De Casteljau subdivision is a procedure by which a Bezier curve is defined. It is also used to subdivide a quadratic Bezier curve into two distinct quadratic Beziers representing individual segments of the original curve. The first Bezier segment is in the interval [0,t] and the second lies in [t,1] for t in (0,1). Some 3D graphics packages use the term refinement to describe to the process of subdividing or splitting a curve into two equivalent segments at some interior point. Refinement is also used to describe a process of extracting the coefficients of an equivalent-degree curve at two interior points. In this context, refining a quadratic Bezier curve produces another quadratic Bezier that represents the original curve in the interval [t1,t2] where t1 < t2 and both t1, t21 are in [0,1].
This is an infrequent operation, although I recently used it in the organic scroller, the boundary of which is a quadratic Bezier spline. The process may also be used to extrude any segment of a path comprised of quadratic Beziers (such as a circle approximation).
The quad refinement algorithm is now in Degrafa. I’m still in the process of conducting some unit tests for possible numerical issues. I wanted to get a release out now, while I’m in between gigs, since I have the time and am thinking about it. Otherwise, you know what happens. It’s onto the next gig and I forget 🙂
A screen shot of the demo program is shown below.
![Quadratic Bezier Refinement in the Interval [0.2, 0.6]](https://algorithmist.files.wordpress.com/2010/11/quadbezierrefine.jpg "QuadBezierRefine")
Not much to the demo. The blue curve is the quad. Bezier that interpolates the three points, P0, P1, and P2 with chord-length parameterization. The red curve is drawn from coefficients returned from the quadRefine() method of the BezierUtils class. Above, the red curve illustrates refining the original quad in the interval [0.2, 0.6].
Drag the points to see how the refinement is modified. Adjust the stepper thingies to modify the interval. [t1,t2] can be extended to a maximum of [0.1,0.9] in increments of 0.1. Note that t2 must be strictly greater than t1. Code has been committed to the Origin branch.
The Algorithmist 
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