NOTE: ITKE’s WordPress software has serious problems with embedding images in a posting, so to get the blog entry below, along with its images, please go to the blog I create for my animation students at the University of Colorado:
In the previous posting of this blog, we considered the problem of uniquely specifying a curve with a minimal amount of information. Curves are a core building block for engineers, product designers, and artists.
A straight line can be specified with two points, but how do we specify a curve, like the ones on the bodies of cars or soda bottles?
The answer is attributed to a French car designer named Bezier.
The naive approach.
Consider the obvious, brute force way of doing it.
You want a curve to start at one point and end at another. How do you uniquely define it? You can’t just supply a beginning and an end point. How about three points, a beginning, a middle, and an end point?
If those three points were spread across a computer display, our immediate response would be that there are almost countless curves that could go through those points, and the only reason it isn’t literally countless is that there are only a finite number of pixels on the display.
To make the curve truly unique, it seems like you’d have to provide every single point along the curve, with the only limiting factor being the total number of pixels in the display.
That is a very tedious way for a designer to convey the shape of a curve to someone building a car. For one thing, how many points are along a curve on a car? It’s not a computer display with pixels. It isn’t at all obvious how many points you would need.
A better, radically simpler way.
But Bezier realized that by providing just a handful of points, along with mathematical expressions called polynomials, a curve could be completely, unambiguously defined.
A popular form of a Bezier curve is is “cubic” curve, which requires only a beginning point, an end point, and two other points, along with two polynomials like the ones below:
The diagram below is characteristic of the Bezier approach, as well as several similar techniques. The four points, including the two end points and the two points above the curve, along a couple of mathematical expressions, are all that is needed to create the curve in the diagram.
Yes, just four points and a little mathematics, and we get this curve:
This curve, by the way, was made with Rhino3D, a powerful and very elegant design application.
Next time, we’ll look closer at the intuition behind this – and at the magic it represents.