THE BEZIER CURVE
"Is (imagination) not the ability to catch barely seen analogies – as Alice in Wonderland did, to go "through the mirror?" (Pierre Bezier)
How Can One Many Change the World
FROM SNOW BANKS TO BEZIER
An Interesting Introduction to the Man Who Changed the World
We, like many parts of the Midwest, got hit with a snowstorm last week. It was quite a workout shoveling the driveway. The snowplow, of course, made it more difficult, dumping wave upon wave of snow at the end of the driveway.
While resting one moment, I looked at the street being plowed, and sympathized with the snow-plow driver. They really have a tough job, when one thinks about it. Get too far away from the curb, and they're criticized for not clearing the streets well enough. Get too close, and they'll likely scrape a front yard, or worse yet, cause damage to the curb - or their plow.
What a no-win situation.
Isn't there a way to mark front yards to make the job easier? For example, what if neighbors put flags in their front yards, along the street, indicating "edge of yard"? Would that work?
It works fine along the east side of our street, but what would happen along the west side?
Obviously, this wouldn't work well. And a thought came to mind. One could create markers along the way indicating a "change of direction".
What's not so obvious is how this simple idea might impact industry. It did - and the world.
Pierre Bezier did something like this in the automotive industry, and the applications have since "changed the world". How does one capture the essence of the shape? Straight lines? That's pretty easy. What about curved lines? Is it possible to capture curved shapes using points?
That's the essence of the Bezier method.
Let's suppose I wanted to draw the following curve, from Point P0 to Point P2:
What Bezier envisioned was a series of control points describing a polygon, and an algorithm creating the curve:
(image from Wikipedia)
Now: what's the algorithm? It can be described algebraically, but also geometrically, and that's where things really get interesting. You may recognize the following:
Many applets on Bezier Curve
I invite you to play around with Bezier-related applets at the phenomenal Wolfram Demonstration Project website ...
The Mathematics of the Bezier Curve
THE MATHEMATICS OF STRING ART
A Tribute to Pierre Bezier (1910-1999)
by Paul Cox
Note: I read this many years ago, and printed it off. The only place I could find it now was the "Way Back Machine". It's by Paul Cox, under "Math Mistakes". The great write-up below is his:
Back in my essay on infinity, I mentioned in passing a French mathematician named Pierre Bezier. A few people mentioned that they have never heard of him or his theories, and I thought it would make a great essay. While doing some research on Bezier and his work, I came across a notice of his obituary. He died only 6 months ago, November 25, 1999. Even though it is slightly belated, I thought I would write a tribute.
Pierre Bezier was one of the first famous mathematician I had ever met in person. The occasion was a conference on Computer Aided Geometric Design, an obscure branch of mathematics that created most of the algorithms used in every drawing program from AutoCAD to 3D Studio Max. Bezier was one of the early pioneers. The conference was held at Arizona State University where many of Bezier's successors in the field teach and do research.
Bezier gave a lecture on how he came up with the Bezier Curve, the thing that made him famous among us computer graphic junkies. Bezier was instantly likable. He reminded me of my grandfather, if my grandfather spoke with a French accent. At 80 years old, he was pretty much enjoying his retirement, his stay in Arizona included a visit to the Grand Canyon. Even at 80 he was as sharp as ever, and he had a great sense of humor. When someone pointed out a flaw in one of his lecture slides, he quipped, "It's OK, I'm already famous."
The story of the Bezier curve was an unusual one in the history of applied mathematics. Most of the time when you find a real world problem needing a mathematical answer, you just find the math you need and apply it. Such is the case with Einstein's General Relativity and Riemann's Non-Euclidean Geometry discovered a century earlier.
Bezier worked as an engineer for a french automaker. To satisfy the needs of manufacturing, they needed a way of describing a curve exactly at every point. In those days, engineers sitting at drafting tables would would mark a starting point and an ending point of the curve they wanted, then pulled out a french curve and drew an approximate best-fit curve.
At the machine shop level, these best fit approximations were not good enough. In order for pieces to fit together the parts could only vary within certain tolerances, many of these approximate curves were outside the tolerances. By 1960, hardware became available that allowed the machining of 3D shapes out of blocks of wood or steel, known today as CAM or Computer Aided Manufacturing. Computer graphics was still in its infancy at the time, so designing a method of describing any curve you wanted was of utmost importance.
Bezier had to come up with a method of describing a best fit curve that would be easy to use and exact enough to meet the demands of manufacturing. Unfortunately, no mathematics existed at the time to do the job adequately(1). After numerous schemes, he came up with a method of describing any 2nd degree curve using only four points.
The method is rather simple. He starts by describing a curve inside a cube (the figure below to the left) using a parametric equation equal to the graph of y = x2. Then by transforming the cube into any kind of parallelepiped (below to the right), the curve will change as well. The four control points are the vertices of an imaginary parallelepiped. In the illustrations, points a and d represent the starting and ending points. Points b and c determine the curves depth and orientation. The slope of line ab is the starting slope of the curve, the slope of cd is the ending slope. Bezier's mathematical representation can be expanded to more than four control points to create curves of higher degrees, but for most uses four is enough.
For you real math junkies out there, the parametric function for Bezier Curve bn(t), where point A is b0, B is b1, etc. and n is the number of points - 1 is:
2nd degree Bezier Curves can be lined up one after another to create all kinds of shapes in two dimensions. But, what was really important to auto manufacturing was describing a whole piece in 3 dimensions. Putting four curves together in a square shape creates a bezier surface with 12 vertices, and creating tiles of these surfaces can create any three dimensional shape you can imagine.
In today's computer aided world, the applications are numerous. Not just in obvious applications like computer graphics and animation (animation often uses bezier curves applied to the fourth dimension to describe smooth motion), but also in robot controlled manufacturing. The Bezier Curve changed the world.
1. At least two mathematicians solved the problem before Bezier: Airplane designer James Ferguson, and engineer Paul de Casteljau who worked for Citroen. The latter's work is mathematically equivalent to Bezier, in fact the formula listed above is De Casteljau's. Unfortunately, their discoveries were closely guarded industrial secrets and were not published until after Bezier.
How do we learn?
Bezier In His Own Words
I would like to finish with a quote by Bezier himself waxing philosophical about what goes into creating a useful mathematical model and about the creative process in general: