{"id":218,"date":"2013-07-18T13:10:57","date_gmt":"2013-07-18T18:10:57","guid":{"rendered":"http:\/\/blogs.ams.org\/blogonmathblogs\/?p=218"},"modified":"2013-07-18T13:13:07","modified_gmt":"2013-07-18T18:13:07","slug":"218","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/blogonmathblogs\/2013\/07\/18\/218\/","title":{"rendered":"String Art, Bezier Curves, Picasso, and me"},"content":{"rendered":"

\"linedrawing2\"<\/a>\u00a0Going through some old boxes at my parents house the other day I ran across some line drawings I made as a teenager.\u00a0 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. \u00a0 \u00a0As 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\u2019t know that they were Bezier curves.\u00a0 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.<\/p>\n

In 1960’s France, Bezier popularized his eponymous curves (actually invented much earlier) by using them to design cars for Renault.\u00a0 Around the same time and place Picasso drew his famous line drawings.\u00a0 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\u2019s works.\u00a0 In his May blog post, Bezier curves and Picasso<\/a>, Jeremy gives an in-depth description of Bezier curves and imitates one of Picasso\u2019s sketches quite handily using nine Bezier curves.<\/p>\n

As I read Jeremy\u2019s post, I realized for the first time that my drawings traced out quadratic Bezier curves.\u00a0 I can\u2019t 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!\u00a0 So Jeremy inspired me to create the picture you see below in Inkscape in which I tile the plane.\u00a0 The Inkscape program has a tool specifically designed for drawing Bezier curves. \u00a0 \u00a0Half the tiles are made using this tool, and the other half are made entirely of straight line segments.<\/p>\n

\"linedrawingoriginal\"<\/a>\"linedrawingoriginal\"<\/a><\/p>\n

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\u2019s excellent explanations.\u00a0 And don\u2019t skip Jason Davies animation of the recursive construction of a cubic and quartic Bezier curves.<\/p>\n

Aside from my own pleasant surprise at the content of this particular post, Jeremy\u2019s blog seems quite unusual in the at the moment due to its accessibility and subject matter (as literally described by his blog\u2019s title) at the intersection of Math and Programming.\u00a0\u00a0 While I have noticed many blogs about Complexity Theory and Theory of Computing, I haven\u2019t seen any like Jeremy\u2019s that are well kept up and structured with both theory and code included.\u00a0 Surely there are more blogs like this?\u00a0 Let me know in the Comments please!<\/p>\n

Speaking of complexity, Jeremy ends his post with some comments about the relationship between complexity and beauty \u2013 the general idea being that the more beautiful an object, the less complex.\u00a0 Might we measure the complexity of a drawing by the number of Bezier curves one needs to draw it?\u00a0 Unfortunately, as he points out, the beautiful circle cannot be perfectly drawn using a Bezier curve.\u00a0 This made me think about other line drawings I\u2019ve seen such as those in this July 2nd<\/sup> post created with nested polygons at Benice equation<\/a>.\u00a0 Are the spirals traced out by nested squares Bezier curves?\u00a0 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.<\/p>\n

For more background, see Bridges, String Art, and Bezier Curves<\/a>, a post from 2012 by Renan Gross, an Israeli student at Technion who has his own blog entitled Sarcastic Resonance<\/a>.\u00a0 Although the latest entry is in Hebrew, those previous are English, so don\u2019t let the first one deter you.<\/p>\n

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\u00a0Going through some old boxes at my parents house the other day I ran across some line drawings I made as a teenager.\u00a0 At the time I found it fascinating that just by drawing straight lines I could create beautiful … Continue reading →<\/span><\/a><\/p>\n

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