The polygonal approximation problem is a primary problem in computer graphics, pattern recognition, CAD/CAM, etc. In $R^2$, the cone intersection method (CIM) is one of the most eÆcient algorithms for approximating polygonal curves. With CIM Eu and Toussaint, by imposing an additional constraint and changing the given error criteria, resolve the three-dimensional weighted minimum number polygonal approximation problem with the parallel-strip error criterion (PS-WMN) under $L_2$ norm. In this paper, without any additional constraint and change of the error criteria, a CIM solution to the same problem with the line segment error criterion (LS-WMN) is presented, which is more frequently encountered than the PS-WMN is. Its time complexity is $O(n^3)$, and the space complexity is $O(n^2)$. An approximation algorithm is also presented, which takes $O(n^2)$ time and $O(n)$ space. Results of some examples are given to illustrate the eÆciency of these algorithms.

In this paper, we introduce a novel method that can generate a sequence of physical transformations between 3D models with different shape and topology. Feasible transformations are realized on a chain structure with connected components that are 3D printed. Collision-free motions are computed to transform between different configurations of the 3D printed chain structure. To realize the transformation between different 3D models, we first voxelize these input models into similar number of voxels. The challenging part of our approach is to generate a simple path --- as a chain configuration to connect most voxels. A layer-based algorithm is developed with theoretical guarantee of the existence and the path length. We find that collision-free motion sequence can always be generated when using a straight line as the intermediate configuration of transformation. The effectiveness of our method is demonstrated by both the simulation and the experimental tests taken on 3D printed chains.

Delaunay meshes (DM) are a special type of triangle mesh where the local Delaunay condition holds everywhere. We present an efficient algorithm to convert an arbitrary manifold triangle mesh M into a Delaunay mesh. We show that the constructed DM has O(Kn) vertices, where n is the number of vertices in M and K is a model-dependent constant. We also develop a novel algorithm to simplify Delaunay meshes, allowing a smooth choice of detail levels. Our methods are conceptually simple, theoretically sound and easy to implement. The DM construction algorithm also scales well due to its O(nK logK) time complexity. Delaunay meshes have many favorable geometric and numerical properties. For example, a DM has exactly the same geometry as the input mesh, and it can be encoded by any mesh data structure. Moreover, the empty geodesic circumcircle property implies that the commonly used cotangent Laplace-Beltrami operator has non-negative weights. Therefore, the existing digital geometry processing algorithms can benefit the numerical stability of DM without changing any codes. We observe that DMs can improve the accuracy of the heat method for computing geodesic distances. Also, popular parameterization techniques, such as discrete harmonic mapping, produce more stable results on the DMs than on the input meshes.

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