mokus@deepbondi.net wrote:
It took me a while to get the intuition right on those, but here's a quick sketch. Let n = number of control points, m = number of knots, and p = degree. For p = 0 (constant segments), each control point corresponds to one span of the knot vector, so n = m - 1. Each time you move up a degree, the basis functions span one more segment of the knot vector. Thus, to keep the same number of control points you have to add a knot when you add a degree, so n = m - 1 + p. Equivalently, n - p = m - 1, which incidentally makes an appearance in the deBoor function below when we zip "spans p us" with "spans 1 ds". This is just the property that ensures that the length of these lists is equal.
How embarrassing, I managed to get this simple math wrong. That's what I get for trying to think in the morning without either my notes or my coffee, I suppose. At least, I have that standard excuse to fall back on, so I'll take advantage of it. "n = m - 1 + p" should read "n + p = m - 1" - I added "p" to the wrong side. It is indeed the property that makes the zip line up, but not by the math I gave. Instantiating n and m as the respective "length" expressions makes the equation:
length ds + p = length us - 1
With subsequent derivation in light of the fact that the 1st control point is being ignored (which introduces a "- 1" on the RHS, if I'm not mistaken again), as mentioned earlier, and the fact that the expression in question is supposed to return a list one shorter than ds:
length ds + p - 1 = length us - 1 = length (tail us) length ds - 1 = length (tail us) - p length (spans 1 ds) = length (spans p (tail us))
(Under the assumption that all the lists involved are long enough for the tail and drop operations) And all is (hopefully) right again. -- James