@article{MahdaviAmiri201612,
title = "Hierarchical grid conversion ",
journal = "Computer-Aided Design ",
volume = "79",
number = "",
pages = "12 - 26",
year = "2016",
note = "",
issn = "0010-4485",
doi = "http://dx.doi.org/10.1016/j.cad.2016.04.005",
url = "http://www.sciencedirect.com/science/article/pii/S0010448516300173",
author = "Ali Mahdavi-Amiri and Erika Harrison and Faramarz Samavati",
keywords = "Refinements",
keywords = "Grid conversion",
keywords = "Patch-based data structures",
keywords = "Transformations",
keywords = "Semiregular",
keywords = "Subdivision ",
abstract = "Abstract Hierarchical grids appear in various applications in computer 
graphics such as subdivision and multiresolution surfaces, and terrain models. Since 
the different grid types perform better at different tasks, it is desired to switch 
between regular grids to take advantages of these grids. Based on a 2D domain obtained 
from the connectivity information of a mesh, we can define simple conversions to switch 
between regular grids. In this paper, we introduce a general framework that can be used 
to convert a given grid to another and we discuss the properties of these refinements such 
as their transformations. This framework is hierarchical meaning that it provides conversions 
between meshes at different level of refinement. To describe the use of this framework, we 
define new regular and near-regular refinements with good properties such as small factors. 
We also describe how grid conversion enables us to use patch-based data structures for hexagonal
cells and near-regular refinements. To do so, meshes are converted to a set of quadrilateral 
patches that can be stored in simple structures. Near-regular refinements are also supported by 
defining two sets of neighborhood vectors that connect a vertex to its neighbors and are useful 
to address connectivity queries. "
}