6 research outputs found
A Hermite interpolatory subdivision scheme for -quintics on the Powell-Sabin 12-split
In order to construct a -quadratic spline over an arbitrary
triangulation, one can split each triangle into 12 subtriangles, resulting in a
finer triangulation known as the Powell-Sabin 12-split. It has been shown
previously that the corresponding spline surface can be plotted quickly by
means of a Hermite subdivision scheme. In this paper we introduce a nodal
macro-element on the 12-split for the space of quintic splines that are locally
and globally . For quickly evaluating any such spline, a Hermite
subdivision scheme is derived, implemented, and tested in the computer algebra
system Sage. Using the available first derivatives for Phong shading, visually
appealing plots can be generated after just a couple of refinements.Comment: 17 pages, 7 figure
B-spline-like bases for cubics on the Powell-Sabin 12-split
For spaces of constant, linear, and quadratic splines of maximal smoothness
on the Powell-Sabin 12-split of a triangle, the so-called S-bases were recently
introduced. These are simplex spline bases with B-spline-like properties on the
12-split of a single triangle, which are tied together across triangles in a
B\'ezier-like manner.
In this paper we give a formal definition of an S-basis in terms of certain
basic properties. We proceed to investigate the existence of S-bases for the
aforementioned spaces and additionally the cubic case, resulting in an
exhaustive list. From their nature as simplex splines, we derive simple
differentiation and recurrence formulas to other S-bases. We establish a
Marsden identity that gives rise to various quasi-interpolants and domain
points forming an intuitive control net, in terms of which conditions for
-, -, and -smoothness are derived
Smooth Macro-Elements on Powell-Sabin-12 Splits
Abstract. Macro-elements of smoothness C r are constructed on Powell-Sabin-12 splits of a triangle for all r ≥ 0. These new elements complement those recently constructed on Powell-Sabin-6 splits [5,12], and can be used to construct convenient superspline spaces with stable local bases and full approximation power that can be applied to the solution of boundary-value problems and for interpolation of Hermite data