Subdivision schemes provide efficient algorithms both for the design and processing of arbitrary control meshes. Theoretically, these techniques are often considered as an elegant algorithmic way to approximate a desired surface from a given surface. In practice, controlling the accuracy of control meshes with regard to the limit surface remains difficult. In this paper, from a bound of the dis...

Subdivision surfaces refer to a class of modelling schemes that define an object through recursive subdivision starting from an initial control mesh. Similar to B-splines, the final surface is defined by the vertices of the initial control mesh. These surfaces were initially conceived as an extension of splines in modelling objects with a control mesh of arbitrary topology. They exhibit a numbe...

Our goal is to find subdivision rules at creases in arbitrary degree subdivision for piece-wise polynomial curves, but without introducing new control points e.g. by knot insertion. Crease rules are well understood for low degree (cubic and lower) curves. We compare three main approaches: knot insertion, ghost points, and modifying subdivision rules. While knot insertion and ghost points work f...

Subdivision provides an efficient method to generate smooth curves and surfaces. Recently matrixvalued subdivision schemes were introduced to provide more flexibility and smaller subdivision templates for curve and surface design. For matrix-valued subdivision, the input is a set of vectors with the first components being the vertices of the control polygon (or the control net for surface subdi...

Starting with values {f 0 j }j∈Z assigned to the integers, a binary subdivision scheme defines recursively values {fk j }j∈Z, respectively assigned to the binary points {2−kj}j∈Z. The purpose of subdivision analysis is to study the convergence of such processes and to establish the existence of a limit function on R and its smoothness class. A general treatment of uniform subdivision can be fou...

Broad-scale destruction and fragmentation of native vegetation is a highly visible result of human land-use throughout the world (Chapter 4). From the Atlantic Forests of South America to the tropical forests of Southeast Asia, and in many other regions on Earth, much of the original vegetation now remains only as fragments amidst expanses of land committed to feeding and housing human beings. ...

‎The distinguishing number (resp. index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$‎ ‎such that $G$ has an vertex labeling (resp. edge labeling) with $d$ labels that is preserved only by a trivial‎ ‎automorphism‎. ‎For any $n in mathbb{N}$‎, ‎the $n$-subdivision of $G$ is a simple graph $G^{frac{1}{n}}$ which is constructed by replacing each edge of $G$ with a path of length $n$...

Haj\'os conjectured that every graph containing no subdivision of the complete $K_{s+1}$ is properly $s$-colorable. This conjecture was disproved by Catlin. Indeed, maximum chromatic number such graphs $\Omega(s^2/\log s)$. We prove $O(s)$ colors are enough for a weakening this only requires monochromatic component to have bounded size (so-called clustered coloring). Our approach leads more res...