Fig. 14 and Fig. 11 respectively redrawn and extracted from: Le Bulletin de la Section d’Histoire des Usines Renault¹

Diagram redawn from http://rocbo.lautre.net/bezier/

In these two images, we see how a three-dimensional modeling method breaks when rounded down to a flat image. Bézier did not imagine describing solely the path of the designs he enabled, he imagined describing a deformed space that would accompany the shape into itself. This makes a lot of sense for the original context of usage, considering his practice of bending metal sheets. Unfortunately though, there has been a lot of interpretation of his established basis for the application in a space where the third dimension is not.

It is crucial to understand that Bézier designed his description system using the mathematics developed by Paul de Casteljau² to support it in a three dimensional space. When flattening a 3D system onto a 2D surface, it forces a perspective onto both the tool, and the resulting drawing. We literally lose a dimension.

Keep in mind that these vertices were designed by Bézier to be used in conjunction with others, not as single self standing elements as is the case in 2D vector drawing tools. This does not mean that we'd be better off with using 3D tools for drawing 2D shapes, we're just pointing out the fact that it might be worth paying attention to the adaptations made to the system. The squashing of 3D onto 2D requires some fine control before it can be frozen.

Furthermore, the notion of perspective is interesting. The graphic representations we focus on are exactly that: representations wherein lies a point of view. Perspective and viewpoints are key to the notion of drawing, therefore when a 3D description environment is the same tool for a 2D view, we can imagine a slice decision on scale.

Joseph-Benoit Suvée, Dibutade ou l’origine du dessin (1791)

Let's consider the notion of projection. Source, shape, interruption, distortion. It is no surprise to see a three-dimensional shape flattened into a long shadow projection under the effect of sunlight. Out of context though, if you were to outline the shape of the shadow and then remove the object that created it, and observed the outline, it might not be so easy to piece back together the original object.

Still, some indications can be read, and with a forensic attitude, you may be able to reestablish the abstraction.

This scenario leads us to believe that while vectors describe shapes, they are given to us with an outside point of view. It induces an observation from the outside in, not one that comprehends surfaces. Could we say that vector lines from the 3D Bézier origin forced onto a 2D plan should be handled more with an idea of containment, of skin to a material, but not as an ultimate description?

Maybe a vector is actually the lining, and not the object. Maybe these lines tell us more about a contained tension than about a direction. All we need to do is to shift the light source, and everything is reshaped.