A normalized basis for cubic super spline space on Powell–Sabin triangulation

A normalized representation of super splines of arbitrary degree on Powell–Sabin triangulations

“…It extends the representation for S 1 2 ( * ) given in [3] and the representation for S 2,3 5 ( * ) presented in [16]. Recently, a similar construction for S 1,2 3 ( * ) was derived in [10]. In this section, these results are extended to a normalized representation for S d ( * ) with arbitrary d ∈ N, and its main properties are proved.…”

“…Similarly as in [24] for d = 2, polynomial splines of arbitrary degree d can be generalized to rational splines. In [9,10,18] quasi-interpolants for splines of degrees 3 and 3r − 1, r ∈ N, were studied. These results can also be adapted for general degree.…”

“…The paper also deals with a construction of normalized bases for the discussed family of spaces. The derived B-spline functions generalize the basis functions of degrees 3 and 3r − 1 provided in [10,17,18]. Since splines in a B-representation play an important role in approximation theory and computer aided geometric design, this result may be interesting from many applicative viewpoints.…”

“…Recently, several papers have dealt with the construction of spline spaces on Powell-Sabin triangulations with bases possessing this property. Namely, a family of spline spaces of arbitrary smoothness was proposed in [17], and three different cubic spline spaces were studied in [5,10,19]. All of them have similar normalized representations that generalize the representation of the quadratic space.…”

A new B-spline representation for cubic splines over Powell–Sabin triangulations

“…Recently, basis functions with similar properties have been constructed for certain Powell-Sabin spline spaces of higher degree and smoothness. In particular, we mention C 1 cubics (Lamnii et al, 2014), C 2 quintics (Speleers, 2010a), and a family of splines of smoothness r and polynomial degree 3r − 1 (Speleers, 2013a). Local super-smoothness has been imposed in order to simplify their construction and to reduce their number of degrees of freedom while maintaining the full order of approximation.…”