# U-Splines and "Bernstein-like" basis

Another question I’ve recently developed relates to the restriction of the applicability of the U-Spline algorithm. Schillinger’s work demonstrates the ability to decompose a B-Spline into Lagrange polynomials and computing an extraction operator, in addition to the more familiar Bezier decomposition/extraction methods.

As I’ve tried to demonstrate in my below picture, my understanding of the U-Spline algorithm is that it’s able to compute the smoothness operator which converts a spline with a “Bernstein-like” basis function into an equivalent B-Spline. As mentioned in the U-Spline paper, this operator is the transpose of the extraction operator. So it would also seem that if I can decompose into a Lagrange spline, computing an extraction operator whose transpose is the subsequent smoothness operator, that I should be able to use the U-Spline algorithm on a Lagrange spline as well.

However the U-Spline paper’s definition of a “Bernstein-like” basis function would seem to preclude its application on a Lagrange spline:

1. for 0 <= i <= n, B_i is positive on ]a, b[.

Which is clearly not true for Lagrange basis. However, I’m struggling to see why the U-Spline algorithm shouldn’t work for a Lagrange basis.

Hi Greg,

Matt forwarded me your email on your presentation. Great work! I was going to come here to make a post to clarify a few things. There is no such thing as a Lagrange spline. You can represent a B-spline in terms of a Lagrange basis locally but this is not really a big deal because it doesn’t change the B-spline basis functions at all (they are still noninterpolatory). It only changes the local representation to be in terms of the Lagrange basis. That local representation becomes interpolatory but that is of little practical use. You could make a similar transformation to any complete polynomial basis such as the power basis (1, x, x^2,…), the Legendre, Hermite, Laguerre, Chebychev or any other polynomial basis. Each of these bases has features that may be nice but the Bernstein basis is the only one with positivity and an ordering of derivatives.

The U-spline algorithm utilizes these special properties to simplify an intractable constraint problem. Once you have used the U-spline algorithm to compute a basis, you could carry out a transformation to a local Lagrange basis. I suppose there might be places where this is helpful but I can’t think of any off the top of my head.

Thanks Derek,

Regarding “Lagrange spline” - I guess my thoughts are that you can certainly make a piecewise polynomial function, with each piece described with a Lagrange basis, and prescribe some level of continuity (as restricted as the continuity may be) between the sections. I would think that would constitute the function as being a “spline” - essentially I’m equating “Composite Lagrange Curve” as being a “Lagrange spline” and a “Composite Bezier Curve” as a “Bezier spline”. I realize these may not be officially splines, but I’ve used these descriptions as a contrast to help describe what a B-Spline is, e.g. really helpful for explaining “minimal support”.

The question that motivated this forum post was (paraphrased): “How smoothly could this technology be integrated within a conventional FEA solver and/or workflow? Specifically, could it take a Lagrange FEM mesh and convert it to a B-Spline via this U-Spline algorithm?”