French speakers generally refer to the mental states of the protagonist in the story, e.g. il est perplexe, il s'apercoit, il s'est dit que… (he's perplexed, he realizes, he told himself that…). The use of the 'modalized conditional' in counterfactual contexts sounds concordant with the subjective pattern in narrative tasks. There are various reasons that might explain the frequent use of the modalized conditional by native speakers of French. First, the modalized conditional results from the combination of two processes: the grammatical activation of a specific past tense, i.e. the past conditional, and the lexical activation of a modal verb in the past participle form. When these two processes converge, the listener may have access to the speaker's subjectivity in a more direct way than through other structural devices such as conditional sentences or even the indicative mood, which generally do not accommodate modal values in the mutation core. These pragmatic implications may be the first factor in explaining the high frequency of this structure in French. The second explanation concerns the economy factor. Unlike conditional constructions, the modalized conditional is generally not subordinated to a main clause. Moreover, in just three syllables (aurait pu / dû) the speaker is opening the door to another possible world by using the same verb tense as they might have used in a main conditional clause while avoiding the production of an antecedent (P). It could be argued that the prominent role of the 'modalized conditional' in French may be explained by the fact that the ta sk instruction itself contains the 'modalized conditional'. If that was the case and the input structures of the mutation task played a real effect on the data elicited, then one might expect the same effect in the Spanish corpus. However, in Spanish, the 'modalized conditional' of the input is not prominently reproduced by the Spanish native speakers. Additional data elicited in Italian L1 [START_REF] Repiso | Parlons de l'irréel. L'expression de la contrefactualité en français, en espagnol et en italien et par des apprenants hispanophones et italophones de français[END_REF]) confirmed no correlation between the 'modalized conditional' of the input and the responses of the native speakers of Italian. The prominent use of the subjunctive in Spanish in the mutation task might be explained because of the frequency in general of this type of structure in Spanish. In fact, the subjunctive in Spanish is semantically associated with the irrealis domain (Real Academia Española 2009: 1866). Among the uses of the irrealis subjunctive, we find the expression of threat related to the future, e.g. Que se quejen ('Let them complain') or Que vengan mañana ('We'll see if they come tomorrow'). We maintain that, unlike the examples mentioned, the occurrences elicited in the mutation task, e.g. Que ella hubiera informado en su trabajo que sufría esa enfermedad ('She might have informed people at her job that she had that disease'), are semantically anchored to the unreal past frame [START_REF] Harris | Th e historical development of si-clauses in romance[END_REF]).

In Spanish, different grammatical devices may allow a counterfactual reading: the past perfect subjunctive presented here, but also the combination of the Spanish imperfect indicative and the modal markers poder 'could' or tener que 'have to ' (17) or the combination of the simple past tense and the modal deber 'should' (18). For example:

(17) Ella tenía que haber informado que sufría esa enfermedad. She should have informed that she had that disease.

(18) El señor Carlson debió preguntar qué prefería comer ella. Mr Carlson should have asked what she preferred to eat.

We interpret the use of the modal markers in ( 17) and ( 18) as evaluatives, since the speakers use them to express a critical judgement within a set of particular conditions (i.e. those established in the stimulus). Even if the counterfactual meaning may well be encoded by the Spanish imperfect indicative, our data showed that the subjunctive mood is significantly more frequently used by Spanish native speakers. In Spanish L1, we elicited 5 responses over 90 (5.5%) combining an indicative tense plus a modal marker as in examples ( 17) and (18). Four responses followed the pattern of example ( 18) and only one response followed the pattern of example (17). The possibility of a link between certain varieties of Spanish and the use of the simple past tense and a modal marker to encode counterfactuality cannot be ruled out, but has not been statistically tested. However, it is interesting to note that three out of the four responses coherent with example (18) were produced by Colombian native speakers. Further studies devoted to the expression of counterfactuality are still necessary to explore possible differences across the American and peninsular varieties of Spanish.

Conclusion

Our objective has been to examine the grammatical devices encoding counterfactuality in French and Spanish. Previous studies dealing with the expression of counterfactuality have approached this category by analysing conditional constructions [START_REF] Bates | Language and Context: The Acquisition of Pragmatics[END_REF][START_REF] Bloom | Th e Linguistic Shaping of Th ought: A Study in the Impact of Language on Th inking in China and the West[END_REF][START_REF] Reilly | The Acquisition of Conditionals in English[END_REF][START_REF] Au | Chinese and English counterfactuals: the Sapir-Whorf hypothesis revisited[END_REF][START_REF] Bowerman | First steps in acquiring conditionals[END_REF][START_REF] Bernini | Le frasi ipotetiche nell'italiano di stranieri[END_REF][START_REF] Chini | Meno male che non elo un topo, se no mi mangia un gatto, pe fi nta. Precursori e genesi dei costruitti ipotetici in una bambina italofona[END_REF][START_REF] Katis | Th e emergence of conditionals in child language: are they really so late?[END_REF][START_REF] Schouten | The Role of the Native Language in the Non-Native Acquisition of Hypothetical Conditional Structures[END_REF][START_REF] Yeh | Reasoning counterfactually in Chinese: picking up the pieces[END_REF]. Our main contribution to these studies has been to show that the mutation cores encoding counterfactual scenarios were rarely marked by bare conditional sentences in Spanish. In French, the most frequent device was the combination of the past conditional and a modal verb (e.g. Il aurait pu choisir une autre assiette, 'He could have chosen another dish'). Significant differences were found between this 'modalized conditional' and coordinate clauses in symmetrical verbal forms, suggesting that either present or past linear narrations are rare when conceptualizing what might have been but was not. In Spanish, simple clauses in the subjunctive mood were found to demonstrate significant differences compared to any of the other mutation cores, including conditional constructions. The non-prominent role of the conditional constructions suggests that the role of conditionality as a conceptualization process within irreality needs to be reconsidered. The major difference when comparing the two languages lay in the presence of a modal marker within the mutation core, significantly higher in French. The use of evaluative modals in the mutation task confirms that French tend to integrate subjective values in narrative texts, in argumentative contexts.

Notes

1. The existence of counterfactuals related to the future is controversial. Some authors claim that counterfactuality cannot be related to the future, since the future has not been realized nor is known at the time of utterance and thus cannot be considered against any factual state of affairs [START_REF] Iatridou | Th e grammatical ingredients of counterfactuality[END_REF].

2. Historically, the subjunctive mood in the protasis comes from Latin but it has been used less and less in French since the thirteenth century [START_REF] Brunot | Précis de grammaire historique de la langue française[END_REF]. Th e canonical system -composed of the pluperfect indicative in the protasis and the conditional perfect in the apodosis -appeared in the sixteenth century.

3. The spoken Ottawa-Hull variety of French resolves this ambiguity by using the conditional form in the protasis as a mark of potentiality and the pluperfect indicative as a mark of counterfactuality (LeBlanc 2009).

4. Example: J'aurais un peu d'argent, je m'achèterais l'intégrale de Mozart ('Had I a little money, I would buy the complete works of Mozart', Riegel et al. 1994: 558).

Karen. Il pensa d'abord aux coquilles Saint-Jacques mais au dernier moment il choisit les moules marinières. Monsieur Carlson ne le savait pas mais les moules marinières étaient cuites dans une sauce à base de vin tandis que les coquilles Saint-Jacques ne contenaient pas de vin. Karen savoura son plat mais elle commença à se sentir mal peu après avoir fi ni. Quelques minutes plus tard elle souffrit de convulsions et elle fut transportée en ambulance. En chemin pour l'hôpital elle mourut.

Proposez trois modifications qui auraient pu empêcher la mort de Karen et expliquez pourquoi elles auraient empêché sa mort.

*** Karen era asistente a la edición en una pequeña empresa de publicidad. Padecía una rara enfermedad hereditaria llamada hemotrisoma de Karpinson, caracterizada por la falta de una enzima que normalmente divide algunas proteínas en el aparato digestivo. Por ello las bebidas fermentadas como el vino o el licor podían causarle graves reacciones alérgicas. Karen había sido ascendida de manera que su superior, el señor Carlson, la llevo a un caro restaurante francés para celebrarlo. El señor Carlson había comido en dicho restaurante varias veces, así que pidió por ambos. Mientras miraba el menú se preguntó que pedir para Karen. Primero pensó en pedir unas vieiras pero en el último momento se decidió por los mejillones a la marinera. Aunque el señor Carlson no lo sabía, los mejillones a la marinera llevaban una salsa a base de vino mientras las vieiras, no. Karen disfrutó de la comida pero empezó a sentirse mal poco después. En cuestión de minutos sufrió convulsiones y fue socorrida en una ambulancia. Karen murió de camino al hospital.

Imagina tres modificaciones de la historia que habrían podido evitar la muerte de Karen y explica por qué habrían evitado su muerte.

  [START_REF]SBJ: au lieu de laisser choisir par lui. Instead of allowing him to choose it[END_REF] We elicited ninety responses per group (three responses per participant). The alternative scenarios to factual past events were encoded by mutation cores produced within four types of clauses in French and Spanish: . Coordinate clauses in verbal forms of the indicative mood, either the present tense (13) or past tenses (14). This type of structure recalls that of a chronological narration because of the recurrent use of the additive conjunctions et in French and y in Spanish (and). Example:(13) Karen est invitée par son patron donc au restaurant mais elle l'avertit qu'elle a une allergie et elle lui demande de choisir elle-même son plat. Karen is invited by her boss to the restaurant but she tells him she's allergic and she asks him if she can choose her dish herself.(14) Karen comió las vieiras, se empezó a sentir mal y en el propio restaurante había un médico […] y entonces finalmente sí salvó su vida. Karen ate the scallops, she felt ill but a doctor was in the restaurant […] and eventually she survived. D. Simple clauses introduced by que -in verbal forms of the subjunctive. Example:

	(15) Qu'il commande autre chose que les moules.
	Let him order something other than the Moules.

A. Conditional constructions. Example:

(9) Si Karen n'avait pas eu de promotion, elle (ne) serait pas morte. If Karen had not got the promotion, she would not have died.

(10) Si no la hubieran ascendido, no la habrían invitado a cenar. If she had not been promoted, she wouldn't have been invited to dinner. B. Clauses combining a conditional tense and a modal marker. Example: (11) Elle aurait pu choisir toute seule son plat. She could have chosen her own dish. (12) Carlson podría haber pedido las vieiras. Carlson could have ordered the scallops. C

Table 2 .

 2 Mutation cores in French (n=90)

	Type A	Type B	Type C	Type D
	19	32	11	19

Table 3 .

 3 Mutation cores in Spanish (n=90)

	Type A	Type B	Type C	Type D
	6	3	7	58

Table 4

 4 compares the ways in which the mutation cores are encoded in French and Spanish. We established a decision threshold of 0.001 to minimize Type I errors.7 The results of the comparison of French and Spanish underlines two salient differences concerning the distribution of the grammatical devices encoding counterfactuality. Spanish speakers preferred the subjunctive mood introduced by que, whereas French speakers favoured the combination of a modal marker and the conditional tense.

	Table 4. Conditional constructions v. other verbal predicates (n=90)
	P-values resulting from the X² analysis
		Type A	Type B	Type C	Type D
	French	19	32	11	19
	Spanish	6	3	7	58
		0.009	1.34e-04*	0.456	1.036e-08*

Acknowledgements

We are grateful to the anonymous reviewers, whose questions contributed to improving our article, and to Professor Yann Kerdilès for assisting us in the final edition of this article.


Jean-Baptiste Bellet

AN EXACT RADON FORMULA FOR LAMBERTIAN TOMOGRAPHY

Keywords: 2020 Mathematics Subject Classification. 44A12, 78A46 Radon transform, tomography, Lambertian diffusion

This letter, based on an extension of the Radon transform on distributions, is a mathematical contribution to the field of reflective tomography in optics. Indeed, we tackle the reconstruction of a Lambertian convex reflector using tomography. In a bi-dimensional setup, we prove that the Lambert's cosine law can be inverted by some Radon formula involving distributions. The associated reconstruction contains the geometry and the physics of the problem: it is a Radon measure supported by the reflector, and its density is the inverse of the albedo.

Introduction

Over the past decades, there has been a considerable interest in developing new imaging modalities based on scattering of light and extensions of the Radon transform, such as Compton scattering tomography [START_REF] Nguyen | Inversion of a new circular-arc Radon transform for Compton scattering tomography[END_REF][START_REF] Rigaud | 3D Compton scattering imaging and contour reconstruction for a class of Radon transforms[END_REF], Bragg scattering tomography [START_REF] Webber | Microlocal Analysis of Generalized Radon Transforms from Scattering Tomography[END_REF], or reflective tomography [START_REF] Berginc | Optical 3D imaging and visualization of concealed objects[END_REF][START_REF] Berginc | Simulation of 3D laser systems[END_REF][START_REF] Gering | Object Modeling using Tomography and Photography[END_REF][START_REF] Johnson | Three-dimensional surface reconstruction of optical Lambertian objects using cone-beam tomography[END_REF][START_REF] Knight | Tomographic Techniques Applied to Laser Radar Reflective Measurements[END_REF][START_REF] Rigaud | Reflective Imaging Solved by the Radon Transform[END_REF]. This paper deals with reflective tomography in visible to near-infrared optics. In this field, one reconstructs a scene from optical images using a Radon inversion. The geometry of such a reconstruction is meaningful [START_REF] Bellet | Heuristic imaging from generic projections: backprojection outside the range of the Radon transform[END_REF], because the Radon transform is a Fourier Integral Operator [START_REF] Krishnan | Microlocal Analysis in Tomography[END_REF]. Nevertheless, to the author's knowledge, reflective tomography is not justified by an exact inversion formula. In this letter, we tackle this problem in the case of uniform diffuse reflection off a strictly convex surface, as in Figure 1. We derive a Radon formula which rigorously inverts a Lambert's cosine law. Here, an isotropic point source z emits light with a power per unit solid angle Φ/(4π). On the surface ∂D, an illuminated point y reflects light uniformly; for an angle of incidence α, the point y reflects a radiance Φ/(4π 2 |z -y| 2 ) ρ(y) cos α in any direction above the tangent plane. The dimensionless coefficient ρ(y) ∈ [0, 1] is the albedo, defined as the percentage of incident irradiance which is reflected.

To begin with, let us describe uniform diffusion of light from an ideal matte opaque surface, called a Lambertian reflector. This modeling is based on radiometric concepts [START_REF] Wolfe | Introduction to radiometry[END_REF], and is an usual model for reflection off surfaces in optics; see for example [START_REF] Ma | An invitation to 3-D Vision[END_REF]Sec. 3.A], [START_REF] Durou | Reconstruction 3D à partir des ombrages[END_REF]. As in Figure 1, we assume that the surface of a Lambertian reflector, denoted by ∂D, is illuminated by an isotropic point source located at z. We denote by Φ the power of this source (energy by time unit, in [W]). The associated power per unit solid angle is Φ/(4π) (in [W.sr -1 ]). Therefore, an illuminated point y ∈ ∂D, with angle of incidence α, receives an irradiance Φ cos α/(4π|z -y| 2 ) (power per unit area, in [W.m -2 ]). By assumption, this incident irradiance is uniformly reflected off the surface; it is assumed that the point y radiates the same radiance (power per unit projected area, and per unit solid angle, in [W.m -2 .sr -1 ]), in any direction above the tangent plane. This uniform radiance is given by the Lambert's cosine law Φ 4π 2 |z -y| 2 ρ(y) cos α; here, the (dimensionless) albedo ρ(y) ∈ [0, 1] represents the percentage of the incident irradiance which is reflected in any direction. This model of radiance is of particular interest, since a light sensor in a camera measures a quantity which is essentially proportional to the radiance of the visible point y [START_REF] Horn | Calculating the reflectance map[END_REF]. Finally, assuming that z is in far field with |z -y| ≈ R a large constant, we consider that the records are given by ρ(y) cos α, up to a constant factor.

In this letter, we restrict our attention to a Lambert's cosine law ρ(y) cos α, in a bi-dimensional setup, in R 2 . We aim at finding an explicit Radon formula which relates the geometry ∂D ⊂ R 2 , the albedo coefficient ρ : ∂D → R, and values of the "radiance" ρ(y) cos α. In comparison, in usual (X-ray) tomography, a record is an attenuated intensity along a line in a semi-transparent medium. Up to a factor, the Beer-Lambert's law models the record of a line L by exp -

L a(x)d ,
where the function a(x) represents an attenuation coefficient of the medium [START_REF] Natterer | Mathematical methods in image reconstruction[END_REF]; in this case, the link with the Radon transform of a, defined by R[a](L) = L a(x)d , is very clear. For a Lambertian reflector, extracting a Radon transform from the cosine law ρ(y) cos α is not so clear. Our main result, Theorem 1, answers to this question for a strictly convex reflector; it is based on an extension of the Radon transform on distributions, as described in [9, Chap. 1], [15, Chap. 2].

Background and notation

Unless otherwise stipulated, we assume throughout the letter: x • θ = s} represents a line at signed distance s from the origin, orthogonal to θ, and oriented by θ ⊥ := (θ 2 , -θ 1 ). For our purpose, intersecting lines with ∂D is of particular interest; we refer to Lemma 1 in Appendix A for intermediate results related to this problem of convex analysis. In

Main result

Theorem 1. Assume (A1-A2). Let R : E (R 2 ) → E (S 1 × R) denote the Radon transform extended on distributions by duality. Then,

R 1 ρ dµ = 1 L[D, ρ](θ, s) + 1 L[D, ρ](-θ, -s) 1 L[D,ρ](θ,s)>0 , (1)
where, by convention, the right member takes the value 0 if (and only if ) L[D, ρ](θ, s) = 0.

Proof. The proof is a combination of the definitions of a Radon measure [START_REF] Bony | Cours d'analyse: théorie des distributions et analyse de Fourier[END_REF]Chap. 4], a Radon transform on distributions [9, Chap. 1], and the formula (5) of Appendix A for integration on ∂D. Indeed, the function 1 ρ is bounded on ∂D; therefore 1 ρ ∈ L 1 (∂D, dµ), and

1 ρ dµ ∈ E (R 2
) is defined as a compactly supported distribution by

1 ρ dµ, ψ = ∂D ψ(y) ρ(y) dµ(y), ψ ∈ E(R 2 ).
Then the Radon transform of 1 ρ dµ is a distribution in E (S 1 × R), defined by

R 1 ρ dµ , φ = 1 ρ dµ, R * φ , φ ∈ E(S 1 × R),
where R * φ ∈ E(R 2 ) is defined by

R * φ(x) = S 1 φ(θ, x • θ)dθ, x ∈ R 2 . Fix φ ∈ E(S 1 × R). By Fubini's theorem, R 1 ρ dµ , φ = ∂D R * φ(y) ρ(y) dµ(y) = S 1 ∂D φ(θ, y • θ) ρ(y) dµ(y)dθ.
For any θ ∈ S 1 , the relation ( 5), with

f (y) = φ(θ,y•θ) ρ(y) ∈ L ∞ (∂D) ⊂ L 1 (∂D), proves that ∂D φ(θ, y • θ) ρ(y) dµ = S(θ) φ(θ, s) ρ(y(θ, s))ν(y(θ, s)) • θ ⊥ + φ(θ, s) ρ(y(-θ, -s))ν(y(-θ, -s)) • (-θ) ⊥ ds. Therefore, R 1 ρ dµ , φ = S 1 R φ(θ, s) 1 s∈S(θ) L[D, ρ](θ, s) + 1 s∈S(θ) L[D, ρ](-θ, -s) dsdθ.
This computation shows that the compactly supported function

1 s∈S(θ) L[D, ρ](θ, s) + 1 s∈S(θ) L[D, ρ](-θ, -s) , with 1 s∈S(θ) = 1 L[D,ρ](θ,s)>0 , is in L 1 (S 1 × R)
and coincides as a distribution with R 1 ρ dµ ∈ E (S 1 × R). This concludes the proof.

Corollary 1. Assume (A1-A2). Then, the distribution 1 ρ dµ ∈ E (R 2 ) is uniquely determined by the Lambertian projection L[D, ρ]; moreover, it can be reconstructed by the formula

1 ρ dµ = 1 2π R * H s ∂ s 1 L[D,ρ](θ,s)>0 L[D, ρ](θ, s) ; (2)
here R * : D (S 1 × R) → D (R 2 ) denotes the backprojection and H : E (R) → D (R) denotes the Hilbert transform (both extended on distributions).

Proof. We apply the Radon inversion formula extended on distributions [9, Theorem 5.5], with

T = 1 ρ dµ ∈ E (R 2 ), T = 1 4π R * H s ∂ s RT, T ∈ E (R 2 ).
Here, by definition of R * : D (S 1 × R) → D (R 2 ), and H : E (R) → D (R), the formula means

1 ρ dµ, ψ = 1 4π S 1 ×R 1 s∈S(θ) L[D, ρ](θ, s) + 1 s∈S(θ) L[D, ρ](-θ, -s) ∂ s H s [Rψ(θ, s)]dθds, ψ ∈ D(R 2 ),
with the usual Radon transform R, Hilbert transform H s (with respect to s), defined by

Rψ(θ, s) = x•θ=s ψ(x) d ∈ D(S 1 × R), ψ ∈ D(R 2 ), H s g(s) = 1 π p.v. R g(t) s -t dt ∈ E(R), g ∈ D(R). Since ∂ s H s [g(-s)] = [∂ s H s g](-s), the change of variable (θ, s) := (-θ, -s) yields S 1 ×R 1 s∈S(θ) L[D, ρ](-θ, -s) ∂ s H s [Rψ(θ, s)]dθds = S 1 ×R 1 -s∈S(-θ) L[D, ρ](θ, s) ∂ s H s [Rψ(-θ, -s)]dθds.
By the way, we have the symmetry S(θ) = -S(θ) and Rψ(θ, s) = Rψ(-θ, -s), and we obtain

1 ρ dµ, ψ = 2 4π S 1 ×R 1 s∈S(θ) L[D, ρ](θ, s) ∂ s H s [Rψ(θ, s)]dθds = 1 2π R * H s ∂ s 1 s∈S(θ) L[D, ρ](θ, s) , ψ , ψ ∈ D(R 2 ).

Conclusion and discussion

Theorem 1 provides a way to extract a Radon transform from a Lambert's cosine law ρ(y) cos α, for a Lambertian convex reflector (D, ρ) in two dimensions. The formula (1) exhibits an appropriate pre-processing to obtain an element in the range of the transform: the cosine law L[D, ρ] must be inverted, then symmetrized. In this case, the relevant mathematical object to represent the scene is the Radon measure dµ ρ . This object contains simultaneously the geometry and the physics of the problem. Indeed, the support of this Radon measure is exactly the boundary ∂D of the reflector, while the density is directly the inverse of the albedo ρ. As a consequence of the Radon formula (1), the representation dµ ρ of the scene can be deduced from Lambertian projections, by the filtered backprojection (2), well-defined mathematically.

To the author's knowledge, it is the first time that such a mathematical proof is obtained for a problem of Lambertian tomography, with an exact formula. This result potentially opens new perspectives concerning reflective tomography in a more general framework. For instance, studying the following question is of particular interest: given a set of bi-dimensional optical images of a three-dimensional scene with occlusions, is there a Radon measure supported by surfaces, whose Radon transform (or X-ray transform) reproduces some transformation of the optical images? Also, the results of this work may open new practical applications in optics. It is worth investigating practical implementation of (2) and testing on real optical data. We use the parametrization of (ii), once for θ, once for -θ. To finish with, we change the variable in the second integral; s := -s, the domain of integration becomes -S(-θ) = S(θ).

Figure 1 .

 1 Figure 1. Diffuse reflection by the Lambert's cosine law on a strictly convex surface.Here, an isotropic point source z emits light with a power per unit solid angle Φ/(4π). On the surface ∂D, an illuminated point y reflects light uniformly; for an angle of incidence α, the point y reflects a radiance Φ/(4π 2 |z -y| 2 ) ρ(y) cos α in any direction above the tangent plane. The dimensionless coefficient ρ(y) ∈ [0, 1] is the albedo, defined as the percentage of incident irradiance which is reflected.

  (A1): D ⊂ R 2 is a bounded open set with C 1 boundary ∂D, such that the closure D is strictly convex, i.e. ∀x, y ∈ D, ∀t ∈ (0, 1), tx + (1 -t)y ∈ D; ν(y) ∈ S 1 denotes the exterior unit normal vector to D at y ∈ ∂D, and µ denotes the length measure on ∂D. (A2): ρ ∈ L ∞ (∂D) is a positive function, bounded and bounded away from zero, i.e. ρ(∂D) ⊂ [c, C] where c, C are two positive constants. As displayed in Figure 2, for any angle θ = (θ 1 , θ 2 ) ∈ S 1 , and for any s ∈ R, the set {x ∈ R 2 :

Figure 2 .

 2 Figure 2. Geometrical setup of Lemma 1 and Definition 1. For any angle θ ∈ S 1 , the set S(θ) contains the values s ∈ R such that the line x • θ = s intersects D. For any s ∈ S(θ), ∂D ∩ {x • θ = s} = {y(θ, s), y(-θ, -s)} with ν(y(θ, s)) • θ ⊥ > 0. For s ∈ ∂S(θ), the line x•θ = s is tangent to ∂D at a unique y(θ, s), and ν(y(θ, s))•θ ⊥ = 0. For any s / ∈ S(θ), D ∩ {x • θ = s} = ∅. The bold curve represents a set of visible points, {y(θ, s), s ∈ S(θ)}.

Appendix A. Convex analysis

For completenness, we prove some intermediate results of convex analysis; they appear clearly in Figure 2.

Lemma 1. Assume (A1) and let θ ∈ S 1 be an angle. (i) Let s ∈ R be such that the line x • θ = s intersects D, i.e. s ∈ S(θ) where S(θ) denote the set

Then, there exists a unique point y ∈ ∂D such that

(ii) The function s ∈ S(θ) → y(θ, s), where y(θ, s) is defined by (4), is a

∂D, dµ) be integrable for the length measure µ on ∂D. Then,

Proof. As in [18, pp. 51-53], fix d :

The exterior normal vector satisfies ν(y) = ∇d Fix s ∈ ∂S(θ). Then the line x • θ = s is a supporting hyperplane of the convex set D (see [5, pp. 50-51]). Hence, D ∩ {x • θ = s} is contained in ∂D, and it contains a unique point y 0 ∈ ∂D, because D is strictly convex. The boundary ∂D is smooth, so the supporting hyperplane x • θ = s is tangent to ∂D at y 0 , i.e. ν(y 0 ) • θ ⊥ = 0. Indeed, the C 1 function g : t → d(sθ + tθ ⊥ ) has a minimum at t 0 = y 0 • θ ⊥ , because g(t 0 ) = 0 and g(t) > 0, t = t 0 ; this implies g (t 0 ) = ∇d(y 0 ) • θ ⊥ = 0.

Fix now s ∈ S(θ). Then the line x • θ = s meets D, and ∂D ∩ {x • θ = s} contains only the two distinct extremities of the line segment D ∩ {x • θ = s}. Denote these extremities by

(ii) We deduce from the proof of (i) that ∂D(θ) = {y(θ, s), s ∈ S(θ)}; indeed, for any s ∈ S(θ), y(θ, s) ∈ ∂D(θ), and for any x ∈ ∂D(θ), x = y(θ, x • θ) with x • θ ∈ S(θ). Furthermore, the set ∂D(θ) satisfies an implicit equation