Analyzing midpoint subdivision

Midpoint subdivision generalizes the Lane–Riesenfeld algorithm for uniform tensor product splines and can also be applied to non-regular meshes. For example, midpoint subdivision of degree 2 is a specific Doo–Sabin algorithm and midpoint subdivision of degree 3 is a specific Catmull–Clark algorithm. In 2001, Zorin and Schröder were able to prove C¹-continuity for midpoint subdivision surfaces analytically up to degree 9. Here, we develop general analysis tools to show that the limiting surfaces under midpoint subdivision of any degree ?2 are C¹-continuous at their extraordinary points.     

Artifact analysis on triangular box-splines and subdivision surfaces defined by triangular polyhedra

Surface artifacts are features in a surface which cannot be avoided by movement of control points. They are present in B-splines, box splines and subdivision surfaces. We showed how the subdivision process can be used as a tool to analyse artifacts in surfaces defined by quadrilateral polyhedra ( [Sabin et al., 2005] and [Augsd?rfer et al., 2011]).In this paper we are utilising the subdivision process to develop a generic expression which can be employed to determine the magnitude of artifacts in surfaces defined by any regular triangular polyhedra. We demonstrate the method by analysing box-splines and regular regions of subdivision surfaces based on triangular meshes: Loop subdivision, Butterfly subdivision and a novel interpolating scheme with two smoothing stages. We compare our results for surfaces defined by triangular polyhedra to those for surfaces defined by quadrilateral polyhedra.     

A direct approach for subdivision surface fitting from a dense triangle mesh

This article presents a new and direct approach for fitting a subdivision surface from an irregular and dense triangle mesh of arbitrary topological type. All feature edges and feature vertices of the original mesh model are first identified. A topology- and feature-preserving mesh simplification algorithm is developed to further simplify the dense triangle mesh into a coarse mesh. A subdivision surface with exactly the same topological structure and sharp features as that of the simplified mesh is finally fitted from a subset of vertices of the original dense mesh. During the fitting process, both the position masks and subdivision rules are used for setting up the fitting equation. Some examples are provided to demonstrate the proposed approach.     

曲线插值约束的LOOP细分曲面

提出基于Loop细分方法的曲线插值方法,不需要修改细分规则,只需以插值曲线的控制多边形为中心多边形,向其两侧构造对称三角网格带,该对称三角网格带将收敛于插值曲线。因此,包含有该三角网格带的多面体网格的极限曲面将经过插值曲线。若要插值多条相交曲线只需在交点处构造全对称三角网格。运用该方法可在三角网格生成的细分曲面中插值多达六条的相交曲线。     

Improved error estimate for extraordinary Catmull–Clark subdivision surface patches

Based on an optimal estimate of the convergence rate of the second order norm, an improved error estimate for extraordinary Catmull–Clark subdivision surface (CCSS) patches is proposed. If the valence of the extraordinary vertex of an extraordinary CCSS patch is even, a tighter error bound and, consequently, a more precise subdivision depth for a given error tolerance, can be obtained. Furthermore, examples of adaptive subdivision illustrate the practicability of the error estimation approach.     

Evaluation of piecewise smooth subdivision surfaces

Published online: 23 July 2002     

Error bounds for Loop subdivision surfaces

In this paper, we obtain the error bounds on the distance between a Loop subdivision surface and its control mesh. Both local and global bounds are derived by means of computing and analysing the control meshes with two rounds of refinement directly. The bounds can be expressed with the maximum edge length of all triangles in the initial control mesh. Our results can be used as posterior estimates and also can be used to predict the subdivision depth for any given tolerance.     

改进三棱柱体剖分算法研究

针对现有矿山储量计算方法计算结果精度低以及边界条件适应能力差的问题,提出了改进三棱柱体剖分算法:结合改进三棱柱顶面、底面及三角形切割截面,按不同规则将三棱柱体剖分为不同的三角锥;根据边界控制条件得到储量范围,结合地质统计学,计算控制范围内所有改进三棱柱体的体积及储量。     

Interpolation by geometric algorithm

We present a novel geometric algorithm to construct a smooth surface that interpolates a triangular or a quadrilateral mesh of arbitrary topological type formed by n vertices. Although our method can be applied to B-spline surfaces and subdivision surfaces of all kinds, we illustrate our algorithm focusing on Loop subdivision surfaces as most of the meshes are in triangular form. We start our algorithm by assuming that the given triangular mesh is a control net of a Loop subdivision surface. The control points are iteratively updated globally by a simple local point-surface distance computation and an offsetting procedure without solving a linear system. The complexity of our algorithm is O(mn) where n is the number of vertices and m is the number of iterations. The number of iterations m depends on the fineness of the mesh and accuracy required.