to point to the summit. “Can we touch that?” C. replied that we were already close to losing our grip. I pointed to the inflection point. Becchia replied: “You’d better not!” My finger then went to the take-off area: “And that!” C. replied: “There’s already wear and tear!” Then there was the ramp of silence: “And that!” For a silence, it was like a funeral! Was this Citroën’s? After two minutes, it was the Deputy Director of Research who broke the silence: “De Casteljau is not an engineer, he’s an academic. It’s normal that he doesn’t know; you have to explain it to him!” How well it was said! My ignorance could not have been better highlighted, and it was the Director of Research himself who came up to the board to give me this very special lesson. With the plan under my arm, I left puzzled. Did they really expect something from me, or were they just using me to make the management understand, who were pestering them: “Cobbler, keep to your last”?<fnref n="63" /> How could I, in two or three months, have competed with masters of the art, who only command the situation after only fifteen years of work! A brief calculation told me that the vibrations corresponded to the 14th harmonic of the cam, it was logical to do the harmonic analysis of the profile with my E 101. Nothing resembles a profile more than the envelope of its harmonic analysis, an inverted bell [curve]. The amplitude went down below the micron as soon as rank 11 or 12, and with rank 13 and especially rank 14 (Here! Here!) we saw a sudden rise. All that was left to do was to invent a machine to shovel the 14th harmonic!

Ten days later, another session at the Studies [Department]! I was in the rather uncomfortable situation akin to a bonesetter invited to give a talk on cancer treatment at the Academy of Medicine.

I explained my harmonic study of the profile of cam no. 4. I was very curtly interrupted: “We know that!” My Head of Service [Department] then intervened, asking them: “Have you already calculated it?” Answer: “No!” Then he literally exploded: “What! You knew about it, and you did nothing!” This allowed me to calmly continue with my initial calculations. – A silence – It was broken by the Director of Research, who simply said: “I don’t believe in the harmonic hypothesis.” Like the crackling of a machine gun, in descending hierarchical order, I heard a flurry of “Me, neither!” The Head of the Calculation Office then took the floor to inform us that all the American works advocated polydyne cams, the very word of which he pronounced with an emphasis that ended up making me deeply dislike them.<fnref n="64" /> So, undeterred, I set to work, by rather tedious trial and error, thanks to the E 101.

A much more rational instance was the subject of my Ulvik presentation (1994).<fnref n="65" /> Compared to a profile using at any point the maximum acceleration, numbered 16, I proposed two sets of 14 poles, the profile being expressed as a function of the cosine of the angle of rotation, over the whole range $-1 + 1$. It was thus a 13th degree curve, totally free of the 14th harmonic. I now believe that a reduction to the 12th degree, and an additional gain of 9% ($\approx 12/13$), could have been attempted, but not beyond. Both profiles had seven poles aligned, equal to 0, for $\cos \varphi = -1$, to respect the kernel, zero lift zone. I never understood why, the Research Director pointed to one of the two profiles, excluding any discussion: “That one!” This profile was given the number 15.

From that moment on I had to endure a barrage of jeers: “You have copied from Bouzitat!” I had never heard of him! They then gave me a document, adding, to show their contempt: “You can keep it!” This study had been carried out with a view to the elaboration of the [Peugeot] 203 cam, on which act no. 4 was based! And it was they who dared to accuse me of plagiarism! The document of Bouzitat is an excellent document, relating to the Fourier transformation.<fnref n="66" /> My disagreement with him is on one point. Bouzitat claims that one can neglect any harmonic whose amplitude is lower than the machining tolerance (then 0,01 mm!). By doing so, he confuses random errors and systematic errors. NO! It is essential for a harmonic of rank $n$ to be brought down to at least the $n^{th}$ part of the tolerance, and even below $1/n^2$, if we take into account the acceleration that it is likely to excite. Moreover, Bouzitat remains silent on how to develop a profile free of these undesirable harmonics. But that is the whole point.

When drawing up act no. 4, the Citroën Studies [Department] had taken measurements on a series cam; a rationalisation had then made it possible to give a table, to the nearest 1/100-th of a micron, every half degree, where the differences had been followed up to the fourth. It was still a good job! But then, a serial cam comes from a large-diameter reproducer, which takes three seconds to read for the complete turn, and which drives a kinematic chain, driving the grinding wheel that rectifies the cam. This is a real vibration amplifier, which in the case of Citroën, had two stages! (gain $g^2$!)

At the time, I was attached to the Tooling Service Department, in charge of machining all these profiles: a few days later, a workshop manager, who had the plan of act [no.] 15 in his hand, came to find me; with his finger, he showed me the undulations which preceded the silence ramp, and with a snide look, he asked me: “Do I also have to machine this?” I was beginning to get sick of cams, and I jumped up and down and told him: “Do what you are told! It is not you who will be ridiculous, but me!” These undulations are there, by virtue of the principle, you have to step back in order to leap forward! Satisfied with my reaction, he left mockingly! Basically, he had been in too much of a hurry to take part in the big joke, and all he had to do was wait a little! It should be added that the Studies [Department] had not been idle. In two or three months, we had already passed the number 40, these 40 profiles being all worse than the others. Some were even geometrically unfeasible! Not one of them even managed to match the performance of act no. 4, with the exception of cam no. 19, without however reaching the fateful 100 HP threshold.

The Tooling colleagues were very amused: like any sailor who sees an iceberg coming dangerously close to the hull of his ship, and who wants nothing to do with the show; I was just another comedian, even more eccentric! It was perhaps the survival of Citroën that was at stake, only this was not visible!

<footnote n="63">⁶³ *À chacun son métier – et les vaches seront bien gardées:* French proverbial rhyme, literally “each to their own profession – and the cows are well looked after”.</footnote>

<footnote n="64">⁶⁴ The polydyne approach goes back to Dudley and Studdard, it combines a polynomial description of a cam with the dynamics of its follower. The approach aims to limit vibrations at the expense of a constant speed (Norton, 2020).</footnote>

<footnote n="65">⁶⁵ cf. de Faget de Casteljau (1995).</footnote>

<footnote n="66">⁶⁶ Jean Bouzitat (1914-1995) was a French mathematician, who was also active in the French society of automotive engineers SIA, where he published a conference paper on harmonic functions (Bouzitat, 1951). De Casteljau mentions the SIA conference notes in his Ulvik paper (de Faget de Casteljau, 1995), where he also dates this episode to 1963. Coincidentally, Bouzitat had studied at ENS together with Pham Tinh Quat (cf. footnote 23) from 1934-1937.</footnote>
The fun part having been postponed to a later date, I was finally able to savour the return to calm. The only information that the Studies [Department] let filter through was that the act [no.] 15 had passed the single-cylinder test without any problems.

Six months later, I was summoned to the Endurance Laboratory. Its director handed me a valve and said: “Very good! Your act has made it possible to gain 12% in rotation speed, thus going from 5,200 revolutions to 5,800, and more than 20% in power – 118 HP instead of 98 HP!” A silence followed and he added: “But here’s the thing – we can’t use it in series! He then showed me the pitting, with metal stripping, on the valve stem. Another silence: “Don’t worry! You’ve calmed down the valve lift so much that it no longer turns on itself to distribute the effort. Up to now it was bad enough to turn on its own! But we know the solution: we’ll have to put valve rotators back in; we haven’t had them since 1920!” And so the DS 21 was able to fulfil its promise.

The Head of the Calculation Office was able to recognise that this was the result of experience. “Your stuff worked, but not for the reasons you claimed!” He had me banned from ever setting foot in the Studies again, and the library from lending me any books whatsoever. He even added: “The way your cam works is contrary to all the laws of physics!” Fifteen years later, when the Peugeot arrived, he tried to silence a smiling coachbuilder as he recounted this adventure to newcomers, and tapped so hard on the edge of the table that he turned his fingers over and appeared to be in great pain.

It wouldn't be long before he was back again, this time to lead the Bodywork.

## The design

At the head of the Design Department, after a period of hostility to pole forms, Flaminio Bertoni,<fnref n="67" /> who had become famous for the DS shape, had rallied with conviction to this new technique. His support in principle was precious to us, but what we could not foresee was his imminent death.

Another Studies session had been scheduled, devoted to the problems posed by the bodywork drawings. We have seen above, that we would have had to be totally unconscious to persist in our particularly hazardous endeavour of rationalising the bodywork designs. My position, now unreservedly adopted by Bx, was that of an individual to whom any possibility of a fightback was impossible, and who, to escape the trap, was forced to practise flight forward. Oh, how we would have preferred to have a period of three to five years to perfect our knowledge. But already a teacher from the Design School had come to find me in order to teach me about pole forms! He had already written three booklets on the subject and wanted to know my opinion. Many of these students, trained at the Citroën Design School, left us immediately to work for our immediate competitors! It would not be for me to decide on the measures to be taken. Citroën had tried to protect the pole forms with a patent, the patent law firms deemed it impossible, outside the “Soleau” envelope.<fnref n="68" /> This precaution has, at least, guaranteed us against legal action from those who later could have accused us of plagiarising what they had borrowed from us! The work of mathematical creation is not protected by any law against counterfeiting; any formula, any algorithm, any method of definition or calculation immediately becomes the property of the whole of humanity, from the moment of discovery. One cannot imagine the mathematician as anything other than a disinterested philanthropist, whereas the smallest advertising slogan can be duly protected. It is easier to convince the competition of the validity of a theory than one's own society, because in the first case one does not want to, and in the second one meets with stubborn resistance. At the Bodywork Session, we found ourselves sitting opposite the same Director of the Calculations Office of Studies.

It would have been inappropriate for the word ‘cam’ to be uttered, and it was not. It was with admirable ease that our former aeronautics officer shared his thoughts with us. The future could not belong to digital computers, too difficult to use; he preferred, and by far, to use the services of analogue computers, developed by small aviation companies, whose precarious financial balance did not worry him at all. Our presence in such a place began to seem, to ourselves, totally incongruous. He had been so kind as to welcome us, to explain in detail what we should have doubted, without disturbing him, nor making him waste his precious time. And he had the exquisite tact not to tell us. For the twentieth time, he reiterated that as long as we respected the tolerance of 0.25 mm imposed, we could do it at our convenience, whatever the means; it was none of his business and he certainly did not want to practice what he would have considered as interfering in our work, neither directly nor indirectly. We all felt a blush rise to our foreheads, due to our confusion, as we tried to explain to him that putting the poles right from the conception of a form remained the best guarantee against drifts and false interpretations. He once again took refuge behind his firmly held point of view and considered the incident closed.

It is extremely rare for a top management representative to take part in a technical discussion, and yet it was he who intervened: “Before there are any Studies problems and Tooling problems, there is Citroën first; this is a Citroën problem! The surface poles must be put in place from the outset, and therefore in the Studies [Department], and they will be! Mr H.... do you want to do it?” he replied curtly: “No!” Deep inside, we felt an immense relief. The management representative continued: “Well then, Citroën is going to set up an office for Bodywork Calculation in Studies, separate from Mechanics Calculation!” And so the unbelievable happened! Citroën had difficulty in recruiting bodywork staff, despite the 30% premium on salaries! The draughtsmen considered this activity to be degrading, far from the technical side, always a matter of guesswork, never of science. However, Bx, who had chosen 4-micron steps for the milling cutter movements, which were noticeable when isolated as a result of the brutal action of the stepper motors,

<footnote n="67">⁶⁷ Flaminio Bertoni (1903-1964) was an Italian sculptor who became automobile designer. While at Citroën, he designed the *Traction Avant*, 2 CV, *Type H*, DS and the *Ami 6*.</footnote>

<footnote n="68">⁶⁸ *The Enveloppe Soleau* is a French procedure to ascertain the date of an idea. The inventor sends two identical copies to the *Institut National de la Propriété Industrielle* (INPI) and receives a registered copy back from there. The INPI keeps a *Soleau* envelope for five years, which can be prolonged for a small fee. De Casteljau's first ideas were registered at INPI (de Casteljau, 1959). As this procedure only operates with papers of less than eight pages, Citroën chose a different procedure for de Casteljau's publication on surfaces of 58 pages, which was registered with the bailiff instead (de Casteljau, 1963).</footnote>
decided on a new standard of 1 micron for the dimensions! The Mechanics [Department] used to use only one hundredth! In turn, it was the coachbuilders who took on the role of scientists, reversing the roles, which in turn caused discomfort in Mechanics!

Citroën set up an avant-garde unit next to the Design Office, equipped with a 3D digitizing machine, a large plotting table, which reassured the designers. Between Friday evening and Monday morning, it was not uncommon to see the papers, even though they had been stabilised with zinc oxide, move by more than a millimetre, over five metres if not more, depending on the humidity in the air. The battery of Burmester sets<fnref n="69" /> had practically disappeared, as well as all kinds of smoothing tools from the draughtsmen's panoply.

A strike by the wood modellers, who considered themselves irreplaceable, allowed the pole forms to be anchored permanently. We were no longer allowed to go backwards and even less to miss our entrance. Two vehicles, the Ami 8 and the GS,<fnref n="70" /> were thus entirely defined by “pole forms” and were manufactured in this way. Each time a new vehicle was released, the manufacturers snatched up the precious wood modellers from one another, and by this means our new technique could not remain unnoticed for long by the automotive competition, before gaining ground in aviation. On the other hand, those draughtsmen who could not adapt to the new techniques joined the ranks of my bitter enemies, along with many other champions of the “fine point (of a pencil) well sharpened” or “metal sculptors”.

## “Attention!”

Although anarchy didn't reign, the rapid development of pole forms had created a climate of freedom; the rules were not established, and this is what had allowed our *corseaire*, Monsieur de la Boixière, to develop further. A certain fantasy reigned, which allowed us to make lightning progress. The first tests, such as the "additions of poles" were promising, and we were not going to stop there.

And already, people were thinking about 5-axis machining, toric milling, and the generation of shapes by algebraic devices on the poles. Some people were already talking about a split between a laboratory that would continue its research and the industrialisation of the new technique. Citroën was no longer in the era of its founder, when a modest worker could, one day, find himself Head of Department, if he had the capacity. From then on, if a technique had been developed at Citroën, it was necessary to go and find a high-level specialist in a company that could not honor itself having success with this genre of research.

This is how I saw a whizz-kid arrive from the Air Force, with him things were going to be put into some order. What a pity, for example, that a film had not been made on the machining of a cam, rebranded "camoid", a sort of cone with a continuously variable slope; its base was bean-shaped, with a significant flatness, and the smaller curve at the top was round. The blank was fixed on a circular plate, itself fixed on a divider, and the machining was carried out in full milling. I was called in to choose the plugs to fix the direction of rotation. If on paper the milling cutter turns to the right, it is because in reality the workpiece turns to the left; at the level of the divider, it was so hair-raising! When we decided to press the button, I was still a little worried. It was our first five-axis machining! Everything went perfectly, and the form took shape before our eyes, without being spoiled by the slightest machining festoon; an unforgettable moment, but so brief!

There was also the rather acrobatic calculation which linked the meridian of a grinding wheel to the straight section of a milling cutter, in order to machine it in a single operation. This lent the specialised worker a degree of freedom that he wasn't yet using, and also led to the standardisation of milling cutters.

Industrialisation put an end to our trials and forced us to fall into line. Back from the U.S.A. our whizz-kid announced to us: "The Americans make TABCYL!"<fnref n="71" /> and "At Renault, they manufacture with a tip!" We were then obliged to feign great admiration for techniques that seemed more than primitive to us. For little, we would have been back to carving flints! Then, one day, our new teacher took the pretext of a doze following a rather heavy digestion, to stop inviting me to his meetings. Shortly afterwards, he took advantage of this to set as his priority research objective the quest to find the book, document, magazine or grimoire where I had been able to unearth the concept of poles, which metamorphosed – like a winged insect emerges from its chrysalis – into "Bézier points, Bézier curves", which seemed to give him satisfaction! There was no longer any doubt that I was being pushed to the sidelines. 'Bx' had reluctantly agreed to study fuel cells designated for the electric car. Would he have been successful? He was in competition with the Batelle Institute in Geneva, with which Citroën had a contract.<fnref n="72" /> My direct 'boss', who was already positioned in the middle to our whizz-kid, suggested that I apply for a transfer to Research. I answered without hesitation: "Transfer me, if you wish; but don't count on me to ask for it!" Shortly afterwards, I met the new director of the Research Department, who had barely shaken my hand in a jovial manner. He said, "Hello, Mr de Casteljau, how are you?" when he had already arrived at the other end of the room! I made no effort to join him there.

<footnote n="69">⁶⁹ cf. French curves, footnote 52.</footnote>

<footnote n="70">⁷⁰ Citroën's Ami 8 was succeeding the successful Ami 6 in 1969, an economy family car positioned between the economy car 2CV and the luxurious DS. The Ami (Friend) series was produced from 1961 to 1979. The GS, which was designed to fill the gap between Ami and DS, was launched in 1970 and produced until 1986. Its production can be reviewed in Louis Malle's 1974 documentary "Humain, trop humain".</footnote>

<footnote n="71">⁷¹ TABCYL is the abbreviation for *tabulated cylinder*, a construction method for curvature continuous joins, cf. Faux and Pratt (1979), first introduced to the programming language APT in 1957 (Ross, 1978, p. 92). It is also related to Wilson-Fowler splines (Fowler and Wilson, 1966).</footnote>

<footnote n="72">⁷² *The Battelle Memorial Institute* is an independent research institution, headquartered in Ohio, US, with a focus on contracted research and development of metals and material sciences. In 1969, Battelle started its Electrochemical Generators Group in Geneva, from where they supported the mentioned Citroën project. These activities led in 1974 to the creation of SORAPEC – *Société de Recherche et d'Applications Électrochimiques*, finally sold to the Suez Group in 1991 (Sunergy, 2019).</footnote>
Then the whizz-kid summoned me to his office, became polite and said to me: "You obviously don't fit into the norms. How could we designate you in the new organisation chart wanted by our management from Peugeot?". Like the rat in the fable,<fnref n="73" /> I thought to myself: "That doesn't tell me anything worthwhile." But he insisted again and again. So I told him that in the yearbook of the École Normale, a former student had coined the term 'advisor', which is not recognised by the Law and which has a certain allure. "That way, he could not be accused of usurping his title!" Two days later, a memorandum by Services informed us of his and a colleague's appointment to the positions of "Advisor". Immediately, I wrote to the École Normale to withdraw my title and not replace it with anything, or, if they so desired, to take minion or so.<fnref n="74" /> This must have triggered a phone call to my boss, who was unaware of this, and who, taken by surprise, bombarded me with a "Graduated in Applied Mathematics", which did not correspond to anything, and which I have been since. I was then 'promoted' to Engineer 2B, a brilliant promotion to the rank of Commander from that of a General. Until then I was "Upper Management".

## The lion

Peugeot shares with Crédit Lyonnais the emblem of the Lion. With the arrival of Peugeot, and of several alumni of the Lycée Victor Hugo, I could have hoped for a breakthrough: I would have had to be ecstatic about Peugeot's technology: it was stronger than me! When I was confronted with insurmountable difficulties, no one told me that they had been surmounted at Peugeot and resolved at Renault! I had my back to the wall, and the signal for the slaughter was imminent. Others knew how to get on the winning side, duly noted – pay tribute to them, no! "The poles are so simple that anyone could have invented them," said one, before adding: "You must be happy, your ideas are triumphing," Yes! But not at Citroën!

And those who had turned my career into an obstacle course were beginning to worry about their own. To prove our precedence, one of them, who did not want to reveal any "secret", asked me if we could communicate our internal booklet to the Germans. Written from notes, in which only one sentence out of every two appeared, and with page inversions, this booklet triggered strong laughter to those who tried to extract knowledge from it. Mischievously, knowing that the Germans would know how to notice triangular patches, other than by the confusion of two vertices of a rectangular patch, as Mr Bézier claimed, I replied that everything was now known; the writing of this booklet was detestable, but constituted an irrefutable fossil.<fnref n="75" />

Henceforth I decided to keep my findings to myself, the others hastening to minimise them at once. However, I made an exception for an engineer of Germanic origin, whom I will refer to later as "my friend", a former disciple of Monsieur Bézier. His desire for conciliation was respectable. A director, who had been subordinated to Peugeot, told me: "Mr Bézier did as well as you, and the Americans much better. Admit that you are a finished man!" Moreover, two memos, one from Citroën and one from Peugeot, forbade me from dealing with CAD (Computer Aided Design)!

Equipping myself with a personal microcomputer at home cost me a small fortune at the time. Even more seriously, the extension of my house by a quarter resulted in an eightfold increase of my local taxes, I emphasize eight, not to mention the taxes linked to the construction. It was too late to have regrets!

The CEO at that time was more courteous towards me. The office I shared with two other engineers became a "company restaurant" for senior executives, and after a day spent in a corridor, I was installed next to a passageway between two buildings, which seemed to be a temporary bridge constructed by the engineers. The boards were so disjointed and noisy that any effort to concentrate would have been an achievement. This CEO, alluding to the lack of legal protection for creative mathematical work, ventured this comparison, which was intended to put him in the picture: "You are not profitable! You don't know how to sell!" The future was cruel for him, as he was at the origin of the construction of the Romanian factory in Craiova, in the land of Dracula, of which the director, who said that I was a finished man, was able to say: "What it manufactures is so much shit, that nobody wants it!"<fnref n="76" />

I was reduced to doing most of my research in the evening at home. At the time, a joint Peugeot-Citroën team had created a synthesis between Unisurf and the American Splines, which materialised in a booklet of a dozen pages and the drawing of a duck. I received a copy to solicit my appreciation, which could only be unreserved admiration. For my part, I had tried to understand what might seem irreconcilable between the poles and the Splines. According to the texts, of American origin, certain complementary claims led to contradictions. One day I said to my friend: "If I can understand this, there will be no more problems." Sometimes they claimed to have mastered interpolation, sometimes it required some smoothing. If you added that to the minimisations of the tensions in an elastic fibre, you're in deep trouble.

<footnote n="73">⁷³ Jean de La Fontaine's fable *Le Lion et le Rat* talks about a rat, that surprisingly pops out of the ground between a lion's paws. As the lion spares its life, the rat later helps the lion to escape from a hunter's net. It ends with the verse "By time and toil we sever / What strength and rage could never." The reader is invited to associate the lion with Peugeot.</footnote>

<footnote n="74">⁷⁴ *grouillet*: squirrel. On the copy that de Casteljau sent to me, he added in pencil (translated from French): "In fact, it was 'bougnoul' (towelhead), which nowadays is considered a racial insult: it means 'man' in a Senegal language, but is improperly attributed to Arabs!".</footnote>

<footnote n="75">⁷⁵ After Wolfgang Boehm (1928-2018) got access to a 1971 article about Citroën's work in CAD/CAM which included a figure of de Casteljau's algorithm (Krautter and Parizot, 1971), he wrote a letter to Citroën in 1975 asking for the author and further documents. In response to his letter, he received a paper copy of the *Enveloppe Soleau* (de Casteljau, 1959) and a 58-page collection, which de Casteljau here refers to as 'the booklet' (de Casteljau, 1963). Boehm was also in possession of the encyclopaedic article by Parizot, which includes a section by Bouffard-Vercelli on *courbes et surfaces à pôles* and introduces their mathematical form with a reference to de Casteljau, but without elaborating on the algorithm (Parizot, 1971-1975; Bouffard-Vercelli, 1971-1975).</footnote>

<footnote n="76">⁷⁶ The Romanian government was interested in manufacturing automobiles for French brands, which in 1976 led to a first joint venture with Citroën called Oltcit, a combination of the Romanian region of Oltenia and Citroën. It produced the Oltcit Special and Oltcit Club (also exported to Western Europe as Citroën Axel), based on the platform that had earlier been planned for the Citroën Visa in France, but was not used for it, as Peugeot preferred the existing 104 platform after its takeover. The joint venture ended in 1990 due to poor quality (Lambru, 2016).</footnote>
If I had lost my way for a brief moment, it was for having accepted the validity of the often contradictory claims attributed to the Splines. The solution jumped out at me, taking up an old idea, if I had pushed it back it was because of the reduced performance of computers in 1960; I had not pursued them, and chased them from my mind. The generalised poles, polar forms extended to several points, instead of being limited to the extremities of the arc, took on consistency. Instead of splitting into the subtle distinctions between “knots and nodes” and getting confused by the duplication of indices, reserved for the ends of an arc, everything became clearer and simpler, once and for all. The main difficulty I had to solve was to manage to concentrate in a particularly noisy office, dominated by the laughter of a neighbour, whose jokes lacked humour. So I gave my friend a few hints, which seemed to me to be quite sufficient to put him on the right track, and to highlight this logical process. This was [happening] just on the eve of a three-day weekend, when moreover he was going to benefit from the inspiration of the Holy Spirit, whose feast day was being celebrated. Convinced that I would find him looking cheerful on Whit Tuesday, but I noticed at a glance that this was not the case; even worse, he had been working. So in less than three minutes, I showed him my solution: that's all it takes! Then I see him completely dumbfounded, and with a silly look on his face he comments softly: “Oh! No! The *Maître*, he is not a professor, but a professor's professor!”<fnref n="77" /> Then he gathered himself up, when I recommended total discretion for a while. Let's not forget that I was forbidden, by memo, to work on CAD! This was the only time I saw him explode! “Do you realise what you are asking of me! Afterwards, people can say to me: ‘You knew about this and you didn't tell us!’” However, he scrupulously respected our conventions. It should be added that, as a result of the reorganisation that followed the Peugeot Citroën merger, the CAD responsibilities were much more concerned with their personal careers than with technical progress! My friend provided me with a rather difficult plan of a marine propeller, to support my theses; the points had been taken with a marking gauge! One could not be more approximate.

## The roar!

By what mystery was I invited to a session intended to share my impressions of the famous duck!? It was an opportunity not to be missed! I was determined to distinguish myself from this study, which was conducted without me. For the first time, I had the opportunity to measure myself against the experts from Peugeot, whose sole aim was to demonstrate their superiority in the field. Starting with “knots and nodes”, I then targeted the doubling of indices, considered, apart from the ends of an arc, as a degrading complication, leading to “subsplines” without interest, except in the case of sharp edges. The mastery of this index-doubling was fundamental, without even revealing the properties of the degree of restitution.<fnref n="78" /> The text on the duck ended with a calculation of the poles of a spline segment of third degree. I had no difficulty in showing that the result was correct, despite a calculation error, which was mysteriously compensated for afterwards! I lifted a corner of the veil of my theory. It was possible for me – in less than ten lines – to carry out these calculations of degree 5 or even 7, and that without any risk of error.

My friend rushed to me at the exit: “You don’t realise! You have humiliated him! He’s your Peugeot equivalent: he’s the one who set up the Unisurf programme!… Renault has done nothing! It’s him who did it all!”… No one had yet told me that this researcher, for whom I had a certain respect, was at the origin of a document, which was presented to me as collective, and moreover under the direction of another. Despite my outburst, this session brought us closer together, strengthening our mutual esteem.

At the same time, a similar misadventure occurred. Two engineers could not find their way out of a seemingly simple problem. They had been working on it for more than a month, even a month and a half. It took someone more than ten days before deciding to come and talk to me about it. In my office, they handed me a sheet of paper where a system of equations of the second degree was spread out, followed by one of the third degree, which they wanted to solve. Listening to them, there were no other possibilities to put it into equations, and the only thing they expected from me was to tell them how to continue the calculation. They answered Yes and No to my questions, nothing else! This system looked like vector algebra and I asked them “Do these three express, that three vectors are unitary?” – “Yes that’s right”; “The next three express scalar products?” – “Yes, that’s right!”; “And the last one is the mixed [scalar triple] product?” – “Yes, that’s right!”; “Why didn’t you tell me right away?”; “So you want to express the condition, starting at cosines of angles, that three vectors are coplanar!” – “Yes!”. Inwardly, I repeated to myself: “Be nice! Don’t make fun of them! Go slowly! Take care of them! Watch your intonations!”… I go to the board and tell them calmly: I pose the matrix *M* of the components, with respect to the still unknown axis system, *Ou Oz* is perpendicular to this plane. Its determinant is zero, since the last column consists of zeros. Do you follow me? – “Yes”. I perform the product *M* *M*\* = 0, but isn’t that the condition you are looking for? – “Yes! Yes!” – “A trick, due to Lagrange”, I hastened to add, as if to excuse them for their ignorance. You multiply zero by zero to find a relation. I then saw them respectfully moving back, step by step, like two crabs, up to the door! My two flabbergasted fellows stared at me with eyes like fried whiting! Once through the door, I saw them take to their heels and quickly clear out! A fortnight later, I was told that one had gone to Renault, the other to EDF (Électricité de France).

I was still a child molester! So, what should I have done well? The ‘finished’ man sometimes allowed himself to roar, and he was blamed for it!

In the meantime, my friend had machined the marine propeller blade. The result was admirable! No, it wasn’t quite perfection, there were still two particularly sensitive areas where one might have wished for a slight shift in the poles of definition. If you look at the piece, very obliquely, “at the end” as practitioners do who want to check if “it works”, and then walk their finger around to detect a slight bump, or hollow, there were still two critical points, barely visible. This machining did not receive the interest it deserved. It was as good as the duck, but perhaps that’s why it was in the way!

<footnote n="77">⁷⁷ De Casteljau was called « *Le Maître* » (the master) by the colleagues in his team.</footnote>

<footnote n="78">⁷⁸ The degree of restitution is part of de Casteljau’s books (1985, 1986, 1990) and his early articles (1992, 1993), also see footnote 96; yet in his eyes, it has not been sufficiently recognised by the CAGD community.</footnote>
The unexpected peace and quiet that resulted allowed me to expand and consolidate my research to an impeccable stage. By this time, the progress of the “American work” had a much deeper psychological impact than one might think. I found the same climate that had prevailed when Monsieur de la Boixière was evicted, but this time it was against the evictors. The ‘Americans’ had not yet finished monopolising the numerical definition, and many were already pledging their allegiance to them. The result was almost general unease. (Mr de la Boixière, when Peugeot arrived, had asked to be dismissed, to take early retirement: it was no more peaceful than his years of activity!)

The politicians were agitating in all directions, anxious to preserve their prerogatives. My serenity contrasted with the atmosphere. An idea was emerging: instead of continuing to bicker, we had to gather our forces in France in order to counter the threat. Peugeot had been wavering between joint production with (or rather at) Renault, foreseen by the Peugeot-Renault agreements, and the use of the modern investments by Citroën, which had been put in such financial difficulty by the oil crisis, that it cost it the absorption by Peugeot. It was now or never to test the solidity of the Peugeot-Renault agreements, which had not been terminated at all.<fnref n="79" /> Moreover, there was a possible area of agreement. The three manufacturers Renault, Peugeot and Citroën held a part of the capital of the *Société Chausson*, and used its services.<fnref n="80" /> The latter had been put on notice to adopt the new techniques of defining forms.

## A glimmer

Two engineers from Chausson – a *Polytechnicien* (?) and a *Centralien* – contacted me directly, and without revealing my batteries, I was able to tell them the kind of problems for which I had a solution. His<fnref n="81" /> investigation extended to all the CAD systems existing in France. His 500-page report earned him special congratulations from his General Management. He distinguished the Peugeot or Citroën systems from the ‘de Casteljau’(!) system, all three of them being different.

It taught me that his study was only one element of a much vaster project, with the support of the government (Raymond Barre). It was about regrouping the whole of French CAD into a single company. It was recognised that this research had been led by the automotive industry, well before the aviation industry began to concern itself with it; so as to protect the sensitivities of the latter; we could have envisaged a double company with an automotive and an aviation branch, but the preference was for a single company.<fnref n="82" />

I saw the possibility of finding an atmosphere more conducive to my research, and soon I began to give him a hint of my agreement to join them. Besides, there was a certain family relationship<fnref n="83" /> with their CEO that we discovered together. On the occasion of a family meeting, I could meet their CEO; he seemingly wanted to avoid me. His discretion honoured him, and I understood very well that such dealings could remain confidential. All I obtained from him was: “We’ll see *that*, in three months!”

## The quake

For a show, it was a show, three months later! The CEO and his two technical and commercial directors were ruthlessly ejected and replaced by R.V.I. (Renault Véhicules Industriels).<fnref n="84" /> As for my two correspondents, overwhelmed, they preferred to resign in turn.

Peugeot, after the 500-page report, whose conclusion was that “all systems are blocked, except for only one, the ‘de Casteljau’ system”, could no longer afford to continue ignoring my research. Well forced this time to accept a session, I made one condition and only one: that the whizz-kid who was responsible for the eviction of Mr de la Boixière would not be invited. This did not seem to pose any problems and the condition was accepted. This session was entitled the solemnity of the General Management Floor in Neuilly, which I was hardly familiar with, and I was able to begin my presentation on generalised poles, the future ‘Blossoming’ already perfectly developed! (Under Raymond Barre, I repeat)<fnref n="85" />

It is obvious that a specialist, stuffed with Bernstein, can only be amazed when he discovers the polar forms! They juggle with the problems with such elegance that one remains stunned! Others experienced this when the ‘Blossoming’ spread. After an hour, I saw our whizz-kid enter, a little pallid, who apologised for his lateness, using the pretext of another appointment, which he would not have had the appropriateness to move. I didn’t have the courage or the presence of mind to put down my piece of chalk, to conclude by thanking the audience and to withdraw with dignity.

On leaving, my ‘boss’, a Polytechnician, sighed: “In the end, it went well!” There are moments when only silence has sufficient dignity. This Polytechnician was not a bad guy, and already in his office, he had proclaimed to me one day: “I will personally see to the development of this idea; I will follow it until it succeeds!” Two hours later he called me back into the small room adjoining

<footnote n="79">⁷⁹ Peugeot had a collaboration agreement with Renault since 1966, which also related to the joint further development of Unisurf 2 and 3. Late 1968, early 1969, Unisurf was capable of designing the entire body of the Peugeot 104 (Poitou, 1989, p. 39). Yet after Peugeot acquired Citroën in 1974, they had access to Unisurf as well as to Sadusca, which allowed taking a well informed decision on the CAD software strategy.</footnote>

<footnote n="80">⁸⁰ Chausson was originally an independent bus manufacturer and car supplier. The company merged with Renault's commercial brand SAVIEM in 1959. From 1971 to 1991, Peugeot and Renault were main shareholders of the production plant (each 49%), where they placed their manufacturing of commercial vehicles and coupés in small volumes. During a larger restructuring in 1993, PSA sold its Chausson shares for a symbolic Franc to Renault. The last formerly Chausson production facility was closed by Renault in 2000.</footnote>

<footnote n="81">⁸¹ De Casteljau hereafter uses the singular although he introduced two engineers.</footnote>

<footnote n="82">⁸² The study is also mentioned by Poitou (1989, p. 41), but the source remains unclear.</footnote>

<footnote n="83">⁸³ On the copy that de Casteljau sent to me, he added in pencil (translated from French): “he bore the same name as my mother-in-law!”.</footnote>

<footnote n="84">⁸⁴ As a consequence of the Citroën takeover, Peugeot sold Citroën's bus business brand Berliet to Renault in 1975, which merged it with its own brand Saviem to form RVI, today branded “Renault Trucks”.</footnote>

<footnote n="85">⁸⁵ Raymond Barre was France's Prime Minister from August 1976 to May 1981, under the presidency of Valéry Giscard d'Estaing. So this meeting must have taken place several years before de Casteljau's book on blossoming got published (1985) and long before Ramshaw (1987) coined the term *Blossoming*.</footnote>
the office and said, hiccupping, “I can’t! I can’t!...” A cough in his office told me that he was just a marionette, and that his boss was ‘personally seeing to it’ that this idea was blocked.

But when, back to my office, my friend came and in turn congratulated that “it went well” by taking over the formulation, I burst out “How! It went well! The generalised poles... it’s all screwed up!” My friend looked at me, bewildered, and then I wrote with a big red pencil on my calendar, on the day’s date<fnref n="86" />: “Dies irae; Dies illa...! X has recovered the generalised poles!... It’s screwed up!”<fnref n="87" />

It would take my friend about six months to understand!

Instead of giving France a substantial advance, which could have enabled us to negotiate partnership agreements that would have benefited us, we had returned to the personal ambitions of people who were much more concerned with their careers than with the general interest of the country. From the idea of regrouping CAD into a single company, or from the possibility given to Peugeot (the choice of the [General] Management Floor seemed to indicate it) to go it alone – nothing, or almost nothing, remained of all that. Moreover, even Raymond Barre, who could have seized the opportunity to manifest himself as a giant of the economy, other than by building the Romanian factory, soon disappeared down the trap door.

It is regrettable that the quakes triggered by “politicians” can get the better of the patient and relentless work of researchers, in one instant. On television, I once saw a researcher from the *Institut Pasteur* who declared: “In France, you find many people who are researching, but we are still researching for a guy who finds!” Another researcher said less skillfully: “In France, you have the right to research on the express condition that you don’t find anything until you retire!”

This, moreover, is not a privilege of France! The confidences of foreigners prove it.

## Normalisation

Of course, I reminded my interlocutors of the service notes, which forbade me to think in CAD! This didn’t seem to disturb them any more than the fact that I didn’t care; at most a smile accompanied by a murmur, supposed to give the reasons for this surprising decision. On the other hand, my momentum was broken, and since I was now back in the ranks, I was asked to write a note to the *Académie des Sciences*.<fnref n="88" /> This kind of publication is strictly codified – number of lines, or characters per line, bilingual abstract, etc… If we were to listen to the rumour, the *Académie des Sciences* had lost the charisma that it had previously emitted abroad. Personally, I had witnessed texts of little value and even erroneous, or of extreme complexity. It was becoming aware of this, and its reorganisation was in progress. First of all, I had to search for an academician who would be willing to present my study, which is not difficult for a former student of *Normale*. It took more than a year to come up with an essay that complied with the rules. Yet the academician who had offered to publish it needed only a few minutes to phone me the references of a work by Mr Bézier, published in Anglo-American. Never in Mr Bézier’s thesis is the idea of a polar form or of generalised poles mentioned in any way! Mr Bézier’s thesis, of which I have a copy, is a veritable patchwork, which contains a serious incorrection and a principle error: bi-parametric surfaces are approached through analysis, using Bernstein functions, via the integral calculus. This has absolutely nothing to do with the purely algebraic approach using polar forms. I refuse to believe that this field had become a private preserve, solely for Mr Bézier; this academician could have shown more foresight! Let us say again that such exclusivity is not claimed by Mr Bézier; he reiterates this at every opportunity.

I was then invited to publish a book,<fnref n="89" /> a solution strongly encouraged by the Peugeot management. I couldn’t get thoughts out of my head that this publication, which had been authorised at a very late date, would only find a very limited audience, in the midst of general indifference. Once again, I was going to hear that it was all too well known! This was the reaction of the publisher, who when contacted by Citroën, from the very first moment, strongly reproached us for our impudence: “You can’t expect us to believe that you are at the origin of the works that led to Unisurf and the B-splines!” I was so expecting to hear this kind of reaction that I did not make the slightest gesture of denial. I was well placed to know that during the first five years of my industrial activity, I had received nothing but rebuffs, not the slightest document or indication concerning promising work by our competitors or research laboratories. The most advanced was Ferranti,<fnref n="90" /> who used Fourier functions. Rumour has it that Peugeot had used the method of proportional parts for the calculation of the 203, and Renault was bogged down in biquadratics – I am talking about 1960. As a conclusion to a meeting organised by Ferranti, during the final discussion, Mr Bézier had declared: “The bodywork is the example of what can never be defined by Mathematics!” When I remind some people of this sentence, they answer me: “Only fools don’t change their mind!” This does not detract from the immense work accomplished by Mr Bézier to overcome resistance and impose upon practitioners to change their habits. It is not enough to wave a magic wand and say: “At Citroën, they did it!”

On the other hand, the haemorrhage of designers, or students of the Citroën School of Design, was almost fatal to Citroën. While all these memories were coming back to me, I began to observe the reactions of our main interlocutor from the Hermès publishing house, listening to the discussion with a distracted ear. It was my friend, a former disciple of Mr Bézier’s, and well placed

<footnote n="86">⁸⁶ On the copy that de Casteljau sent to me, he added in pencil (translated from French). *Tuesday 5 July 1983*.</footnote>

<footnote n="87">⁸⁷ “Dies irae, dies illa” is the first verse of a medieval Latin poem on the Last Judgement, “Day of wrath and doom impending!” It has long been part of the Catholic liturgy in the Requiem Mass.</footnote>

<footnote n="88">⁸⁸ De Casteljau produced a handwritten document on *Formes à Pôles* in February 1983, 32 pages, unpublished until now (de Faget de Casteljau, 1983).</footnote>

<footnote n="89">⁸⁹ “Formes à Pôles” (de Faget de Casteljau, 1985) is the first original book of de Casteljau after being mentioned before only in the early 1970s (Krautter and Parizot, 1971; Parizot, 1971–1975), it includes a solid introduction to blossoming. The English translation was published as “Shape Mathematics and CAD” in 1985.</footnote>

<footnote n="90">⁹⁰ Ferranti was a British electronics company, producing the first supercomputers such as the Mark I in 1951 and the Atlas in 1962. The success of Ferranti’s analogue computers declined as the third generation of computers emerged at the end of the sixties (Poitou, 1989, p. 51).</footnote>
to distinguish all the nuances between the two approaches, who then patiently intervened. He began to explain, in an angelic voice, the gulf that separated Bernstein's functions from the polar forms.

At first, our interlocutor had struggled immensely to suppress his anger in the face of so much effrontery on our part, turning crimson and gesticulating; gradually he began to blush, as my friend calmly continued his demonstration. It is not my intention to seek out Mr Bézier's sources; he sometimes reports on a mysterious scholar, completely unknown in libraries.<fnref n="91" /> And he has always been unable to give any details to those who have asked him the question. The only source, to my knowledge, concerning the poles, which could be claimed as anterior, is a text dating from 1870, taken from the "Mathematische Annalen" by J.C.F Haase of Munich, of less than twenty lines, and limited to curves.<fnref n="92" /> At the end of our conversation, our friend had wrung the editor's conviction. It is true that the explanations from which my friend had benefited were beginning to bear fruit. By dint of comparing the difficult and scholarly approach by Bernstein-Bézier functions with the one inspired by polar forms, my friend had understood the extraordinary improvement in comprehension by using the latter, without forgetting the extreme simplification of the calculations, and the possibility of going infinitely further in the elaboration of the theory. Every specialist remains convinced that we are still far from having exhausted all the resources offered by polar forms. Let us quote, amongst others, the algebraic additions of poles, which have remained at the laboratory stage, despite promising tests; the evolution of forms in slices of polar volumes; the separation between even and odd terms of a neutral axis, thus giving thickness to a form, etc... etc... One might even add that arithmetic is more useful to the elaboration of this theory than analysis is. As proof, the table of pole differences, which is also the table of pole derivatives, about which the international scientific community has hardly followed me.<fnref n="93" />

All that was left to do was to provide the publisher with a diskette of my work in word processing! It was the first time that Citroën had tried mathematical editing. I was assigned the services of a charming typist, of Algerian origin, respondent of the kind first name of Soraya, for whom it proved an extremely difficult task, with the half-interlaces necessary for the indices or exponents, especially to return to the initial state! You must control yourself, if you do not want to send to hell the first draws of the printing press; the sheets were crossed out with formulae ascending or descending by 45°! And you still can't help but think: "It's once again screwed!" However, after a year, we were out of the woods, the character police that turned my t-indexes into cemetery alignments had been replaced by a less funereal one. There was no longer any doubt that, after all those wasted years, we were going to arrive like the carabinieri.<fnref n="94" />

The decision-makers then commanded a preprint of the diskette in some dozen copies, destined for privileged few: our whizz-kid himself had drawn up a list of hand-picked personalities, deemed worthy of receiving a copy. Perhaps he was forgetting about the photocopy, capable of producing numerous clones? It was becoming urgent to make the polar forms official, and the surprise effect became irrelevant.

No one had yet really become aware of the existence of the degree of restitution, associated with the notion of characteristic, which draws the borderline between interpolation and smoothing – two aspects of the same idea, which until then had been distinguished, not to say opposed. These ideas were further clarified in my book on "Smoothing".<fnref n="95" /> A Polytechnician then established that the degree of restitution of Splines was ONE, which I made him repeat THREE times, believing that he was joking!<fnref n="96" />

## To merit!

A Society on the other side of the Atlantic was interested in my work, had established contacts at practitioner level. But how to ensure exclusivity in the mathematical field, when the work of Mathematical Creation is not recognised by any Law, and the discovery is not safeguarded by a legal protection? There was indeed the possibility for me to leave the Peugeot-Citroën group, and the Management even reproached me, afterwards, for not having done so! Peugeot did not have the same perspective on research as Citroën, and its management added: "It is far too expensive to conduct research. If a new technique is needed, we will buy it, paying the price that is necessary!" (Nothing, in this case). Finally, at the request of the American company, Peugeot (not Citroën) signed cooperation agreements and received as a gift a complete CAD system, worth three million Francs. That year, our CEO was awarded the "l'Expansion" prize, thus crowning his good leadership, for having received a tribute from an American company, which he had not even requested.

Those who had hoped that the release of my book would be a flop and a commercial failure were in for a surprise, and this seemed to dismay the Director, who saw in me a finished man. At the end of the year, the engineers were given a substantial readjustment, making up for the delay they had suffered. He had the pleasure of announcing a homeopathic salary increase to me, but not being

<footnote n="91">⁹¹ Pierre Bézier invented an imaginary professor, Onésine Durand, which he had used inside Renault to convince the organisation of the seriousness of his own approach by virtue of a (non-existent) third party, cf. (Laurent and Sablonnière, 2001).</footnote>

<footnote n="92">⁹² Haase's algorithm for the construction of a point on a rational planar curve (Haase, 1870, p. 543) was discovered by Boehm and presented in Oberwolfach in 1992, where he encountered de Casteljau (Boehm, 1993).</footnote>

<footnote n="93">⁹³ de Casteljau prefers finite differences from arithmetics over differential calculus over analysis, cf. (de Faget de Casteljau, 1990).</footnote>

<footnote n="94">⁹⁴ arriver comme les carabiniers (French proverb): to come too late.</footnote>

<footnote n="95">⁹⁵ *Le Lissage* (de Faget de Casteljau, 1990) deals with various approaches on smoothing, including the above mentioned aspects of finite differences, but also including trigonometric, algebraic, variational approaches. It has been published in French only.</footnote>

<footnote n="96">⁹⁶ Let us assume, that a set of points originates from a curve of polynomial degree *r* or higher. De Casteljau then calls *r* the degree of restitution the smallest degree of an interpolant *C*, such that this interpolant reproduces the original curve (by its given point set). This interpolant can easily be found by Lagrange interpolation. If *r* equals one, the interpolant and therefore the 'curve' is a line. Instead, the degree of restitution for a spline of piecewise degree *n* is *n*. The details are described in *Formes d'Pôles* (1985), more subtly in *Le Lissage* (1990) or in Ramshaw's course notes (1991). An overview is also given by Müller (2024) in this special issue.</footnote>
able to ignore my efforts, he added: “Plus a hundred Francs, because you have worked well!” That year, our CEO had insisted that raises would be made on merit in order to explain their greater disparity.

Several leaders from companies in *Franche-Comté* later told me the reasons for our management's attitude. At least four times, I heard: “To succeed at Peugeot, you have to be *Centralien, Franc-Comtois* and Protestant!” (“and” not “or”).<fnref n="97" />

However, I was told that access to the computers was no longer forbidden, making it official that until then it had been forbidden!!! The CAD system was protected by coded locks; I did not have the key.

## The quaternions

The [polar] forms, informatics and all the informal formalisms were beginning to make me nauseous. If our management didn't care, why should I? Once again, we were going to wait for my theses to be applied elsewhere before recognising their merits and using them...! It was no longer prudence, nor obstinacy, but blindness. If the money, swallowed by Raymond Barre in the Romanian mirages, in the country of Dracula, had been devoted to the development, in due time, of CAD and Robotics, we could probably have made our investment profitable, by marketing our know-how...! Instead of that...! Crash...!

So, as a challenge, exclusively using my personal computer at home, which legally, assured me the full property, I started to study quaternions from a purely arithmetical point of view! A finding was bothering me, and an idea seemed so simple and obvious that I was convinced that it must have already been explored, and that it could not have succeeded without anyone knowing. So I had to look elsewhere, while I checked that all other directions led to dead ends. At the time, I was unaware that Minkowsky had explored this path; fortunately for me, I only found out later. Indeed, Minkowsky accepted negative matrices, performing space reversals, for which the notion of differential continuity cannot exist; and as a corollary, he accepted negative squares (see his $ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2$).

By refusing, a priori, these aberrant elements, it was possible for me to prove a superb theorem, which the existence of negative terms would have made false! If I had started by following Minkowsky's path, I would not have been able to highlight this detail. Here again, my approach clarified and simplified certain obscure passages.

To conclude, I was able to show, from a formal point of view, that, presented from a differential viewpoint, this theorem represented Maxwell's equations, as relativistically presented by Lorentz! This opened the way to further research on the generalisation of the theory of functions of the complex variable, applied to the conformal transformation in $E^3$ or $E^4$. All at once, one understood the reasons for the failure of the previous attempts. Let us simply say that the non-existence theorems, proved for one function, do not apply to the differential domain, defined by a group of several functions. After the non-success of functions with a hypercomplex variable, due to non-commutativity, it was necessary to find a way around the obstacle. In spite of these promising results, I did not push my investigations in this field any further, and this is very regrettable. Since then, it seemed to me that this kind of research should serve as a crowning achievement to a study of Metric Geometry.

The quaternion adventure was just as fertile in memorable anecdotes, as the polar forms, the cams, or the generalised poles, and sometimes even more crisp! As long as I forgot about CAD, it didn't bother anyone, and I didn't even have the slightest thought of the kind: “What can this be used for?” The answer was all ready: “Nothing!” My friend was well aware of this fact, when one day he entered my office waddling from one foot to the other. I felt that he had something to tell me, but he didn't dare to say it. I invited him to do so, only to be told, “Well, there you go! The Americans say that, in order to carry out the calculations of Robotics, nothing beats the use of Quaternions. Nobody knows what they are!... Not even the Polytechnicians. Could you give us a briefing on the subject?” I almost burst out laughing. For six months now, as a provocation, I had been trying to work on something completely useless and “unprofitable” and this is now the needed technique!

“No more than an hour, right!” the whizz-kid shouted at me, as he had his secretariat schedule a session. My audience, made up of Heads of Department, Peugeot or Citroën, and the most brilliant Polytechnicians, listened to my explanations, and with the questions, which crackled, made the session last for three hours!

I undertook to write a more substantial work than the little booklet that summarised my six months of work. A Polytechnician, in charge of clearing up the inconsistencies, distributed it, after having only completed it with a bibliography, and had it typed.

Now that this work was officialized with delay, I felt it was prudent to take some precautions. The most basic one was to ask the Documentation Service what could be found on the subject. After patiently spelling out the words ‘Quaternions’ and ‘Hamilton’, which she was hearing for the first time, our documentalist questioned, with equal failure, the technical bookshops, SVP,<fnref n="98" /> and the AFRI (Association Française de Robotique Industrielle). In the latter, she thought she would find a competent person. Having started her investigations early in the morning, she had at the end of the line, in the early evening, the director of the AFRI in person, from whom she could learn the usefulness of quaternions in Robotics. When, six months later, a correspondent wanted to go into the question of quaternions in greater depth, ‘AFRI’ replied: ‘Oh, we won’t do this to ourselves twice! There is a guy at Citroën who knows them well, a certain de Casteljau, but we would like to warn you that he is not easily accessible!’

When my manuscript seemed to be sufficiently fleshed out, I presented it to my publisher, who without hesitation said: “I’ll take it!” Had he not spoken too quickly? Eight days later, a university professor, the chair of his reading committee, and the Director of the Robotics Section of his publishing house were gathered in his office, each hiding at first behind a paravent of modesty and admitting

<footnote n="97">⁹⁷ The roots of the Peugeot family lie in *Franche-Comté*, they are of Protestant religion, and several family figures studied at the École Centrale de Paris (e.g. Jules, Armand, Robert).</footnote>

<footnote n="98">⁹⁸ SVP is a French B2B expert service and information broker. It associates with the abbreviation of *s'il vous plaît*: 'please'.</footnote>
their lack of competence. In fact, no one at AFRI or SVP knew more about quaternions than the three lines in the dictionary. The Head of the Robotics branch spoke up: “Oh me! My mathematical literacy stops at the level of the *License*!”<fnref n="99" /> Realising that I would not get another word out of him, I turned to the visibly disturbed professor: “You take responsibility for this, don’t you?”, which he repeated vigorously. This reaction said more than all the speeches. The illusion that I had cherished for a moment, of hearing: “It’s really not bad, your thing” collapsed in an instant. A few material details remained to be settled. Never before had Éditions Hermès, specialised in technical or even practical books, ventured into the purely scientific field with so many mathematical symbols. I had to explain the meaning of the word “Nabla”, which is better known in Great Britain than in France, where symbolic calculation remains unknown. The publisher thought it was a good idea not to go into scientific publishing on a single work, and accepted another one, by a very established author. Then I found myself in front of the master typographer, who had disturbed himself to tell me that I was using signs that were not among the 5,000 symbols used in printing! They had to be specially cast. And yet there were the symbols of the card games, hearts or clubs, as well as the cabalistic signs of astrology, the Genii or Jupiter! But the little five-pointed star symbol distinguishing the quaternion from the antiquaternion was not there, nor were a few others.<fnref n="100" />

What I had just done was double or nothing! I didn't need quaternions for my reputation. I was taking a risk with them. Finally, the sale of the book, in the absence of quantity, was honoured by quality. The buyers, few in number, were recruited amongst university professors, and this flattered the pride of my publisher, without contributing much to filling his pocketbook. It allowed him to widen his audience towards scientific horizons.

Having written this book, using personal means, there was none of this that belonged to Peugeot, who had not asked me for anything, except afterwards. It was therefore in a private capacity that I presented a copy of my text to the Seymour Cray France Competition. I knew that the President of the Jury, Mr Aigrain, former Minister of Scientific Research, was above all an Atomist; therefore a study on Quaternions would attract his attention.<fnref n="101" /> This gave me some chances, even if the subject was a little different from the themes proposed for the Competition. However, Peugeot was a customer of Cray computers, and they made the acceptance of my application conditional on written authorisation from my directors, who once again could thus recover my work. However, one of the directors, well aware of the depth of my thoughts, reassured me: “We don’t want to rob you!” When the results were due to be announced, several independent sources told me that I was to receive the first prize. This meant that I would be reimbursed for the purchase of my computer, which I was already thinking of replacing with a more powerful one. However, a short sentence uttered by an engineer from Peugeot bothered me a little, when he announced me the first place – adding: “This will allow us to renegotiate the contract with Cray!” The official award ceremony was postponed for three months, and it was then that I received the fifth prize: “Special Jury Prize” and a sum of money, which did not even cover half the price of my computer. Let’s skip the details and confidences that I was given: people always feel the need to explain options that their conscience does not unreservedly approve. My revered boss became a member of the Jury the following year. And it was he himself who insisted on pointing out to me the ‘merit’ of the winners of this new competition: a ‘first prize out-of-competition’ awarded to his former colleague and fellow student, and another ‘first prize’ of the same amount to the thesis supervisor of his daughter. Why did he insist on justifying these choices to me, when I didn’t ask him anything? A candidate from my year, a candid Normalien, then came to me and said: “I hear it’s a prize reserved for Cray’s clients!” He had been predicted a prize the previous year, and he had got nothing! What candour, after all!

On the subject of the Quaternions, I would like to warmly thank the Peugeot-Citroën company, which enabled me to obtain a particularly exceptional concentration of mind! In the new building, they had allocated me a fairly large office, with opaque partitions, generally reserved for several occupants, but where, despite the regulations, I was alone!

This contrasted with the two previous offices, and even more so with the next one: a cramped and dark square with partial glass partitions, which allowed me to record and learn by heart the numbers of all kinds of machines used throughout the factory, whose references were phoned in all day long to the neighbouring office; an atmosphere particularly conducive to reflection, when on the other side a corridor allows the perpetual passers-by to verify that you are still trying to concentrate. Then I was transferred to Vélizy<fnref n="102" /> in a collective office of about fifty people, arranged according to the latest American standards! This gregarious aspect reminds you at all times that all individuals are equal in law, according to and by virtue of the generous democratic principles of the Republic, unless they are the Republican principles of democracy! In the surroundings, the background noise of so many conversations was such that it was rigorously impossible to be distracted by any one of them, amidst the crackling of all kinds of printers and the various and tainted telephone calls.

<footnote n="99">⁹⁹ The French higher education in that time knew two main degrees, the shorter *License* and the longer *Maîtrise*. The Bologna reforms introduced Bachelor and Master, yet the equivalent to the Bachelor is still called *License* so that it cannot be confused with the school's *Baccalauréat*.</footnote>

<footnote n="100">¹⁰⁰ In his book *Les Quaternions*, de Casteljau introduces a matrix quaternion $q$ and its antiquaternion $q^*$ with a commutative quaternion product $qq^* = q^*q$. The antiquaternion is neither identical with the inverse nor with the conjugate quaternion.</footnote>

<footnote n="101">¹⁰¹ Pierre Aigrain (1924-2002) had studied at École Navale, held a Master's degree from Carnegie Mellon, and finished his PhD at Paris University. He co-founded a research lab on solid state physics at the École Normale Supérieure, and was a promotor of Pierre-Gilles de Gennes (cf. footnote 35) who prepared his thesis in the lab. He also worked with Jacques Friedel, co-author of de Casteljau's very first publication (de Faget de Casteljau and Friedel, 1956). Aigrain made a career in academia as well as in research politics, where he was member of the French research committee DGRST and later became Secretary of State under Raymond Barre (cf. footnote 85) responsible for Scientific Research. Since 1988, he was member of the Académie de Sciences.</footnote>

<footnote n="102">¹⁰² Vélizy, some 20 km outside Paris, had hosted Citroën's technical design centre since the end of the 1960s. After the takeover by Peugeot, more and more departments moved into the newly erected buildings, including the Direction Systèmes d'Information (DSI), but also the commercial divisions. The former Citroën buildings at Quai de Javel in Paris were closed in 1975, the production was moved to a newer factory in Aulnay-sous-Bois. The Javel site turned into the Parc André Citroën, which was opened to the public in 1992.</footnote>
It was in this atmosphere, particularly conducive to meditation, that I set in motion my first reflections on Metric Geometry. There is no doubt that René Descartes' attempt to lay the foundations of Analytical Geometry ended in a flagrant failure! He thought to oppose the treatises on Perspective, or on Projective Geometry, according to the thinking of Desargues; not to mention the odious attitude of the former towards the latter, Analytical Geometry ended in an apotheosis of Projective Geometry: this is the conclusion that must be drawn after Cayley's work on invariants. Analytical Geometry raises the notion of algebraic degree to the level of a dogma. If Descartes had possessed enough genius, he would have quickly realised that his undertaking would very quickly favour the alignment of points, the convergence of lines or other points of the same conic; all this belongs to Projective Geometry. Analytical Geometry had to deal with the graphical representation of the paradoxes of Algebra, mainly the problems of degeneration of curves, a property which recognises that a curve of degree $m+n$ is the juxtaposition of a curve of degree $n$ with another of degree $m$. This is not Geometry!

In Physics or Mechanics, one seeks to describe the relative situation between elements, mutual distances, or angles, and Descartes was aware of this. In his mind, his calculations would define the ideal shape of lenses, to favour the boom of Optics, which the Dutch were in the process of developing.

As always, your thinking immediately turns to endless complications, when the solution is precisely to get rid of the coating that surrounds the jewel. Already with the poles, or the quaternions, it was necessary to highlight the essential detail!

What does nature teach us? Our vision is binocular, which should value the bipolar coordinates, as well as the notion of proximity. For a long time, people were ecstatic in front of the geometric regularity of the wax combs constructed by bees, with their hexagonal symmetry. No wonder! The vision of these insects is carried out as if through a kaleidoscope, using the facets of their eyes! What would have been particularly surprising would have been to see them constructing checkerboards!

The study of bipolar coordinates is not new, but the books say little about it! At the very most, a brief notice at the end of a chapter, written in small print, strongly advises against their use; unless you want to complicate your life unnecessarily! Dead end! As usual, for several months I found myself like Buridan's ass,<fnref n="103" /> which I have never ceased to be, not knowing how to choose between orthogonal circle bundles or homofocal conic systems, the latter having the advantage of being able to be generalised in $E^3$...! However, the answer was there, however, screaming, so blindingly obvious that one did not even realise it! There was a kind of metric duality here, which imposed the simultaneous choice of both at the same time!

The result was a geometrical figure of such admirable aesthetics that it would be immediately deflowered as soon as a line was missing or one more was added. Having had to undergo surgery in early 1996, I sent a canvas of 80 pages to the University of Brunswick, who seemed to be the only ones to pay any attention to it.

This was just in case I would have been unable to regain the level of concentration that is essential to pursue this research. It is therefore more of an enumeration of properties written in a telegraphic style, difficult to follow, rather than a satisfactory writing.

It starts with the simplest curves, from the metric point of view – the straight line, the circle, the focal properties of conics, the diametrical Strophoid, the Cartesian Ovals – to continue with the intersecting forms of aplanatic conjugate mirrors, already considered by Schwarzschild, but I believe with a differential definition. It would have been desirable to add the conformal transformations, linked to the properties of the Quaternions, which I sensed were, at least, confined to the differential domain, as it has been shown that there can be no integral forms of them: this was the unavowed objective of my research.

But let us return to Vélizy! It is difficult for me to put myself in the shoes of the managers of Peugeot or Citroën. It is understandable that they are sufficiently busy with current problems, and that it is already difficult to find the time necessary to apply the improvements developed in the laboratory. This may explain why progress is made in successive stages, carried out in turn by competing companies, to resemble a tennis match, where the essential problem is to catch the ball. In spite of this, the Management could have provided a better welcome to the generalised poles. As for the Peugeot technicians, they spoke of “formes à poils” [hairy shapes], while those of Citroën sought to tone down into “formes à Paul” [Paul's shapes].

Not only had I totally given up hope of taking part in the discussions concerning CAD, but I had also given up all claims concerning it, contenting myself with answering the questions that could be put to me. It was out of disgust that I tried to escape from this cesspool, to occupy myself with thoughts far removed from automotive concerns.

## The trolls

So the bristly hairs on my surfaces emitted Saint Elmo fires, quite easily, and I became quite difficult to approach, when I sensed that my interlocutor had no other aim than to make me mad! For example, by coming to talk to me about hairy surfaces.

When an employee of the Rennes factory phoned me to tell me about the University of Oslo, I wondered what kind of trick was being played on me. I began to procrastinate. What else did they want from me? Well, he really was the brother of Mrs Tom Lyche. It took Tom Lyche a lot of convincing to get me on the plane to Oslo.<fnref n="104" /> I only have to mention that my identity card dated back to 1948, and having lost it, it was a mixed identity and large family card dating from 1942, which I presented to the Town Hall for renewal! Persuading them that it was the truth was quite epic.

<footnote n="103">¹⁰³ Buridan's ass is a paradox whereby a hungry and thirsty donkey, placed between a bundle of hay and a pail of water, would die of hunger and thirst because there was no reason for him to choose one resource over the other. It is said to have been constructed by the French philosopher Jean Buridan (c.1295–c.1358), who studied under William of Ockham (excerpt from Knowles, 2000).</footnote>

<footnote n="104">¹⁰⁴ De Casteljau was invited to the conference on “Curves, Surfaces, CAGD, and Image Processing” in Biri, Norway, which was held in the week of June 20-25, 1991. His French presentation resulted in de Casteljau's first conference publication (1992).</footnote>
It was as if an evil genius was working on me to make me lose all countenance. This, down to those small details of life, which everyone laughs about, except the one to whom it happened. No one wanted to take the risk of translating me, fearing a catastrophe, so as not to be held responsible for it! And so, for the first time, I acquired the right to express myself in French, a privilege that was later extended to the writings of the conference proceedings, subsequently published in the U.S.A.<fnref n="105" />

The previous presentations had made me understand, above all, not to reel off for the fifteenth time the algorithm defining the generalised poles, as I had planned. Even worse, I had never had the opportunity to use transparencies (I swear!) Tom Lyche managed to get them for me the day before, with the necessary markers. I stripped my presentation down to its skeleton, so as to put the emphasis on the claims about my originality! The rest they knew by heart, and had just been repeated to them, as if by a prayer wheel.

The chapter of split pants, however “wrinkle-free, special travel”, or of undesirable digestions, seemed closed to me, when at sunset – late in these regions at the summer solstice – a magnificent rainbow of such beauty as one seldom sees, shone in the sky.<fnref n="106" />

No one would have dared to bet a single kopek on me; Tom Lyche's concern was painful to see. He had spent his time looking for a translator, trying to decide whether it would be better to choose one in the language of departure or in that of arrival; and the last to be found, an American, had promptly found a mad desire to visit the fjords!

With slow gestures, I took out of his briefcase the famous ‘Flying Saucer’, “The suitcase is English, but its content is French!” Then evoking the Blossoming: “You all know this flower, lovingly cultivated in the gardens of California; we will now see this flower, in its wild state in nature, moreover in danger of extinction!” And finally, I limited myself to showing the transition between an interpolation and a smoothing, thanks to the subtle play of the degree of restitution, which, associated with the algebraic degree and the degree of continuity, constituted the characteristic of any algebraic approximation of an arc.

Comparing the happy face of the French, the laughing mouth up to the ears and the slightly frowning face of our American friends, I knew that I had exceeded the wildest expectations, which was amply confirmed by various comments at the exit. I had indeed become the “Professor of Professors”, for one time only! The following month, during a mini-congress in the U.S.A., the specialists scrutinised my presentation. This warranted a superb report, only two months after Biri(!) with a whole page of dedication, written jointly by two researchers, all written in French, a language that neither of them had any practice in!<fnref n="107" />

These unforgettable moments are engraved in the memory with all the precise details, alongside the first *colle* of Pham Tinh Quat, or the machining of the first pole form. However, after a few moments' reflection, one realises to have come so close to the abyss that there was no longer the thickness of a sheet of cigarette paper, as a safety gap; it feels, as if guided by some invisible thread, whose subtle presence we sense, and which we only have to follow. What I have done in my career, I had to do, without ever having really prepared it in advance. No, it is not at all what your entourage imagines, who tended to equate your success with luck, particularly insolent with a game of chance!

## Shoo!

A superb drakkar of honour,<fnref n="108" /> crafted by a pewter goldsmith's work, was there to witness the distinction I had been awarded in Norway. What a welcome Citroën would give me, who had paid for this trip, while also fearing that my inexperience could be the cause of some mishaps. I couldn't get the idea out of my mind that they were going to take advantage of it to get rid of me. This became a reality within the first few moments of my return: A man from the Personnel Department wanted to meet me. He then asked me to claim my pension rights. The inscriptions in runic characters, but in French, on the drakkar, disarmed him; my interlocutor, a former Citroën employee, did not insist! A few years earlier, I had already been trained to resist these attempts at forced persuasion. Having had to undergo six three-hour sessions, which were very hard on my morale, I had only arrived at a state of calmest sea after having said “I am willing to sign, but on the condition that someone wants to tell me what I am accused of!” The only thing that came to my mind at that time had nothing to do with my work, and concerned a decision of the town where I lived, which I had not managed to reverse. It had nothing to do with Citroën's car business, and at most concerned the comfort of its management.

Every morning, before a new assault, my ‘boss’ called me into his office for an hour, and again in the afternoon, to mine for a good quarter of an hour. The donkey would simply swing his head, which would cause his ears to twitch. Let there be no mistake about it, when a donkey dedicates his “eeyore!” to you. It is the man that he is addressing by telling you what he thinks of you. “Ah, how stupid he is! Like a human!” However, the tally-ho, the end of the hunt, was approaching. So I was dismissed by an individual, hired for this role, who knew it was him or me. In exchange for the bonus, I had to sign a document that waived my right to bring

<footnote n="105">¹⁰⁵ De Casteljau's conference papers (1994, 1995, 1997), as well the ones published after this autobiography (2000, 2001a, 2001b), were all published in French, partly as a copy of his handwritten manuscript. His first contribution in Biri (1992) was translated by Hans-Peter Seidel, a subsequent article on the history of CAD (1993) was translated by Michel Menard and Jack Gregory, whilst the translator of his Dagstuhl presentation (1998) remains unknown.</footnote>

<footnote n="106">¹⁰⁶ Unfortunately, his suit trousers tore, but could be temporarily sewn up. Tom Lyche adds (personal communication, February 2023): “On Midsummer day evening June 23 we had rented *Skibladner*, the world’s oldest paddle steamer to see bonfires in the evening. However it was foggy so we did not see a thing.” He also recalls that de Casteljau did not feel too well.</footnote>

<footnote n="107">¹⁰⁷ SIGGRAPH 1991 was held in Las Vegas 18 July - 2 August 1991, where Hans-Peter Seidel organized a course on ‘Blossoming - The New Polar-Form Approach to Spline Curves and Surfaces’, together with Phillip Barry, Ron Goldman, and Lyle Ramshaw. Afterwards, Ramshaw and Seidel sent a copy of the course notes together with a dedication in French to de Casteljau, “*une expression modeste de notre estime pour l’homme qui a introduit les formes polaires à CAGD*” – “a modest expression of esteem to the man who introduced polar forms to CAGD”.</footnote>

<footnote n="108">¹⁰⁸ A drakkar is a Viking longship. Its data set was subject to a contest at the Biri conference, as Tom Lyche mentioned in a personal communication to the annotating author in February 2023.</footnote>
the 'dispute' before an authorised court. Not being much of a litigator, this seemed an unnecessary precaution, but it convinced me that they were not quite sure of their right!

Free!

From then on, my situation was free of all constraints. Retirement finally allows you to work, with no targets to meet, no deadlines to satisfy the whims of a captain of industry. No school imposed its directives on me, and especially not the programs drawn up by a ministry. In return, I made sure to bring a touch of originality to my active participation in CAD, CAGD or other Geometric Modelling conferences. What the participants expect is not the seventh presentation of a well-known algorithm, nor the third variant of a solution method, if none of them brings decisive elements.

Industrial experience quickly taught me that mathematical culture is measured, above all, by the quality of the workpiece being machined. If people start saying about you: "He's a scientist!", don't take any pride in it. It is high time for you to provide material proof of your practical efficiency. It is up to you to make the effort to adapt to the language of the practitioner, and not the other way round, when he talks to you about radii (arcs of a circle), double slope straight lines or an egg-shaped (spherical) cutter.<fnref n="109" /> Mathematical culture allows you to dismiss anyone who pretends to dazzle you with his knowledge of elliptic functions, tensors, the Banach-Tarski-Robinson theorem, or Stieltjes integrals, as long as he does not bring tangible proof of their effectiveness in a practical implementation.

The adventure of the cam is an example of the reflections that can inspire a streamlining of a profile. The failure of orthogonal functions stems from the fact that smoothing in mathematics is seen as minimising the squared errors of a function with its approximation. This results in the Gibbs phenomenon. In physics, a distinction is made between the variables of voltage and current, and there is a perpetual exchange of energy between the potential energy of one and the kinetic energy of the other. A satisfactory smoothing requires the simultaneous regularisation of the function and its derivative. The analysis I gave in Ulvik (Norway) indicates a solution.<fnref n="110" /> The metric duality, which associates the point and tangential viewpoints, is in the domain of Metric Geometry.

No one in 1958 could have foreseen that one day mathematicians would be looking for the equation of an artistic or aesthetic form. Until then, they limited themselves to drawing the curves representing the equations, while being fascinated, above all, by the singularities. Now the singularity has become an obstacle. Without the insistence of Monsieur de la Boixière, and my reluctance to leave Citroën without having been able to offer them a tangible solution, what would have been the history of CAD?

To conclude, I would like to quote Mr de la Boixière's aphorism: "Flattery makes you drowsy, hostility makes you stronger, but there is nothing worse than indifference."

<table><tr><td rowspan="7">Congresses</td><td>Biri (Norway)</td><td>1991</td><td>Degree of Restitution and Characteristics</td></tr><tr><td>Oberwolfach (D)</td><td>1992</td><td>Logic of the Parameter</td></tr><tr><td>Chamonix (F)</td><td>1993</td><td>Focal Splines</td></tr><tr><td>Ulvik (N)</td><td>1994</td><td>Aesthetic Curves and Profiles versus Orthogonal Function (Real Life Story)</td></tr><tr><td>Schloß Dagstuhl</td><td>1996</td><td>Point Ω: Projective Intersection of Two Conics</td></tr><tr><td>Chamonix (F)</td><td>1996</td><td>Machining Tolerance</td></tr><tr><td>Formes à pôles</td><td>1985</td><td></td></tr><tr><td rowspan="3">Books</td><td>Les Quaternions</td><td>1987</td><td></td></tr><tr><td>Le Lissage</td><td>1990</td><td></td></tr></table>

<table><tr><td rowspan="3">Unpublished Studies<sup>111</sup></td><td>Euclid's algorithm in several variables (Arithmetics)</td></tr><tr><td>Numbers, consisting of very small prime factors: (Musical) Scales</td></tr><tr><td>Metric Geometry: A definition attempt</td></tr></table>

## 7. Post scriptum (translation)

End of March 1997

This is the story of my adventure, necessarily incomplete and seen through the distorting prism of my sensitivity. Often, I hesitated between an easy-to-use all-purpose word (e.g.: "chose" [thing], "faire" [to do]...) and an accurate word, of uncommon use, with the same concern about the idioms that can cause difficulties to a translator.

<footnote n="109">¹⁰⁹ De Casteljau mentioned further expressions (personal communication to the author, December 2001): On vous parle de "rayons" (circles), chamfreins (angle abbattu), de "droite en double pente" (non parallèle à un projection principale), d'anse de panier (ellipse en arcs de cercles), d'angle dépouille (pour que ça démoule), de fraisage "en avalant" ou on vous dit "ça file" (il n'a pas de bosse ou de plat). They talk to you about "radii" (circles), chamfers (flattened angle), "double slope straight line" (not parallel to a main projection), basket handle (ellipse in arcs of circles), draft angle (so that it comes out of the mould), "swallowing" milling or they tell you "it works" (it doesn't have a hump or a flat).</footnote>

<footnote n="110">¹¹⁰ The Ulvik presentation reviews the mathematical details of the cam design mentioned in the section A New Eye of this autobiography (de Faget de Casteljau, 1995).</footnote>

<footnote n="111">¹¹¹ Unpublished until 1997, when de Casteljau compiled the list.</footnote>
<table><caption>**Table 1**
Timeline, as assumed by the translating author.</caption><thead><tr><th>Section</th><th>Time</th><th>Event</th></tr></thead><tbody><tr><td>The Roots</td><td>1930-1946</td><td>Seven years at school after 1939</td></tr><tr><td>The Student</td><td>1949-1951</td><td>Two years preparatory courses</td></tr><tr><td>The Normalien</td><td>1951-1958</td><td>Four years ENS, three years military service</td></tr><tr><td>A Corsair</td><td>1958</td><td>Start on April 1</td></tr><tr><td>Utopia</td><td>1958-1961</td><td>First machined part (Poitou, 1989)</td></tr><tr><td>The Bodywork</td><td>1962</td><td>Flying Saucer (de Faget de Casteljau, 1992)</td></tr><tr><td>A New Eye</td><td>1962</td><td>DS 21</td></tr><tr><td>The Design</td><td>mid-1960s</td><td>Strike of wood modelers</td></tr><tr><td>"Attention!"</td><td>early 1970s</td><td>Peugeot takeover</td></tr><tr><td>The Lion</td><td>1975-1979</td><td>Boehm discovery, pentecost reference</td></tr><tr><td>The Roar!</td><td>early 1980s</td><td>Turbine blade (de Faget de Casteljau, 1992)</td></tr><tr><td>A Glimmer</td><td>1980-1981</td><td>Late Barre presidency</td></tr><tr><td>The Quake</td><td>1981</td><td>Chausson restructured towards MCA</td></tr><tr><td>Normalisation</td><td>1982-1985</td><td>Manuscript and first book (de Faget de Casteljau, 1985)</td></tr><tr><td>To Merit!</td><td>1985</td><td>CEO manager of the year</td></tr><tr><td>The Quaternions</td><td>1981-1987</td><td>Manuscript and second book (de Casteljau, 1987)</td></tr><tr><td>The Trolls</td><td>1991</td><td>Biri conference (de Faget de Casteljau, 1992)</td></tr><tr><td>Shoo!</td><td>1991-1992</td><td>Oberwolfach; 40 years of social security contribution</td></tr><tr><td>Free!</td><td>1994</td><td>Ulvik</td></tr></tbody></table>

Should I make the text lively, or limit myself to the dryness of an enumeration? By opting for the first solution, the longer the text, the more credible it is. In spite of a fertile imagination, there are details that cannot be invented! Having had the opportunity to recount certain episodes to a science journalist, he kept whispering "I can't write that!"

So when I quote a proper name in plain language, it is more to pay him a deserved tribute, even and especially if my critics do not spare him!<fnref n="112" />

Is it necessary to add, that I do not wish in any way that a premature dissemination, especially in French, could provoke malicious reactions from people who may feel defamed. As far as possible, overly political (hum!), philosophical, or religious issues have been excluded, as have certain matters of big money (hum!) or family.

For a long time, I had been thinking of writing an outline of my memoirs, which could have served as a basis for more elaborate essays later on. Your letter came at the right moment to remind me of this, which made me decide to start working on it, longer than I had imagined.

This text is still much more than a CV and I hope that its length will not wear on your patience, and that it will give you complete satisfaction.

## CRediT authorship contribution statement

Andreas Müller: Writing – original draft, Writing – review & editing, Investigation, Methodology.

## Declaration of competing interest

The author declares that he has no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Appendix A. Supplementary material

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.cagd.2024.102278.

## References

Boehm, W., 1993. An affine representation of de Casteljau's and de Boor's rational algorithms. Comput. Aided Geom. Des. 10, 175-180.

Bouffard-Vercelli, R., 1971-1975. Courbes à pôles. In: Encyclopédie des Métiers (1971-1975), vol. I, pp. 257-259.

Bourdieu, P., 1981. Epreuve scolaire et consécration sociale. Actes Rech. Sci. Soc. 39, 3-70.

Bouzitat, J., 1951. Méthodes générales de Calcul des harmoniques d'une fonction périodique. Application à l'analyse harmonique des mouvements commandés par certaines cames. In: SIA (Ed.), Conférence SIA et Centre de Documentation Universitaire. Paris.

Clynes, T., 2016. Where Nobel winners get their start. Nature 538, 152.

de Faget de Casteljau, P., Friedel, J., 1956. Étude de la résistivité et du pouvoir thermoélectrique des impuretés dissoutes dans les métaux nobles. J. Phys. Radium 17, 27-32.

<footnote n="112">¹¹² Due to the intention of Paul de Casteljau, these annotations do not disclose the real names of involved persons either, although partly known to the translator. However, the reader will find a timeline with references to the assumed years for each section, see Table 1.</footnote>
de Casteljau, P., 1959. Enveloppe Soleau 40.040. Document no P 2108, registered at INPI Paris. Written in the department *Outillage Méthodes Calcul* at S.A. André Citroën.

de Casteljau, P., 1963. Courbes et Surfaces à Pôles. Document no P.4147, registered with the bailiff. Written in the department *Outillage Méthodes Calcul* at S.A. André Citroën.

de Faget de Casteljau, P., 1983. Formes à Pôles: Courbes et Surfaces à Pôles. Manuscript of 34 pages. Possibly submitted to the Académie des Sciences.

de Faget de Casteljau, P., 1985. Formes à Pôles. Mathématiques et CAO, vol. 2. Hermès, Paris.

de Faget de Casteljau, P., 1986. Shape Mathematics and CAD. Mathematics, vol. 2. Kogan Page, London.

de Casteljau, P., 1987. Les Quaternions, Traité des Nouvelles Technologies, Série Mathématiques appliquées. Hermès, Paris.

de Faget de Casteljau, P., 1990. Le Lissage. Traité des Nouvelles Technologies, Série Mathématiques appliquées. Hermès, Paris.

de Faget de Casteljau, P., 1992. Polynomials, Polar forms, and InterPOLation. In: Lyche, T., Schumaker, L.L. (Eds.), Mathematical Methods in Computer Aided Geometric Design II. Academic Press Professional Inc., pp. 57–68. Translated by Hans–Peter Seidel.

de Casteljau, P., 1993. Polar forms for curve and surface modeling as used at Citroën. In: Piegl, L. (Ed.), Fundamental Developments of Computer-Aided Geometric Modeling. Academic Press, pp. 1–12.

de Faget de Casteljau, P., 1994. Splines Focales. In: Laurent, P., le Méhauté, A., Schumaker, L. (Eds.), Curves and Surfaces in Computer Aided Geometric Design II. AK Peters, pp. 91–103.

de Faget de Casteljau, P., 1995. Courbes et Profils Esthétiques contre Fonctions Orthogonales (Histoire Vécue). In: Dæhlen, M., Lyche, T., Schumaker, L.L. (Eds.), Mathematical Methods for Curves and Surfaces. Vanderbilt University Press, pp. 73–82.

de Faget de Casteljau, P., 1997. La Tolérance d'Usinage chez Citroën dans les Années (19)60. In: le Méhauté, A., Schumaker, L.L., Rabut, C. (Eds.), Curves and Surfaces with Applications in CAGD. Vanderbilt University Press, pp. 69–76.

de Casteljau, P., 1998. Intersection methods of convergence. In: Farin, G., Bieri, H., Brunnett, G., De Rose, T. (Eds.), Geometric Modelling. Springer Vienna, Vienna, pp. 77–80.

de Faget de Casteljau, P., 1999. De Casteljau's autobiography: my time at Citroën. Comput. Aided Geom. Des. 16, 583–586. https://doi.org/10.1016/S0167-8396(99)00024-2. Translation based on Müller and Schultze (1995).

de Faget de Casteljau, P., 2000. Intersections et Convergence. In: Laurent, P., Sablonnière, P., Schumaker, L.L. (Eds.), Curve and Surface Design: Saint-Malo 99. Vanderbilt University Press, pp. 9–15.

de Faget de Casteljau, P., 2001a. Au delà du Nombre d'Or. Rev. Int. CFAO Inf. Graph. 16, 19–31.

de Faget de Casteljau, P., 2001b. Fantastique strophoïde rectangle. Rev. Int. CFAO Inf. Graph. 16, 357–370.

Encyclopédie des Métiers (Ed.), 1971–1975. La Carrosserie, vol. I. Librairie du Compagnonnage, Paris, pp. 1971–1975.

Faux, I., Pratt, M., 1979. Computational Geometry for Design and Manufacture. Halsted Press, Chichester.

Fowler, A.H., Wilson, C.W., 1966. Cubic Spline, a Curve Fitting Routine. AEC Research and Development Report Y-1400. Union Carbide Co.: Nuclear Division, Oak Ridge, TN. https://www.osti.gov/servlets/purl/4524486.

Haase, J., 1870. Zur Theorie der ebenen Curven n-ter Ordnung mit $\frac{(n-1)(n-2)}{2}$ Doppel- und Rückkehrpunkten. Math. Ann. 2, 515–548.

Knowles, E. (Ed.), 2000. The Oxford Dictionary of Phrase and Fable. Oxford University Press, Oxford, New York.

Krautter, J., Parizot, S., 1971. Système d'aide à la définition et à l'usinage de surfaces de carrosserie (SADUSCA). Ing. Automob. 44, 581–587.

Lambru, S., 2016. Oltcit ou la Citroën fabriquée en Olténie. https://www.rii.ro/fr/fr/oltcit_ou_la_citroen_fabriquee_en_oltenie-2542099. (Accessed 4 May 2023).

Langins, J., 1980. Sur la première organisation de l'Ecole polytechnique. Texte de l'arrêté du 6 primaire an III. Rev. Hist. Sci. XXXIII, 289–313.

Laurent, P.-J., Sablonnière, P., 2001. Pierre Bézier: an engineer and a mathematician. Comput. Aided Geom. Des. 18, 609–617. https://doi.org/10.1016/S0167-8396(01)00056-5.

Müller, A., 1995. Neuere Gedanken des Monsieur Paul de Faget de Casteljau. Diploma thesis. TU Braunschweig.

Müller, A., 2024. A Tour d'Horizon of de Casteljau's Work. Comput. Aided Geom. Des. Submitted for publication.

Müller, A., Schultze, P., 1995. Paul de Faget de Casteljau – Meine Zeit bei Citroën. Appendix B of Müller's Diploma Thesis. Translation of a French autobiographical text of 1992 into German.

N.Y. Times, 1968. De Gaulle drives a pun into Citroën deal. N.Y. Times, 49–58.

Norton, R.L., 2020. Cam Design and Maufacturing Handbook, 3 ed. Norton Associates LLC, Naples FL.

Paraponaris, C., 2019. Vers une monopolisation de la conception industrielle. L'Exemple de la market place de Dassault Systèmes. In: 36ème Colloque INFormatique des ORganisations et Systèmes d'Information et de Decision. INFORSID, pp. 247–268.

Parizot, S., 1971–1975. Application à l'étude et à la réalisation de la carrosserie de la Citroën GS. In: Encyclopédie des Métiers (1971–1975). pp. 247–268.

Plévert, L., 2011. Pierre-Gilles de Gennes: A Life in Science. World Scientific Publishing, Singapore.

Poitou, J.-P., 1989. Trente ans de CAO en France ou les petits enfants de Gaspard Monge. Hermès, Paris.

Porch, D., 2016. Breaking the Gustav line. Q. J. Mil. Hist. 28, 46–55.

Pouget, J.-P., Levelut, A.-M., Sadoc, J.-F., 2019. André guinier: local order in condensed matter. C. R. Phys. 20, 725–745. La science en mouvement 2: de 1940 aux premières années 1980 – Avancées en physique.

Ramshaw, L., 1987. Blossoming: A Connect-the-Dots Approach to Splines. Research Report #19. Digital Systems Research Center, Palo Alto.

Ramshaw, L., 1989. Blossoms are polar forms. Comput. Aided Geom. Des. 6, 323–359.

Ramshaw, L., 1991. The de Casteljau approach to quasi-interpolants. In: Blossoming: The New Polar Form Approach to Spline Curves and Surfaces, 18th International Conference on Computer Graphics and Interactive Techniques: Las Vegas Convention Center. 28 July–2 August. In: Course Notes: SIGGRAPH 1991, vol. 26. ACM SIGGRAPH, pp. 6.1–6.22.

Ross, D.T., 1978. Origins of the APT language for automatically programmed tools. ACM SIGPLAN Not. 13, 61–99.

Sunergy, 2019. We celebrate our 50th anniversary! https://www.sunergybattery.com/news/we-celebrate-our-50th-anniversary.

Xia, Z.C., Cao, J., 2019. Springback. In: CIRP Encyclopedia of Production Engineering, pp. 1133–1138.