Talk:Bernstein blending function

Undefined terms

What are the various symbols, N, W, μ, etc supposed to represent? I suppose that i is an index, since it says this is an iterative process. The rest are mysteries to me.

This article needs more definition, else it just doesn't make any sense. DavidCBryant 11:35, 12 April 2007 (UTC)

Note from the Wikipedia Department of Redundancy Department

Good questions Dave; I have a more fundamental one. Does this article convey anything above and beyond what has been (reasonably) explicated at Bernstein polynomial? First some background, then your questions.

Barycentric combinations

This article presents the Bernstein polynomial weighting functions. These are designed to blend collections of geometric points into "centers of gravity" or barycenters.

There is a suble problem with finding the center of gravity of of a collection of points. When ensembles of points move with respect to the origin, their "addresses" change and sums of these change too. To properly find the "center of gravity" of a collection of points, all weightings of input points should add up to one. So constrained, the input points form a barycentric combination resulting in a "center of gravity", a "barycenter", or an "average location". Without this constraint on the weights, the relative position of the barycenter will shift under translations of the input points.

Partitions of unity

The straightforward averaging of points is never a problem: 50% of the position of a first point added to 50% of the position of a second point finds the midpoint between the two. The two weights, 0.5 and 0.5 add up to one, or, to use a fancy five dollar gold-plated expression, they partition unity. The midpoint remains equidistant between the two given points no matter where they move. If one takes 80% of one point's position and blends it with 10% of the other, then the position of this weighted average will change with respect to the input points. Unless the weights partition unity, the distance of the origin factors into the calculation of the barycenter. Move the origin and the barycenter moves as well.

Life gets a little tricky if changes in the weightings of the input points are to shift the barycenter, a mechanism to draw a curve, say. Here, each point on the curve is a weighted blend of the input points. Comprised entirely of weighted blends, the shape of such a curve would ape the positions of the input points and change shape as the input points move with respect to one aother. But the points can't be weighted arbitrarily, otherwise the shape of the curve will, in part, depend upon where the input points are located with respect to the origin of the system.

Weighting functions that partition unity

The linear functions $\quad x$ and $\quad 1-x$ llustrate the idea. The position of point $\quad P(t)=(1-x)P_{0}+xP_{1}$ defines a line between $\quad P_{0}$ and $\quad P_{1}$ as the parameter $\quad x$ varies from zero to one; The shape of the line varies only when one input point moves with respect to the other, not when both move with espect to the origin; that is because these two linear weighting functions partition unity. The sum of $\quad x$ and $\quad 1-x$ partitions unity for all values of $\quad x$.

Decastlejau's algorithm and Bernstein blending functions

Lines are not very interesting; we'd like to model the general case of loops and arabesques. DeCastlejau approached the general case with a recusive method. Setting $\quad x$ to a particular value, he pairwise blended input points using the linear weighting functions $\quad x$ and $\quad 1-x$, then pairwise blended the resultant midpoints, still using the same weighting functions, then pairwise blended the midpoints of the midpoints, still using the same functions, and so on, until he had a single result; Performing this repeatedly for a good many values of $\quad x\quad$ visualized a curve invariant under the translation of the input points.

In the 1960s, Pierre Bezier independently developed this recursive approach — he was unaware of deCastlejau's work — and grounded the recursive blending procedure onto the Bernstein polynomial functions. The recursive pair-wise blending of points weighted by the simple linear functions $\quad x$ and $\quad 1-x$ — de Casteljau's algorithm — reduces to the Bernstein polynomials when one does the necessary substitutions at each stage of the recursion. The question arises, where do the Bernstein polynomial functions come from?

In 1912, Sergei Bernstein was working in the field of approximation theory and needed sets of polynomials of varying degrees that partitioned unity. He began with a very basic identity.

$\qquad 1=1$

and then embroidered the right hand side by introducing a self-cancelling parameter.

$\qquad 1=x+1-x$

Both sides of the identity remain just different representations of one. Bernstein furher embroidered the righthand side by raising it to an arbitrary power, n:, The identity still remains valid because one raised to any power remains one.

$\qquad 1=(x+(1-x))^{n}$

Now presto! Applying the binomial theorem to the righthand side yields a sum of polynomial terms that together partition unity, as desired.

$1=(x+(1-x))^{n}=\sum _{\nu =0}^{n}{n \choose \nu }x^{\nu }(1-x)^{n-\nu },\qquad \nu =0,\ldots ,n.$

This series of polynomials are the Bernstein polynomials of degree n. These are weighting functions which, when multiplied by $\quad n+1$ geometric points $\quad P_{0},\dots ,P_{n}$ generate a barycentric combination. As the parameter $\quad x$ ranges over $[0,\dots ,1)$ the barycenter of the combination shifts, "drawing" a curve.

$P(t)=\sum _{\nu =0}^{n}P_{\nu }{n \choose \nu }x^{\nu }(1-x)^{n-\nu },\qquad \nu =0,\ldots ,n.$

The cubic Bernstein blending functions are th most common; they are the basis for directly computing degree three Bezier curves, the four control point form, making two control handles, which figure almost universally in many vector drawing packages.

Notation, notation, notation!

The development above differs from the presentation of Bernstein's blending function in this article only in the matter of notation. First, note that there are a handful of minor differences. The main namespace article uses:

1. $\quad \mu$ to represent the parameter; this note uses $\quad x$: $\quad u$, $\quad s$, and $\quad t$ are also common notation for parameters.
2. $\quad N$ to indicate the degree of the Bernstein polynomials; this note uses the miniscule letter $\quad n$
3. $\quad i$ to index individual Bernstein functions; this note uses $\quad \nu$.
4. $\quad W(N,i,\mu )$ to represent the ith Bernstein polynomial of degree $\quad N$. This note uses no equivalent notation, but the Bernstein polynomial article uses $\quad b_{\nu ,n}(x)$ and most my texts use various 'Big B' notations: $\quad B_{(}n,\nu )$ or $B_{\nu }^{n}$ to represent the $\quad \nu \quad$^th instance of a degree $\quad n$ Bernstein polynomial.
5. ${n \choose \nu }$ is the binomial coefficient. It expands to $\qquad {n \choose \nu }={\frac {n!}{\nu !\,(n-\nu )!}}$ and is similar to the expansion in the mainspace article. Perform the substitutions, above, on main article version to obtain the exact mainspace presentation. Curiously, this article uses ${}^{N}\!C_{i}$ to denote a binomial coefficient compactly. I am unfamiliar with this form and find the choose form more familiar..

So, tell me this article isn't a fork

So what does this article give me that Bernstein polynomial doesn't? The earlier version offered some odd use of terms, calling 'control points' 'knots' for example. And not a whole lot of explanation of terms could be found, which wacked Dave's head. And there were typos in the math. The technical explanations are still thin and have better versions in de Casteljau's algorithm, Bernstein polynomial, Bezier curve, De Boor's algorithm and related articles. User talk:Marvinklein had just landed on Wikipedialand when he kicked this article off, and I think he just started typing without noticing the other material on the far side of the hills, older, better developed articles, in my opinion. In good faith, he made a content fork. Accidents happen. His notation is a bit idiosyncratic, so similarities with this and related material is not drop-dead obvious. Nobody's seen Marvin since last November; he's walked off the edge of Wikipedialand,so I don't know where he was going to take this piece. After putting these notes together, however, I wouldn't know what area to develop that isn't already (better) established in Bernstein polynomials. Take care Gosgood (talk) 19:41, 3 July 2008 (UTC) (revised post)

I'm a tad embarrassed to follow up on your long analysis with such a short post. I agree that this article contributes little, if anything, above what Bernstein polynomial provides, so I replace it with a redirect (it's a tad embarrassing to follow up on your long analysis with such a short post). -- Jitse Niesen (talk) 13:24, 29 August 2008 (UTC)