This paper presents a new scheme for subdivision surfaces based on four-directional meshes. It combines geometry-sensitive refinement with convolution smoothing. The scheme has a simple, efficient implementation and generates smooth well-shaped meshes.

Let ∆ be the area of the fundamental triangle ABC of barycentric coordinates and let α = cotA, β = cotB, γ = cotC. The vectors vi = [xi, yi, zi] (i = 1, 2) have the scalar product 2∆(αx1x2 + βy1y2 + γz1z2). This fact implies all important formulas about metrical relations of points and lines. The main and probably new results are Theorems 1 and 8.

In this paper, we introduce a new subdivision scheme which we call honeycomb subdivision. After one iteration of the scheme each vertex becomes exactly 3-valent and with consecutive applications regular regions strongly resembles a honeycomb. This scheme can be considered as a dual for triangle schemes. The major advantage of the new scheme is that it creates a natural looking mesh structure.

A vertex coloring of a graph G=(V,E) is called an exact square G if any pair vertices at distance 2 receive distinct colors. The minimum number colors required by the chromatic and denoted χ[#2](G). set clique are 2. G, ω[#2](G), maximum cardinality clearly, ω[#2](G)≤χ[#2](G). In this article, we give tight upper bounds various standard operations, including Cartesian product, strong lexicograp...

For each infinite cardinal k we give several necessary and sufficient conditions for a graph not to contain a minor isomorphic to the infinite kbranching tree in terms of a certain kind of a "tree-decomposition," in terms of a "path-decomposition," and also in terms of a "cops-and-robber game." We also give necessary and sufficient conditions for a graph not to contain a subgraph isomorphic to ...

It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical points. The proof is stated in terms of discrete Morse functions on posets.

Subdivision surface intersections can be costly to compute. They require the intersection of high resolution meshes in order to obtain accurate results, which can lead to slow performance and high memory usage. In this paper we show how the strong convex hull property can lead to a method for efficiently computing intersections at high resolutions. Consequently, the method can be used with any ...

Given a digraphH, anH-subdivision is any digraph obtained by replacing each arc uv of H with a (directed) u− v path of arbitrary length. A directed graph D is H-linked if every injective functionf : V (H) → V (D) extends to an H-subdivision in G. The H-linkage property has been well-studied in undirected graphs, and in both the directed and undirected case generalizes the notions of k-linked an...

A graph is planar if and only if it does not contain a Kuratowski subdivision. Hence such a subdivision can be used as a witness for non-planarity. Modern planarity testing algorithms allow to extract a single such witness in linear time. We present the first linear time algorithm which is able to extract multiple Kuratowski subdivisions at once. This is of particular interest for, e.g., Branch...