An Ode to the Bezier Curve
How a Mathematical Equation Can Shape Modern Art Aesthetics for the Masses
Changes in the course of art history have been catalyzed by cultural revolutions and complete shifts in human thought; rarely are they attribute to a mathematical equation.
But arguably, a 4 variable algorithm has sculpted the modern-day aesthetics of illustrative design. From cereal boxes to the gallery space, it facilitates an aesthetic of simple shapes, highly-decisive color palettes, and a focus on visual functionality. Its efficiency and optimized format has warranted its position in the toolbox of every digital artist today. And though it may be a trick to pronounce, it contributes to most of the visual works we interact with every day, even the letters you are reading right now would not be as clean and smooth without it.
Rarely can a mathematical equation be attributed to catalyzing a new moment in art language. But rarely are the solutions created as simple and elegant as the Bezier curve.
Bezier curves are curves created from the de Casteljau algorithm. Only requiring four points to be set, the algorithm produces smooth, organic lines or polynomials, which have found use in many software applications. In animation, it creates smooth transitions between point A and point B; and further in this essay, we will see how these lines became as universal in art as using a paintbrush.
The history of Bezier curves, unlike the curves themselves, had a relatively-flatlined start in the 1912. Contrary to the name, the mathematical concept of our smoothly-curved polynomial was originally developed by Sergei Natanovich Bernstein in 1912: when Pierre Bezier was only 2 years old.[1] Without the invention of computers, the Bernstein basis couldn’t find a strong industry application, until a little over 50 years later, a mathematician and an engineer at two competing French car manufacturers had a problem to tackle.
While the United States enjoyed the luxury of low gas prices in the mid-60s, European car manufacturers had to determine new ways to increase fuel efficiency in their vehicles.[2] Mathematician Paul de Casteljau at Citroen and engineer Pierre Bezier at Renault, both independently of one another, looked to the Bernstein basis as a solution, combining the mathematical basis function with Computer Aided Design to model smooth, curvy car bodies. And although de Casteljau discovered this application for the Bernstein basis in 1959 [3], a few years earlier than Bezier, Citroen did not allow him to publish his work. As Bezier had the freedom to publicly patent and publish his findings, our wonderful polynomial gets the slightly less-difficult-to-pronounce title: the Bezier curve.
As aforementioned, the Bezier curve has since contributed to a number of software applications. But it was most visually introduced to software during a time when software was destined for office work and word processing.
In 1982, 3 computer scientists at the Xerox Palo Alto Research Center found themselves frustrated to the point of quitting their jobs. Xerox PARC was focused at the time on inventing the office of the future through computer programs. They invented text editors, networks, laser printers, and to John Warnock’s frustration, a lack of focus on graphics. With the denial from Xerox to commercialize a new controlled printing graphics language, Warnock, along with Geschke and Putman, decided to leave Xerox and start their own company in Silicon Valley named Adobe. [4]
Their first product was a technology equivalent to Interpress called PostScript. It was a universal language to be used by printers in order to interpret text documents and print the correct content. But rather than being a language which was based on bitmaps, PostScript was a language based on control points and an algorithm that connected these control points with smooth, organic curves.
“I’d already been exposed to the mathematics of Bézier, but not to the wonderful geometric constructions that are associated with it, and the very, very simple computer implementations associated with Béziers. And so once I learned about that it because very natural, very easy to program, very easy to deal with in the computer. I thought all other approaches were silly, after that!” — Adobe co-founder John Warnock in 2004 Interview [5]
Adobe’s implementation of Bezier curves to represent text lettering was revolutionary. Texts could be of any size, large or small, squashed or stretched. PostScript was scalable, brought to market in 1985 through Apple’s Laser Printers and going on to become a universal graphics language for rendering font on printers and desktops of any make and any model.
It could do images and type, but as PostScript was a graphics language, you still had to be a computer programmer in order to create these graphics. So in 1985, Adobe started development on a product that would emit PostScript code from a drawing interface. With PostScript being based on vectors and curves, the user could click to create anchor points and then drag to set the second control point which would define the curve. The end result was Adobe Illustrator.
“…after PostScript got established we said, Gee, we really need a font drawing program. I said, let’s be very straightforward about it, and have people put down the control points, and allow them to control the control points, and control the curves that way, and that’s where that came from. Mike Schuster did a little prototype called Picasso, which was the precursor name. It had several names: ArtiFactory was one of them. It worked very well. Once people got used to it, they could control it and then we started using that for type design.” — John Warnock in 2004 Interview [5]
Illustrator would stem far past being used for type design. Prior to illustrator, graphic design as a profession was a laborious task. It was a career which depended on having technical education and patience. Imagine modern day graphics being done completely by hand. The analog process included sheets and sheets of Letraset dry-transfer letters to transfer type onto a piece of paper for design. Illustrators and other graphical elements were done with Rapidograph pens, which required consistent maintenance, refilling, and handwashing from ink explosions.
“The process of getting camera-ready art was pretty involved, back in the day. You had multiple elements… The text was spit out from a typesetter, which was cut up and paste on the board.” — Digital Painter Bert Monroy [6]
“One page of set type, it might take a day or two days to actually do.” — Illustrator Ron Chan [6]
Graphic design was a classical trained art form, bent on precision and accuracy to create works that could be processed through a camera and distributed through print. Warnock, whose wife Marva was a classically trainer graphic designer, focused on these issues that graphic designers had. Radial corners, differing stroke sizes, perfect curves: he would identify these problems to transfer graphic design from a specialized hand-done process to a process that could be done through a computer program.
The computer program was quickly adopted by the print industry, replacing the way that new illustrations were done, covers, print ads, all adopted this new form of drawing. With its direct influence from classical graphic design, many of the features directly translated from the terminology of graphic designer. And a task that previously required the perfect stroke of a French Curve was now precise, adjustable, and required two clicks.
“It was that ability to manipulate curves that really got me because curves are very important, especially if you’re designing type. Traditionally, I wanted that curve, and I wanted to give it a little bit more curve, and if it wasn’t right, I would have to white it out and start all over again whereas in Illustrator, I could just grab it and move it.” — Digital Painter Bert Monroy [6]
With the adoption from traditional graphic designers, there was some fear of how much easier it made the graphic design trade. A trade which required years of training and education, proper equipment and expertise was rapidly reduced to requiring a seat in front of a computer screen and a copy of Adobe Illustrator. It created fear in the industry that graphic design would be reduced itself. With the quantity of training no longer being required, the quality of the designer would also be a consequence of Illustrator.
“Everybody said that we’re [Adobe] going to ruin good design because now anybody could do it. But, the cream rises to the top.” — Adobe co-founder John Warnock in 2004 Interview
Accessibility has often played the role of the threat to traditional practices and thinking. There is a fear of allowing public access to tradition, as the exclusivity and necessity of those who are already taking part in it can potentially self-identify themselves as obsolete.
And in art history, the gallery space is a solid parallel to this tradition. In the past, attending exhibits in the gallery space required years of education in order to understand works of art. Without a solid understanding of Homer’s works, The Slaughter of the Suitors at the End of the Odyssey is deprived of meaning and just becomes an old pottery vessel.
The gallery space utilized a barrier of entry through education, pieces depending on their intertextuality and references to concepts and stories read in higher education. This concept was highly popularized through the Lowbrow art movement, which contributed in blurring the lines between gallery space and mass-produced content. Intertextuality and references of pieces did not allude to higher education, but either pop culture or visceral human knowledge. Comparatively, Adobe Illustrator achieved a similar effect, but rather than removing the barrier of entry to understand art, it removed the barrier of entry to produce art for mass consumption.
With its contribution to print ads, most of the graphic design work which Illustrator took over was focused on mass audience. Print agencies and companies were focused on efficiency and cost, and a computer software compared to the equipment of traditional graphic design was a no-brainer to shift their in-house design staff to Adobe Illustrator. With this shift occurring in companies driven by profit and not artistic incentive, Illustrator became a tool trade, even though it carried most of its design and features from a classically trained art form. This combination of accessible fine art tools created an interesting phenomenon; where time to creation becomes less, while the influence of the tool rather than the artist takes more weight.
With the increased efficiency that comes with software, traveling from initial idea to final production changes from a road trip to a plane ride.
Ahead of a road trip, you have to understand the maps, plan out your stops, and spend more hours behind the wheel. The necessity of one’s own control is paramount, and therefore, the road trip requires more foresight and preparation of the traveler.
Ahead of a plane ride, the actions required after you know where you want to go are less demanding. Past purchasing a ticket, the preparation does not include knowledge of the space between getting on the plane and landing. Just like with graphic design, the space between thinking up an idea and shipping a final design no longer requires the knowledge of how to put pen to paper and typeset to film. The mode of travel demands less, reducing two factors: a traveler’s overall time and a traveler’s need for preparation.
When time is reduced in the act of creating art, the spectrum of influence shifts. This spectrum is defined by two sides: the influence of the art vs. the influence of the medium itself. Two contrasting examples of the effect of this spectrum are French history painter Theodore Gericault and pop artist from Kutztown, Pennsylvania, Keith Haring.
These two artists have well-cemented themselves into the art history canon, lying on two opposite sides of history.
Prior to the invention of the photograph, painting was a means to record events in history. Gericault’s most-well-known pieces, The Raft of Medusa, depicts the aftermath of a French naval ship wreck. Gericault’s piece is remembered for its brilliant composition which carries the eye, haunting tone that leaves a haunting feeling, and sheer size of the piece. But the painting can hardly be described as showing evidence of the medium of paint. The sheer size of the painting requires years of work, eliminating the visual side effects of using a paintbrush and becoming more like a giant photograph. The time spent on the canvas reduced the influence of the medium, and in addition, Gericault’s preparation was an effort to create an image as accurate as possible to what a photograph could provide.
“The artist did an enormous amount of preparatory work and research: interviewing survivors and actually constructing a scale model of the raft in his studio. He went to local hospitals and morgues where he would observe firsthand the color and texture of the skin of dying and dead patients. He also created countless oil studies and drawings.” — Quote about Géricault preparation process [7]
The fact that Gericault was using oil painting as his medium did not fully determine the aesthetic of his final piece. With the amount of time invested, the medium would have bent to his sheer will to recreate the image which he had spent so many years creating in his mind.
Hundreds of years later across the Atlantic Ocean, Keith Haring would be riding the New York City Metro in 1982. With a stick of chalk in hand, Haring would search for a blank advertising poster. In just a minute or two, he would draw a cartoon-like picture on the black backing of the blank poster, leaving it for millions of subway goers. He had to act quickly, as what he was doing what technically illegal graffiti, which warranted his arrest a number of times. [8]
“The leap from subway trains to empty subway ads, in other words, was a short one, and Haring’s technique⎯fast, public, improvisational, virtuosic⎯ certainly owes a lot to that of early graffiti…It is both temporal⎯chalk, of course, being subject to erasure, especially in such a public place⎯and “tactile,” which is to say that his line is more emphatically indexical to its own act of creation (its creator, its time and place of making). It is a line that demands to be understood as (Haring writes) an “instant response to pure life.” — Charlie Gaillard Thesis on Haring’s Subway Art [9]
His style is legendary in modern art history, remembered for characters made of simple shapes and uniformly-sized stroke. And when we hone in on his medium of chalk and blank ad boards, we can infer where this style derived from. The aesthetic and the medium are more closely tied. And though the subways was where Haring developed his distinct style, it matured in his paintings, still keeping consistent with outlined characters and uniform stroke.
And of course, Haring must be credited for his thought and ideas behind his pieces, but when compared to the forethought exercised by Gericault, Haring’s work required much less. Haring worked with what he had: pressed time and chalk outlines, while Gericault worked with years and research. Haring could define how he shaped these lines, but not exactly how the lines appeared. He worked with the rawest forms which his medium could create, and therefore, his medium defined his aesthetic.
Now let’s return to the story of Illustrator. A number of in-house designers being pressed on deadlines to produce print ads and graphic art. In addition, the trade of graphic design now required less technical education and expertise, allowing fresh minds to contribute to the industry without the baggage of ideas that comes with classical training. Just like Haring but on an industry scale, the work became more closely tied to the medium, and in this case, that medium was Adobe Illustrator.
The Pen tool encourages clean edges, natural curves, and straight lines, similar to how chalk encourages bold uniform lines and primitive shapes. It is possible to create a work as detailed as Gericault, fine details that would require a far zoom-in to see the sharp edges of the pen tool. This work would require time, a detailed plan of execution, and in-depth knowledge of the software.
My argument is not that Illustrator revolutionized the world of art by bringing sharp edges and natural curves to the forefront. Observe works of artists like Piet Mondrian and the Memphis Group to see that the use of primitive shapes with sharp edges has been explored in the realm of fine art for years. Illustrator did not start these ideas; there were many artists and movements that revolutionized these ideas in the areas of art, design, and architecture. With this in account, my conclusion is, in fact, that the features of Illustrator revitalized the relevance of exactly these movements to have a greater impact on the modern graphic design aesthetic.
The modern graphic design aesthetic I am referring to are styles of graphic design which are used in fields such as print and software visuals, which I will refer to as “Digital Design”.
Works of illustrators like Gregory Hartman, Ryan Putnam, and Nick Slater are examples of this definition of the Digital Design aesthetic. All of the art have incredibly distinct styles, but all which are heavily depend on the use of Adobe Illustrator. Recreating the radial curves of Gregory Hartman’s Duolingo People with any other technique would not achieve as precise of a look.
Creating a production-ready version of Ryan Putnam’s Dropbox logo would require no mistakes, and even though impossible to recreate his consistent line width and color and opacity palette. With Nick Slater’s work, the aesthetic deviates further from the mathematical roots of Illustrator, but that is done through more complex use of Bezier curves. The style is still defined by the limited color palette and harsh lines.
Each of these artists created digital illustrations for top tech companies Duolingo, Dropbox, and Asana/Slack. The illustrator role in tech companies has a good amount of creativity, but also requires brand consistency and to serve purposes of advertising ideas of the company.
Digital Design is a distinguishable style, and though the artists used in the example are artists with less time-constraints and more creative freedom, the style has been developing since the creation of Illustrator, and at the time of these illustration, determined the skills and techniques to use the tool to create specific aesthetics.
Digital Design can be related most generally to the art movements of Bauhaus and the Memphis Group, characterized by sharp edges and flat design.
Yet movements such as Bauhaus and The Memphis Group had, in scale, incredibly short life spans, lasting 14 and 8 years respectively. Their influence on modern design is undeniable. Educational concepts from Bauhaus shaped ideas of modern day design schooling, focusing on functionality versus aesthetic. The isometric form of Bauhaus, based on architectural and engineering drawings, can be stylistically replicated quickly with the pen and perspective grid tool in Adobe Illustrator. The Memphis Project has had a resurgence in advertising and interface design, with bold colors and patterns that many allude to as the “80s aesthetic”. The shapes and loud but limited color palette of the Memphis Project can add flavor to vector images made in Illustrator, especially with the ability to change the color of multiple elements with the Color Palette tool.
Both movements have aesthetic characteristics that thrive in the environment of Illustrator. They can be recreated quickly while still having a playground for creative ideas and concepts. Extending this idea can capture reasons why Minimalism, Art Deco, Constructivism, Art Nouveau, and Modernism have maintain relevance in modern graphic design. [10] And as these movements have been revamped with visual relevance, they carry into the modern age their ideas and additional insights. These ideas are further adopted by our artists in question, graphic designers who are producing visual works to be consumed by a mass audience.
This is where the importance of this influence lies. The influence isn’t directly affecting the aesthetics of visual art in the gallery. If this were the case, the aesthetic would be traded among those who venture into the gallery space and those who create art for the gallery space.
The accessibility-to-art concept which emerged from the Lowbrow art movement defines that art made for mass consumption can have a greater, inherit impact on modern culture at large. Rather than focusing on the confines of the gallery space, design is art which is created for public consumption, and with it, it carries the ideas of these revitalized art movements into the public. The visual art carries the ideas of these art movements as well, which has the immense power to be able to affect the way a culture thinks, behaves, and creates art.
These art movements were chosen through the modern day tool for illustration: Adobe Illustrator. Adobe Illustrator was created to advance the field of graphic design past pen and paper, building off the foundation of the Bezier curve. The Bezier curve was invented to interpolant smooth curves using four points of control. Illustrator’s shift of the graphic design industry brought about resurgence of old art movements, which were referred to and republished into the public. This connection from a four point equation to the establishment of today’s fundamental understanding of images and art is how a simple mathematical equation has changed the course of art history.
