This paper presents a new scheme for subdivision surfaces that is based on four-directional meshes. It combines geometry-sensitive refinement with convolution smoothing. The scheme has a simple, efficient implementation and generates well-shaped meshes which converge to smooth surfaces. keywords: subdivision schemes, four-directional meshes, quincunx lattice, refinenement, smoothing.

We study the positive Bergman complex B(M) of an oriented matroid M , which is a certain subcomplex of the Bergman complex B(M) of the underlying unoriented matroid M . The positive Bergman complex is defined so that given a linear ideal I with associated oriented matroid MI , the positive tropical variety associated to I is equal to the fan over B(MI). Our main result is that a certain “fine” ...

The flow polytope F G̃ is the set of nonnegative unit flows on the graph G̃. The subdivision algebra of flow polytopes prescribes a way to dissect a flow polytope F G̃ into simplices. Such a dissection is encoded by the terms of the so called reduced form of the monomial ∏ (i,j)∈E(G) xij . We prove that we can use the subdivision algebra of flow polytopes to construct not only dissections, but als...

A Roman dominating function of a graph G is a labeling f : V (G) −→ {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number γR(G) of G is the minimum of ∑ v∈V (G) f(v) over such functions. The Roman domination subdivision number sdγR(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order t...

The minimum weight feedback vertex set problem (FVS) on series-parallel graphs can be solved in O(n) time by dynamic programming. This solution, however, does not provide a “nice” certificate of optimality. We prove a min-max relation for FVS on series-parallel graphs with no induced subdivision of K2,3 (a class of graphs containing the outerplanar graphs), thereby establishing the existence of...

A new adaptive tessellation method for general CatmullClark subdivision surfaces is presented. Development of the new method is based on the observation that optimum adaptive tessellation for rendering purpose is a recursive error evaluation and globalization process. The adaptive tessellation process is done by generating an inscribing polyhedron to approximate the limit surface for each indiv...

Subdivision rules formesheswith boundary are essential for practical applications of subdivision surfaces. These rules have to result in piecewise C -continuous boundary limit curves and ensure C -continuity of the surface itself. Extending the theory of Zorin (Constr Approx 16(3):359–397, 2000), we present in this paper general necessary and sufficient conditions for C -continuity of subdivisi...

We examine the smoothness properties of the principal curvatures of subdivision surfaces near extraordinary points. In particular we give an estimate of their Lp class based on the eigen structure of the subdivision matrix. As a result we can show that the popular Loop and Catmull-Clark schemes (among many others) have square integrable principal curvatures justifying their use as shape functio...

A second order forward differences based subdivision depth computation technique for extra-ordinary Catmull-Clark subdivision surface (CCSS) patches is presented. The new technique improves a previous technique in that the computation of the subdivision depth is based on the patch’s curvature distribution, instead of its dimension. Hence, with the new technique, no excessive subdivision is need...

Quadrilateral subdivision schemes come in primal and dual varieties, splitting faces or respectively vertices. The scheme of Catmull-Clark is an example of the former, while the Doo-Sabin scheme exemplifies the latter. In this paper we consider the construction of an increasing sequence of alternating primal/dual quadrilateral subdivision schemes based on a simple averaging approach. Beginning ...