Thursday, October 22, 2015

The History of Bezier's Curve

Of all the things in graphic design, I never realized how pervasive Pierre Bézier is, so I wanted to write about his history.

As it turns our the story is quite long and complicated.
it all begins with Ship building. One of the earliest uses of curves to construct something (other than greek and roman amphitheaters)

Almost every boat that has ever been designed, has had some kind of curved bottom. Not only does this give the boat support and buoyancy but it also helps to streamline the vessel. By making the bottoms of boats curved, they minimize drag and increase resistance to lateral motion allowing for faster transport.
Can you see how curves could be particularly useful in making these designs?
These boats needed to be planned out on paper, and long flexible rulers like the ones below were often used to help position a curved line to trace onto the page. These plans were often drawn or lofted in large shipbuilding attics (or lofts) where ships would be built. Think about how long the curved wood needs to be and the amount of space needed to draw usually at a 1 to 1 scale. These shipbuilding lofts were HUGE!

The weights are often referred to as Ducks due to their bill like shape. An optimally smooth, attractive, and mechanically sound curvature is guaranteed by the uniform distribution of stress throughout the long elastic spline as it tries to regain its original straightness.

As ship building designs became more common,  Ludwig Burmester standardized a set of stencils that became known as French Curves. (Yes he was german, no I don't know why they are considered french)
This set of 28 curves was used for years to draw schematics and became a practical alternative for drawing other curved surfaces due to their size (only about 6-8 inches) and versatility. This was how most curves were made for almost 400 years. With paper, pencils, and stencils.

During wartimes, security became a larger priority when it came to tactical schematics. The issue with modeling curved surfaces on paper means it becomes very difficult to securely encrypt the designs, and even harder to reproduce them if they get stolen or destroyed(bombs, rips, fire, etc...). Often times, after a design was completed, a mathematical model of it was derived by hand and those numbers were used to build the ships. By adding the extra step to the ship lofting process, it made it harder for a design to be stolen (each ship building company could store numbers in their own order) but it made ship writes more specialized (One person could interpolate the numbers and get the curve while others built it). 

With this in mind, it stands to reason that storing curves as numbers/equations would make it difficult to adjust them because of how complicated the modeling equations tended to be. 

The birth of modern Computer Aided Design (CAD) can be traced back to French automaker Citroën. At the end of the 1950s, Citroën had adopted early analog computers connected to machinery for milling the stamps and dies that repetitively press parts of the bodywork from sheet metal. But the computers only took coordinates, so lines, circles, parabolas, and other regular geometrical functions could be accurately inputted from the designer’s blueprints into the machinery, but a reliable method for freeform curves did not.

Paul de Faget de Casteljau (1930) was the one who found the link first. A curve could be repented by control points and ratios:

Here is a control polygon.

Each polygon segment is now divided in the ratio of t ( for example: t=1/2) . By doing so a next polygon level is created.

Each segment between the new points is divided in the ratio of t.

Here is the solution for the point t=1/4
A higher order curve represented in a similar way.

Unfortunately, the automaker Citroën didn't listen to De Casteljau's algorithm, and he ultimately was never published until 1970.
"The modelers were unwilling to cooperate. De Casteljaus’ theory was also met with incredulity elsewhere within Citroën. How could a mere ten lines achieve what others before him had unsuccessfully attempted in sixty pages of equations? The modelers scoffed at de Casteljau’s poles, which seemed overly intuitive, claiming that anyone could have invented them. How could this be a worthy method? If valid, it threatened to replace the black arts of auto production with cold hard computation and, in doing so, make the influential role of master modelers obsolete." (cite)
 As it turns out, Citroën's direct competitor "Renault" hired a man by the name of Pierre Bézier just about at the same time. Knowing that an algorithm had been developed but not knowing how, he derived his algorithm from his own observations about how a curve warps in 3 dimensions. In the end, he conceded with the exact same algorithm described above. and the reason why we know the curve as his is because he was allowed to publish his findings.
Pierre Bézier's original observations that lead to the same algorithm

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