h-Blossoming: A new approach to algorithms and identities for h-Bernstein bases and h-Bézier curves

Polynomial degree reduction in the discrete $$L_2$$ L 2 -norm equals best Euclidean approximation of h-Bézier coefficients

“…One way is through the use of the notion of h-blossoming introduced in [18]. Namely, given a polynomial P of degree at most n and a parameter h, there exists a unique multi-affine symmetric function f (u 1 , u 2 , .…”

“…Values of the polynomial P can be computed using an h-de Casteljau algorithm [18]. Moreover, the h-blossoms introduced in [18], as a generalization of the classical polynomial blossoms by altering the diagonal property, allow for the understanding of many concepts related to h-Bézier curves with rather considerable ease.…”

“…Denote by P n the linear space of polynomials of degree at most n. The h-Bernstein basis over the unit interval [0, 1] of the space P n is defined by [18] An h-Bézier curve is a parametric polynomial of the form…”

A generalization of the peano kernel and its applications

“…Their work constructs not only a new type of de Boor algorithm but also novel identities via blossoms. Actually the underlying idea in [17] goes back to the work [16] which aimed to …nd a new form of blossoms to represent q-Bernstein polynomials. Just like classical Bernstein polynomials, the q-Bernstein polynomials possesses remarkable geometric and analytic properties, see [11,13].…”

q-Poisson Bases and q-Poisson Curves

“…In 2003, q-Bernstein operator [9] was used to construct q-Bezier curve due to its fine properties. In 2012, Simeonov and Goldman [10] defined q-B-splines (quantum B-splines), which were based on q-blossoming in [11].…”