Culture
Words by Véronique Vienne
Maths vs. Machines
From cars to curves to computer graphics — the making of the de Casteljau algorithm.
From cars to curves to computer graphics — the making of the de Casteljau algorithm.
This 1942 photograph shows the interior of the Ford Motor Bomber factory in Michigan.In the past, to draw curves, engineers used splines. These flexible curves were held in place with lead weights called “ducks. These awkward tools required you kneel or lie down on your blueprint Credits: HB-07074-G, © Chicago History Museum, Hedrich-Blessing Collection
Some ideas are so simple, so intuitive, anyone could have devised them — or so it seemed. Such is the case of an elegant little algorithm developed by a 28-year-old French mathematician, Paul de Faget de Casteljau. His invention, originally conceived for the modeling of car parts, spearheaded the development of computer graphics. Oddly enough, he never got proper recognition for it. The credit usually goes to Pierre Bézier, an engineer, a much older man who was trying to solve the same problem at the same time, but who came up with a similar solution three years after de Casteljau.
Bézier didn’t “steal” de Casteljau’s idea. He owed his fame only to circumstances. Yet a small difference between the two men might have played a role in the way they are remembered: it’s the difference between a mathematician and an engineer.
Bézier didn’t “steal” de Casteljau’s idea. He owed his fame only to circumstances. Yet a small difference between the two men might have played a role in the way they are remembered: it’s the difference between a mathematician and an engineer.
Left page: Principle of curve drawing. Sketch by the author. A spread from a sketchbook used during the research process of this article. The drawings are geometric representations of the de Casteljau’s algorithm.
Right page: Flavio Bertoni in a workshop. Courtesy Amicale Flaminio Bertoni. The vintage photographs feature different stages of the modeling of the body of two cars, the Citroën Traction Avant (top) and the Citroën Ami 6 (bottom). In the background, one can see a prototype of one of Citroën most iconic cars, the 2CV.
Right page: Flavio Bertoni in a workshop. Courtesy Amicale Flaminio Bertoni. The vintage photographs feature different stages of the modeling of the body of two cars, the Citroën Traction Avant (top) and the Citroën Ami 6 (bottom). In the background, one can see a prototype of one of Citroën most iconic cars, the 2CV.
In 1958, Citroën, the automaker, had hired de Casteljau. Primitive computers had recently been introduced in the plant to help modernize the manufacturing of car parts, but someone had to translate construction blueprints into mathematical equations. Initially, the computers were not suppose to compute complex shapes but simply to produce the coded information necessary to drive milling machines. It was de Casteljau’s job to turn the complex profiles of hoods, door panels and dashboards into numbers.
At first, he didn’t have a clue how to proceed. “I was overcome by an attack of dizziness,» he admitted in a short memoir describing his time at Citroën [1]
1. De Casteljau’s autobiography: My Time at Citroën, from Computer Aided Geometric Design Journal, Volume 16, Issue 7, August 1999, pp 583-586
2. Paul de Casteljau made fun of what he called the ÑAQUA problem solving method of his coworkers. It is a phonetic transcription of Il n’y a qu’à (just fudge it) — a favorite expression of Citroen model makers.
« There was a rumor going round that a university degree was not, compared to good practical workshop experience, a fruitful investment for a man of my age, » he quipped with the kind of wry humor he developed as an isolated white-collar employee in a blue-collar world.
De Casteljau’s task didn’t look like much of a challenge at first. It only affected that small percentage of a car’s body parts that couldn’t be described by traditional geometry. The pieces were often only the ”size of a postal stamp” but “they were peppered with characteristics which contradicted any kind of mathematics,” he complained. But complaints didn’t interest his employers: they wanted expediency.
Flavio Bertoni in a workshop. Courtesy Amicale Flaminio Bertoni. 
This undated photograph shows a corner of the studio of the famous car designer Flaminio Bertoni doing some preliminary research on the shape of the Ami 6. Before the invention of the de Casteljau’s method for drawing curves, engineers the world over used splines, flexible curves held in place with lead weights called “ducks” because of their shape. These awkward drafting tools required you kneel or even lie down on your blueprint. 
The Breakthrough
De Casteljau was teased around the office for being a “polytechnicien” — a graduate of the prestigious French elitist École Polytechnique engineering school. As it turned out, the prefix “poly”, meaning “many”, was key to the solution he was searching. It was staring him in the face, but it took a slightly disparaging remark from his boss, Monsieur de la Boixière, to nudge the idea out of his brain. As de Casteljau was trying to explain to him the meaning of the mathematical terms “polynomial”, “polar forms”, and “interpolation,” the impatient supervisor interrupted, saying that he didn’t need a “pole-technician” to tell him where to find the North pole.
That sneering remark triggered a breakthrough — the kind of breakthrough only mathematicians can appreciate. For the rest of us, suffice it to say that polynomials are mathematical expressions that feature more than one variable [3]. What de Casteljau envisioned at that moment was how to use a particular kind of polynomials, the “Bernstein polynomials”, to derive curves from the geometry of their variable tangents.
In other words, de Casteljeau had the brilliant idea of going off on a tangent. And indeed, the computer-assisted curves he was then able to generate from that point on matched exactly the drawings submitted by the Citroën designers. Smooth and seamless, the curves could have been drawn by a human hand.
Originally de Casteljau had been given three months to figure out how to “express components parts by equations.” Three months “before obtaining admission to participate in the much more serious work of the Research Department.” But, as he wrote, “these three months lasted for more than thirty years!”
His job was to translate existing drawings into numerical data. Period. At no point during the 30 years of his employment at Citroën was he expected to influence the design of the auto body parts or “to change it only slightly”. His role was to implement, not interfere. At Citroën, numbers were at the service of the hand, not the other way around.
That sneering remark triggered a breakthrough — the kind of breakthrough only mathematicians can appreciate. For the rest of us, suffice it to say that polynomials are mathematical expressions that feature more than one variable [3]
3. A polynomial is a monomial that has more than one variable. Or example, “5x” is a mono- mial, while “5x+z6” is a polynomial.
In other words, de Casteljeau had the brilliant idea of going off on a tangent. And indeed, the computer-assisted curves he was then able to generate from that point on matched exactly the drawings submitted by the Citroën designers. Smooth and seamless, the curves could have been drawn by a human hand.
Originally de Casteljau had been given three months to figure out how to “express components parts by equations.” Three months “before obtaining admission to participate in the much more serious work of the Research Department.” But, as he wrote, “these three months lasted for more than thirty years!”
His job was to translate existing drawings into numerical data. Period. At no point during the 30 years of his employment at Citroën was he expected to influence the design of the auto body parts or “to change it only slightly”. His role was to implement, not interfere. At Citroën, numbers were at the service of the hand, not the other way around.
DEFINITION OF A TANGENT
Tangents are those straight lines that hug curves but do not cross them. They constitute an invisible field force that shapes the contours of a curve. By changing the direction of the tangents, you can change the shape of the curves. De Casteljau developed an algorithm based on this principle.He proposed to teach computers how to shape curves by changing the angle of their tangents.
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Citroën vs. Renault
The situation was different at Renault, Citroën’s main competitor, where management style encouraged innovations. Pierre Bézier had been hired there to facilitate the job of the car designers and modelers. His invention was supposed to “assist” designers. What he proposed had to be easy enough to be adopted by all people in the design department, even those with no number sense.
The Citroën and Renault plants were located less than five kilometers from each other, yet their corporate cultures were miles apart. Citroën, a privately held enterprise, was in the southwest corner of 15th arrondissement. Renault, a state-owned automobile company, was headquartered on an island in the middle of the Seine, in Boulogne-Billancourt. While at Citroën fabrication secrets were closely guarded, at Renault, transparency was the rule. Whereas de Casteljau remained anonymous, Bézier was allowed to publish the result of his research and share his findings with colleagues in the industry. In all likelihood, corporate culture is the main reason why de Casteljau is but a footnote to Bézier’s name.
But there is more to it. The differences between the two men are less striking than their similarities. The engineer and the mathematician both came up with conclusions that were not only identical but rested on the same intuition, namely that curves are shaped by external forces, not internal ones.
The Citroën and Renault plants were located less than five kilometers from each other, yet their corporate cultures were miles apart. Citroën, a privately held enterprise, was in the southwest corner of 15th arrondissement. Renault, a state-owned automobile company, was headquartered on an island in the middle of the Seine, in Boulogne-Billancourt. While at Citroën fabrication secrets were closely guarded, at Renault, transparency was the rule. Whereas de Casteljau remained anonymous, Bézier was allowed to publish the result of his research and share his findings with colleagues in the industry. In all likelihood, corporate culture is the main reason why de Casteljau is but a footnote to Bézier’s name.
But there is more to it. The differences between the two men are less striking than their similarities. The engineer and the mathematician both came up with conclusions that were not only identical but rested on the same intuition, namely that curves are shaped by external forces, not internal ones.
“Just a tool”
However, Bézier was a better communicator than de Casteljau. His terminology to describe his invention was user-friendly, with terms like “handles” and “nodes” instead of “ variables” or “control points” — algebraic expressions used to demonstrate de Casteljau’s algorithm. Ultimately, this stylistic difference helps explain why Bézier curves are the preferred formula in graphics programs, from PostScript and TrueType to Adobe Illustrator and Adobe Flash.
Also critical are considerations regarding the role of computer in the creative process. At Renault, Bézier had enough influence to develop the UNISURF application, a surface system that made it possible for each step of the styling, modeling, drafting, and tool-making process to be computer-assisted. This approach was considered radical at the time — a total departure from a tradition upheld at Citroën where people were in control, and computers were treated like mere servants.
As a result, Bézier, acclaimed as the inventor of his eponym curves, is also mentioned as one of the pioneers of CAD technology — Computer Assisted Design, a term that suggests that computers have creative capabilities of their own. In point of fact, today, a majority of designers have adopted computers as full-fledged partners, even though some are holding out against the trend, claiming that the intelligent machine on their desk is “just a tool.”
Also critical are considerations regarding the role of computer in the creative process. At Renault, Bézier had enough influence to develop the UNISURF application, a surface system that made it possible for each step of the styling, modeling, drafting, and tool-making process to be computer-assisted. This approach was considered radical at the time — a total departure from a tradition upheld at Citroën where people were in control, and computers were treated like mere servants.
As a result, Bézier, acclaimed as the inventor of his eponym curves, is also mentioned as one of the pioneers of CAD technology — Computer Assisted Design, a term that suggests that computers have creative capabilities of their own. In point of fact, today, a majority of designers have adopted computers as full-fledged partners, even though some are holding out against the trend, claiming that the intelligent machine on their desk is “just a tool.”
HOW TO SHAPE A CURVE
Bézier’s concept was to represent a curve as the intersection of two elliptic cylinders defined inside a parallelepiped. Altering this parallelepiped would then result in alterations of the curve.
Bézier envisioned shaping curves by grabbing and moving the corners of a virtual box containing them. De Casteljau’s solution was comparable — though a little bit simpler. To shape curves, all you had to do is manipulate their tangents. Without ever consulting each other, the two men agreed: in order to handle complex curves, you first must design a handle for them.
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Ten lines
In his autobiography, de Casteljau expresses his frustration in a light, self-deprecating tone. In spite of his banter, one gets the feeling that his interaction with other employees at Citroën was always on a confrontational mode. The work environment did not support individual idiosyncrasies. Monsieur de la Boixière used to say that flattery makes you drowsy while hostility makes you strong. “Fortunately, there were only very few flatterers at Citroën!” de Casteljau noted with reversed irony.
A brilliant mathematician, he remained to the end a misfit among the Citroën car designers, engineers, technicians, modelers, draftsmen and machinists. As he tells it “I was greeted with a sonorous ‘The plans are wrong’ instead of ‘Good morning’ in all the places where I dared to enter.” His colleagues were quick to blame him when something went wrong, often expressing misgivings about the way he transformed their beautiful prototypes and elaborate blueprints into cryptic strings of equations. “My refusing to adopt certain disciplines was interpreted as laziness,” he writes. “However, it was nothing but some kind of rejection of sterile knowledge. [4]”
Unlike Bézier, who gave lectures at conferences and seminars, de Casteljau was a man of a few words. “My specialty [was] to demonstrate something in ten lines for which others filled sixty pages,” he explained.
Ten lines are all he needed to plot a curve through a series of elementary linear interpolations between control points. So elementary indeed that de Casteljau’s supervisors, upon watching the demonstration of his algorithm, exclaimed: “These polynomials are so simple that anyone could have invented them!”
Child’s play? So is E=mc2. The fact that you can understand this equation does make it less significant.
Likewise, the geometric expression of de Casteljau’s algorithm is a thing of effortless beauty. It allows you to trace the most efficient curve inside an area defined by four points. All you need to do is connect these four points (though 5 or 6 points give more striking results) and mark the middle of each segment. In turn, you connect these midpoints with another round of straight lines. You repeat the process, joining together midpoints, maybe using a different color for each layer. Soon, you begin to see the profile of a crescent. It is formed by a web of tangents that progressively fan out into a graceful arc on the page.
An elegant cluster of consecutive iterations, it is as compelling as the eye of a storm.
A brilliant mathematician, he remained to the end a misfit among the Citroën car designers, engineers, technicians, modelers, draftsmen and machinists. As he tells it “I was greeted with a sonorous ‘The plans are wrong’ instead of ‘Good morning’ in all the places where I dared to enter.” His colleagues were quick to blame him when something went wrong, often expressing misgivings about the way he transformed their beautiful prototypes and elaborate blueprints into cryptic strings of equations. “My refusing to adopt certain disciplines was interpreted as laziness,” he writes. “However, it was nothing but some kind of rejection of sterile knowledge.
4. De Casteljau retired from Citroen in 1989 and became active in publishing. He is the author of Courbes à poles (1959) and Surfaces à poles (1963).
Unlike Bézier, who gave lectures at conferences and seminars, de Casteljau was a man of a few words. “My specialty [was] to demonstrate something in ten lines for which others filled sixty pages,” he explained.
Ten lines are all he needed to plot a curve through a series of elementary linear interpolations between control points. So elementary indeed that de Casteljau’s supervisors, upon watching the demonstration of his algorithm, exclaimed: “These polynomials are so simple that anyone could have invented them!”
Child’s play? So is E=mc2. The fact that you can understand this equation does make it less significant.
Likewise, the geometric expression of de Casteljau’s algorithm is a thing of effortless beauty. It allows you to trace the most efficient curve inside an area defined by four points. All you need to do is connect these four points (though 5 or 6 points give more striking results) and mark the middle of each segment. In turn, you connect these midpoints with another round of straight lines. You repeat the process, joining together midpoints, maybe using a different color for each layer. Soon, you begin to see the profile of a crescent. It is formed by a web of tangents that progressively fan out into a graceful arc on the page.
An elegant cluster of consecutive iterations, it is as compelling as the eye of a storm.
Initially published in 10/2017 for Minotaur type specimen, ISBN 9791093578057.
