A survey of partial differential equations in geometric design

YesComputer aided geometric design is an area where the improvement of surface generation techniques is an everlasting demand since faster and more accurate geometric models are required. Traditional methods for generating surfaces were initially mainly based upon interpolation algorithms. Recently, partial differential equations (PDE) were introduced as a valuable tool for geometric modelling since they offer a number of features from which these areas can benefit. This work summarises the uses given to PDE surfaces as a surface generation technique togethe

Error-Bounded and Feature Preserving Surface Remeshing with Minimal Angle Improvement

The typical goal of surface remeshing consists in finding a mesh that is (1) geometrically faithful to the original geometry, (2) as coarse as possible to obtain a low-complexity representation and (3) free of bad elements that would hamper the desired application. In this paper, we design an algorithm to address all three optimization goals simultaneously. The user specifies desired bounds on approximation error {\delta}, minimal interior angle {\theta} and maximum mesh complexity N (number of vertices). Since such a desired mesh might not even exist, our optimization framework treats only the approximation error bound {\delta} as a hard constraint and the other two criteria as optimization goals. More specifically, we iteratively perform carefully prioritized local operators, whenever they do not violate the approximation error bound and improve the mesh otherwise. In this way our optimization framework greedily searches for the coarsest mesh with minimal interior angle above {\theta} and approximation error bounded by {\delta}. Fast runtime is enabled by a local approximation error estimation, while implicit feature preservation is obtained by specifically designed vertex relocation operators. Experiments show that our approach delivers high-quality meshes with implicitly preserved features and better balances between geometric fidelity, mesh complexity and element quality than the state-of-the-art.Comment: 14 pages, 20 figures. Submitted to IEEE Transactions on Visualization and Computer Graphic

Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces

Let $(M,g)$ be a connected, closed, orientable Riemannian surface and denote by $\lambda_k(M,g)$ the $k$-th eigenvalue of the Laplace-Beltrami operator on $(M,g)$. In this paper, we consider the mapping $(M, g)\mapsto \lambda_k(M,g)$. We propose a computational method for finding the conformal spectrum $\Lambda^c_k(M,[g_0])$, which is defined by the eigenvalue optimization problem of maximizing $\lambda_k(M,g)$ for $k$ fixed as $g$ varies within a conformal class $[g_0]$ of fixed volume $textrm{vol}(M,g) = 1$. We also propose a computational method for the problem where $M$ is additionally allowed to vary over surfaces with fixed genus, $\gamma$. This is known as the topological spectrum for genus $\gamma$ and denoted by $\Lambda^t_k(\gamma)$. Our computations support a conjecture of N. Nadirashvili (2002) that $\Lambda^t_k(0) = 8 \pi k$, attained by a sequence of surfaces degenerating to a union of $k$ identical round spheres. Furthermore, based on our computations, we conjecture that $\Lambda^t_k(1) = \frac{8\pi^2}{\sqrt{3}} + 8\pi (k-1)$, attained by a sequence of surfaces degenerating into a union of an equilateral flat torus and $k-1$ identical round spheres. The values are compared to several surfaces where the Laplace-Beltrami eigenvalues are well-known, including spheres, flat tori, and embedded tori. In particular, we show that among flat tori of volume one, the $k$-th Laplace-Beltrami eigenvalue has a local maximum with value $\lambda_k = 4\pi^2 \left\lceil \frac{k}{2} \right\rceil^2 \left( \left\lceil \frac{k}{2} \right\rceil^2 - \frac{1}{4}\right)^{-\frac{1}{2}}$. Several properties are also studied computationally, including uniqueness, symmetry, and eigenvalue multiplicity.Comment: 43 pages, 18 figure

A Concept For Surface Reconstruction From Digitised Data

Reverse engineering and in particular the reconstruction of surfaces from digitized data is an important task in industry. With the development of new digitizing technologies such as laser or photogrammetry, real objects can be measured or digitized quickly and cost effectively. The result of the digitizing process is a set of discrete 3D sample points. These sample points have to be converted into a mathematical, continuous surface description, which can be further processed in different computer applications. The main goal of this work is to develop a concept for such a computer aided surface generation tool, that supports the new scanning technologies and meets the requirements in industry towards such a product. Therefore first, the requirements to be met by a surface reconstruction tool are determined. This marketing study has been done by analysing different departments of several companies. As a result, a catalogue of requirements is developed. The number of tasks and applications shows the importance of a fast and precise computer aided reconstruction tool in industry. The main result from the analysis is, that many important applications such as stereolithographie, copy milling etc. are based on triangular meshes or they are able to handle these polygonal surfaces. Secondly the digitizer, currently available on the market and used in industry are analysed. Any scanning system has its strength and weaknesses. A typical problem in digitizing is, that some areas of a model cannot be digitized due to occlusion or obstruction. The systems are also different in terms of accuracy, flexibility etc. The analysis of the systems leads to a second catalogue of requirements and tasks, which have to be solved in order to provide a complete and effective software tool. The analysis also shows, that the reconstruction problem cannot be solved fully automatically due to many limitations of the scanning technologies. Based on the two requirements, a concept for a software tool in order to process digitized data is developed and presented. The concept is restricted to the generation of polygonal surfaces. It combines automatic processes, such as the generation of triangular meshes from digitized data, as well as user interactive tools such as the reconstruction of sharp corners or the compensation of the scanning probe radius in tactile measured data. The most difficult problem in this reconstruction process is the automatic generation of a surface from discrete measured sample points. Hence, an algorithm for generating triangular meshes from digitized data has been developed. The algorithm is based on the principle of multiple view combination. The proposed approach is able to handle large numbers of data points (examples with up to 20 million data points were processed). Two pre-processing algorithm for triangle decimation and surface smoothing are also presented and part of the mesh generation process. Several practical examples, which show the effectiveness, robustness and reliability of the algorithm are presented

Survey of two-dimensional acute triangulations

AbstractWe give a brief introduction to the topic of two-dimensional acute triangulations, mention results on related areas, survey existing achievements–with emphasis on recent activity–and list related open problems, both concrete and conceptual