and Applied Analysis 3 For the analysis of subdivision scheme with mask α, it is very practical to consider the z-transform of the mask,

Bivariate interpolatory Hermite subdivision schemes have recently been applied to build free-form subdivision surfaces. It is well known to geometric modelling practitioners that interpolatory schemes typically lead to “unfair” surfaces—surfaces with unwanted wiggles or undulations—and noninterpolatory (a.k.a. approximating in the CAGD community) schemes are much preferred in geometric modellin...

Problem statement: Subdivision surfaces were applied to the entire meshes in order to produce smooth surfaces refinement from coarse mesh. Several schemes had been introduced in this area to provide a set of rules to converge smooth surfaces. However, to compute and render all the vertices are really inconvenient in terms of memory consumption and runtime during the subdivision process. It will...

In this paper, we introduce two generalizations of midpoint subdivision and analyze the smoothness of the resulting subdivision surfaces at regular and extraordinary points. The smoothing operators used in midpoint and mid-edge subdivision connect the midpoints of adjacent faces or of adjacent edges, respectively. An arbitrary combination of these two operators and the refinement operator that ...

Subdivision has gained popularity in computer graphics and shape modeling during the past two decades, yet volumetric subdivision has received much less attention. In this paper, we develop a new subdivision scheme which can interpolate all of the initial control points in 3D and generate a continuous volume in the limit. We devise a set of solid subdivision rules to facilitate a simple subdivi...

‎in this paper we prove that a simplicial complex is determined‎ ‎uniquely up to isomorphism by its barycentric subdivision as well as‎ ‎its comparability graph‎. ‎we also put together several algebraic‎, ‎combinatorial and topological invariants of simplicial complexes‎.

We show that any rectilinear polygonal subdivision in the plane can be converted into a “guillotine” subdivision whose length is at most twice that of the original subdivision. “Guillotine” subdivisions have a simple recursive structure that allows one to search for “optimal” such subdivisions in polynomial time, using dynamic programming. In particular, a consequence of our main theorem is a v...

Linear stationary subdivision rules take a sequence of input data and produce ever denser sequences of subdivided data from it. They are employed in multiresolution modeling and have intimate connections with wavelet and more general pyramid transforms. Data which naturally do not live in a vector space, but in a nonlinear geometry like a surface, symmetric space, or a Lie group (e.g. motion ca...

The concept of ε-exact path planning is a theoretically sound alternative to the standard exact algorithms, and provides much stronger guarantees than probabilistic or sampling algorithms. It opens the way for the introduction of soft predicates in the context of subdivision algorithm. Taking a leaf from the great success of the Probabilistic Road Map (PRM) framework, we formulate an analogous ...

1 Laboratoire des Images et des Signaux, 961 rue de la Houille Blanche, Domaine Universitaire, 38000 Grenoble, France [email protected] 2 Computer Graphics Group, RWTH-Aachen, Ahornstrasse 55, 52074 Aachen, Germany [email protected] 3 University of Cambridge Computer Laboratory, William Gates Building, JJ Thomson Avenue, Cambridge CB3 0FD, England [email protected] 4 Numerical Geom...