Gang Mei

The Modified Direct Method: An Iterative Approach for Smoothing Planar Meshes☆☆☆

Abstract The Modified Direct Method (MDM) is an iterative method for smoothing planar triangular, quadrilateral and tri-quad meshes, which has been developed from the non-iterative smoothing method for planar meshes that was originated by Balendran [1] . Its simple aim is to make triangular elements as close to equilateral as possible and quadrilateral elements as close to square as possible. The MDM is the computationally simpler of the two methods. The performance of the MDM for both triangular and quadrilateral meshes is effectively identical to that of Laplacian smoothing; however, it outperforms Laplacian smoothing for tri-quad meshes. Test examples show that the MDM is always convergent for triangular, quadrilateral and tri-quad meshes.

Ear-Clipping Based Algorithms of Generating High-Quality Polygon Triangulation

A basic and an improved ear-clipping based algorithm for triangulating simple polygons and polygons with holes are presented. In the basic version, the ear with smallest interior angle is always selected to be cut in order to create fewer sliver triangles. To reduce sliver triangles in further, a bound of angle is set to determine whether a newly formed triangle has sharp angles, and edge swapping is accepted when the triangle is sharp. To apply the two algorithms on polygons with holes, ‘Bridge’ edges are created to transform a polygon with holes to a degenerate polygon which can be triangulated by the two algorithms. Applications show that the basic algorithm can avoid creating sliver triangles and obtain better triangulations than the traditional ear-clipping algorithm, and the improved algorithm can in further reduce sliver triangles effectively. Both of the algorithms run in O(n 2) time and O(n) space.

An algorithm for finding convex hulls of planar point sets

This paper presents an alternate choice of computing the convex hulls (CHs) for planar point sets. We firstly discard the interior points and then sort the remaining vertices by x-/y- coordinates separately, and later create a group of quadrilaterals (e-Quads) recursively according to the sequences of the sorted lists of points. Finally, the desired CH is built based on a simple polygon derived from all e-Quads. Besides the preprocessing for original planar point sets, this algorithm has another mechanism of discarding interior point when form e-Quads and assemble the simple polygon. Compared with three popular CH algorithms, the proposed algorithm can generate CHs faster than the three but has a penalty in space cost.

3D geological modeling and visualization of rock masses based on Google Earth: A case study

Google Earth (GE) has become a powerful tool for geological modeling and visualization. An interesting and useful feature of GE, Google Street View, can allow the GE users to view geological structure such as layers of rock masses at a field site. In this paper, we introduce a practical solution for building 3D geological models for rock masses based on the data acquired by use with GE. A real study case at Haut-Barr, France is presented to demonstrate our solution. We first locate the position of Haut-Barr in GE, and then determine the shape and scale of the rock masses in the study area, and thirdly acquire the layout of layers of rock masses in the Google Street View, and finally create the approximate 3D geological models by extruding and intersecting. The generated 3D geological models can simply reflect the basic structure of the rock masses at Haut-Barr, and can be used for visualizing the rock bodies interactively.

T-Base: A Triangle-Based Iterative Algorithm for Smoothing Quadrilateral Meshes

We present a novel approach named T-Base for smoothing planar and surface quadrilateral meshes. Our motivation is that the best shape of quadrilateral element—square—can be virtually divided into a pair of equilateral right triangles by any of its diagonals. When move a node to smooth a quadrilateral, it is optimal to make a pair of triangles divided by a diagonal be equilateral right triangles separately. The finally smoothed position is obtained by weighting all individual optimal positions. Three variants are produced according to the determination of weights. Tests by the T-Base are given and compared with Laplacian smoothing: The Vari.1 of T-Base is effectively identical to Laplacian smoothing for planar quad meshes, while Vari.2 is the best. For the quad mesh on underlying parametric surface and interpolation surface, Vari.2 and Vari.1 are best, respectively.

A hybrid approach for optimizing planar triangular meshes

Modified Direct Method (MDM) is an iterative scheme based on Jacobi iterations for smoothing planar meshes [4]. The basic idea behind MDM is to make any triangular element be as close to an equilateral triangle as possible. Based on the MDM, a length-weighted MDM is proposed and then combined with edge swapping. In length-weighted MDM, weights of each neighboring node of a smoothed node are determined by the length of its opposite edge. Also, the MDM, Laplacian smoothing and length-weighted MDM are all combined with edge swapping, and then implemented and compared on both structured and unstructured triangular meshes. Examples show that length-weighted MDM is better than the MDM and Laplacian smoothing for structured mesh but worse for unstructured mesh. The hybrid approach of combining length-weighted MDM and edge swapping is much better and can obtain more even optimized meshes than other two hybrid approaches.

Discrete surface modeling based on Google Earth: A case study

Google Earth (GE) has become a powerful tool for geological, geophysical and geographical modeling; yet GE can be accepted to acquire elevation data of terrain. In this paper, we present a real study case of building the discrete surface model (DSM) at Haut-Barr Castle in France based on the elevation data of terrain points extracted from GE using the COM API. We first locate the position of Haut-Barr Castle and determine the region of the study area, then extract elevation data of terrain at Haut-Barr, and thirdly create a planar triangular mesh that covers the study area and finally generate the desired DSM by calculating the elevation of vertices in the planar mesh via interpolating with Universal Kriging (UK) and Inverse Distance Weighting (IDW). The generated DSM can reflect the features of the ground surface at Haut-Barr well, and can be used for constructing the Sealed Engineering Geological Model (SEGM) in further step.