Going through some old boxes at my parents house the other day I ran across some line drawings I made as a teenager. At the time I found it fascinating that just by drawing straight lines I could create beautiful curves like the boundary of the shape to the left, and I must have drawn hundreds of these pictures. As a kid my artistic and mathematical tendencies ran parallel without my making many explicit connections between them. Although I realized these curves were parabolas, I didn’t know that they were Bezier curves. More to the point, I had no idea that all of the drawing programs I liked to use were exploiting the very types of curves that I was drawing by hand.
In 1960’s France, Bezier popularized his eponymous curves (actually invented much earlier) by using them to design cars for Renault. Around the same time and place Picasso drew his famous line drawings. After staring at these line drawings, Jeremy Kun, a current graduate student at University of Illinois at Chicago, saw a visual connection between these Frenchmen’s works. In his May blog post, Bezier curves and Picasso, Jeremy gives an in-depth description of Bezier curves and imitates one of Picasso’s sketches quite handily using nine Bezier curves.
As I read Jeremy’s post, I realized for the first time that my drawings traced out quadratic Bezier curves. I can’t believe that I never noticed this connection, and now I see a really great and natural way to insert line drawings into college calculus class! So Jeremy inspired me to create the picture you see below in Inkscape in which I tile the plane. The Inkscape program has a tool specifically designed for drawing Bezier curves. Half the tiles are made using this tool, and the other half are made entirely of straight line segments.
For a better understanding of the recursive nature of higher order Bezier curves and for an understanding of how to decompose these curves, see Jeremy’s excellent explanations. And don’t skip Jason Davies animation of the recursive construction of a cubic and quartic Bezier curves.
Aside from my own pleasant surprise at the content of this particular post, Jeremy’s blog seems quite unusual in the at the moment due to its accessibility and subject matter (as literally described by his blog’s title) at the intersection of Math and Programming. While I have noticed many blogs about Complexity Theory and Theory of Computing, I haven’t seen any like Jeremy’s that are well kept up and structured with both theory and code included. Surely there are more blogs like this? Let me know in the Comments please!
Speaking of complexity, Jeremy ends his post with some comments about the relationship between complexity and beauty – the general idea being that the more beautiful an object, the less complex. Might we measure the complexity of a drawing by the number of Bezier curves one needs to draw it? Unfortunately, as he points out, the beautiful circle cannot be perfectly drawn using a Bezier curve. This made me think about other line drawings I’ve seen such as those in this July 2nd post created with nested polygons at Benice equation. Are the spirals traced out by nested squares Bezier curves? The animations are particularly nice as they allow you to see the way the drawing is generated. There is little or no description of these drawings, but the Geogebra file used to make these drawings is made available.
For more background, see Bridges, String Art, and Bezier Curves, a post from 2012 by Renan Gross, an Israeli student at Technion who has his own blog entitled Sarcastic Resonance. Although the latest entry is in Hebrew, those previous are English, so don’t let the first one deter you.