A bivariate C1 subdivision scheme based on cubic half-box splines

Abstract

Among the bivariate subdivision schemes available, spline-based schemes, such as Catmull-Clark and Loop, are the most commonly used ones. These schemes have known continuity and can be evaluated at arbitrary parameter values. In this work, we develop a C 1 spline-based scheme based on cubic half-box splines. Although the individual surface patches are triangular, the associated control net is three-valent and thus consists in general of mostly hexagons. In addition to introducing stencils that can be applied in extraordinary regions of the mesh, we also consider boundaries. Moreover, we show that the scheme exhibits ineffective eigenvectors. Finally, we briefly consider architectural geometry and isogeometric analysis as selected applications.

Mixed honeycomb pushing refinement

“…Thus all polyhedra P i have equally many non-valence-6 faces and roughly four times as many hexagonal faces as their predecessors P i−1 . This is a typical property shared by other subdivision algorithms as, e. g., the half box spline subdivision algorithm [18] and its extension [2]. Topologically, they generate the same edge graphs.…”

“…These facts: (1) that honeycomb surfaces have interesting special shape characteristics or artefacts, (2) that the primal honeycomb algorithm has been the only interpolatory convexity preserving C 1 scheme for closed surfaces, and 3that the dual honeycomb algorithm has been the only candidate for a surface corner cutting scheme inspired us to investigate the honeycomb algorithm (1) to develop tools that could help to (better) understand corner cutting for surfaces, (2) to modify the honeycomb algorithm so as to get rid of the planar and line segments, and (3) to come up with first genuine corner cutting schemes for polyhedra.…”

Completeness characterization of Type-I box splines

“…Similarly, taking the triangles 2 and 3 , we get ST( 2 , 4 ) = {2, 3} and I = 3 i=2 I d i = {(0, 0, k) : k = 0, 1}. The polynomials f 2 and f 4 and also their derivatives ∂(f i )/∂(x − y), for i = 2 and 3, have the same value at (1,1).…”

“…Other subdivision schemes has been explored in [15]. A C 1 -continuous scheme based on cubic half-box splines was presented in [1].…”

Volume reconstruction based on the six-direction cubic box-spline

“…Convolving the characteristic function of half of a box in 2D, i.e. of a triangle, yields half-box spline spaces with properties akin to box splines [55,23,3,54,1]. Alternatively, one can juxtapose non-centered boxes to form the Voronoi cell of a lattice, i.e.…”