Impressive introduction to bezier curves. I still remember my first exposure to them; I was looking for a way to model a ship hull without making a full blown 3d-editor. After a bit of googling I found the article on bezier curves and spent the rest of the day digesting it all. Very interesting stuff. I couldn't figure out bezier surfaces for my life though, so instead I opted for having the 3 sets of bezier curves to represent the different profiles of the ship, then rasterizing them together. I'm pretty happy with how it ended up coming out; http://i.imgur.com/qv7WAJ7.png
Every time I read one of this guys blog posts, I want to post it a million places. I just finished reading through "Graph Theory (for the math-phobic)" and it was really enjoyable.
He seems to have a real knack for distilling reality down into simpler and simpler concepts until it disappears and you're only left with the abstract, barely having noticed it happened.
Thanks for this link. I found the article to be fascinating (albeit the math got a bit over my head). I was particularly interested in the "The Bull" series by Picasso pictured mid-way through. I ducked around a bit, and found an interesting write up on the back story of these sketches. Check it out: http://drawingdownthevision.com/the-bull-how-picasso-practic...
Fond memories. One of my first job out of college was to approximate splines by a sequence of straightlines and arcs. I rolled my own dynamic programming hack for that. On getting my first paycheck my first reaction was, "Whoa! what ? I can write code that people are willing to pay real money for". Given that I was not trained in CS, that was an empowering feeling.
On Picasso, I recall he was deeply affected by the cave paintings of Altamira. I think the influence clearly shows in his bull. He also said something to the effect that painting as a craft has not seen much innoavation since those Altamira artists. I dont know whether to agree or disagree, but the Altamira paintings do have this something very powerful and visceral about it.
This is a good article. One can read about Bezier curves in a lot of places. I took a Computer Graphics course a long time ago and every so often I get introduced to a new graphics programming framework that provides some ways to draw Bezier curves. But I think the real trick is how to choose the curves to approximate a certain drawing. In this article the author provides an approximation for a Picaso drawing. I wish he would have detailed how he chose the curves that he did. He promises to describe an algorithm in the next article. Can't wait for it.
I concur with all the Kudos to Jeremy, this is another very nice piece.
Bezier curves show up in some unexpected places. I was pretty taken aback when I traced through Java's 2D library [1] and saw that it uses them for circles instead of doing the more processor-intensive sin/cos or r^2 = x^2 + y^2 math.
A nice feature of Bezier curves (and B-splines) is the convex hull property: since the curve must lie within the convex hull of the control points, any point outside the convex hull cannot be within the area bounded by the curve.
So if you have a set of points and need to test which are inside a curve, you can immediately eliminate any that are outside the convex hull, potentially saving a lot of computation.
Damn!!, That is really cool. Now I want to spend next week drawing pictures on the computer. There is a special place in the next realm for folks who bring math to life for us simple folks.
