Computer Aided Geometric Design最新文献

An ellipse can be uniquely determined by five tangents. Given a convex quadrilateral, there are infinitely many ellipses inscribed in it, but the one with maximum area is unique. In this paper, we give a concise and effective solution of this problem. Our solution is composed of three steps: First, we transform the problem from the maximal ellipse construction problem into the minimal quadrilateral construction problem by an affine transformation. And then, we convert the construction problem into a conditional extremum problem by analyzing the key angles. At last, we derive the solution of the conditional extremum problem with Lagrangian multiplier. Based on the conclusion, we designed an algorithm to achieve the construction. The numerical experiment shows that the ellipse constructed by our algorithm has the maximum area. It is interesting and surprising that our constructions only need to solve quadratic equations, which means the geometric information of the ellipse can even be derived with ruler and compass constructions. The solution of this problem means all the construction problems of conic with extremum area from given pure tangents are solved, which is a necessary step to solve more problems of constructing ellipses with extremum areas. Our work also provides a useful conclusion to solve the maximal inscribed ellipse problem for an arbitrary polygon in Computational Geometry.

Surface geometry plays a central role in the design of bridges, vaults and shells, using various techniques for generating a geometry which aims to balance structural, spatial, aesthetic and construction requirements.

In this paper we propose the use of surfaces defined such that given closed curves subtend a constant solid angle at all points on the surface and form its boundary. Constant solid angle surfaces enable one to control the boundary slope and hence achieve an approximately constant span-to-height ratio as the span varies, making them structurally viable for shell structures. In addition, when the entire surface boundary is in the same plane, the slope of the surface around the boundary is constant and thus follows a principal curvature direction. Such surfaces are suitable for surface grids where planar quadrilaterals meet the surface boundaries. They can also be used as the Airy stress function in the form finding of shells having forces concentrated at the corners.

Our technique employs the Gauss-Bonnet theorem to calculate the solid angle of a point in space and Newton's method to move the point onto the constant solid angle surface. We use the Biot-Savart law to find the gradient of the solid angle. The technique can be applied in parallel to each surface point without an initial mesh, opening up for future studies and other applications when boundary curves are known but the initial topology is unknown.

We show the geometrical properties, possibilities and limitations of surfaces of constant solid angle using examples in three dimensions.

Tessellating CAD models into triangular meshes is a long-lasting problem. Size field is widely used to accommodate varieties of requirements in remeshing, and it is usually discretized and optimized on a prescribed background mesh and kept constant in the subsequent remeshing procedure. Instead, we propose optimizing the size field on the current mesh, then using it as guidance to generate the next mesh. This simple strategy eliminates the need of building a proper background mesh and greatly simplifies the size field query. For better quality and convergence, we also propose a geodesic distance based initialization and adaptive re-weighting strategy in size field optimization. Similar to existing methods, we also view the remeshing of a CAD model as the remeshing of its parameterization domain, which guarantees that all the vertices lie exactly on the CAD surfaces and eliminates the need for costly and error-prone projection operations. However, for vertex smoothing which is important for mesh quality, we carefully optimize the vertex's location in the parameterization domain for the optimal Delaunay triangulation condition, along with a high-order cubature scheme for better accuracy. Experiments show that our method is fast, accurate and controllable. Compared with state-of-the-art methods, our approach is fast and usually generates meshes with smaller Hausdorff error, larger minimal angle with a comparable number of triangles.

In this paper, we address the computation of the symmetries of polynomial (and thus also rational) planar vector fields using elements from Computer Algebra. We show that they can be recovered from the symmetries of the roots of an associated univariate complex polynomial which is constructed as a generator of a certain elimination ideal. Computing symmetries of the roots of the auxiliary polynomial is a task considerably simpler than the original problem, which can be done efficiently working with classical Computer Algebra tools. Special cases, in which the group of symmetries of the polynomial roots is infinite, are separately considered and investigated. The presented theory is complemented by illustrative examples. The main steps of the procedure for investigating the symmetries of a given polynomial vector field are summarized in a flow chart for clarity.

In recent years, there has been growing interest in the representation of volumes within the field of geometric modeling (GM). While polygonal patches for surface modeling have been extensively studied, there has been little focus on the representation of polyhedral volumes. Inspired by the polygonal representation of the Generalized Bézier (GB) patch proposed by Várady et al. (2016), this paper introduces a novel method for polyhedral volumetric modeling called the Generalized Bézier (GB) volume.

GB volumes are defined over simple convex polyhedra using generalized barycentric coordinates (GBCs), with the control nets which are a direct generalization of those of tensor-product Bézier volumes. GB volumes can be smoothly connected to adjacent tensor-product Bézier or GB volumes with $G^1$ or $G^2$ continuity. Besides, when the parametric polyhedron becomes a prism, the GB volume also degenerates into a tensor-product form. We provide some practical examples to demonstrate the advantages of GB volumes. Suggestions for future work are also discussed.

The conventional wisdom in point cloud analysis predominantly explores 3D geometries. It is often achieved through the introduction of intricate learnable geometric extractors in the encoder or by deepening networks with repeated blocks. However, these methods contain a significant number of learnable parameters, resulting in substantial computational costs and imposing memory burdens on CPU/GPU. Moreover, they are primarily tailored for object-level point cloud classification and segmentation tasks, with limited extensions to crucial scene-level applications, such as autonomous driving. To this end, we introduce PointeNet, an efficient network designed specifically for point cloud analysis. PointeNet distinguishes itself with its lightweight architecture, low training cost, and plug-and-play capability, while also effectively capturing representative features. The network consists of a Multivariate Geometric Encoding (MGE) module and an optional Distance-aware Semantic Enhancement (DSE) module. MGE employs operations of sampling, grouping, pooling, and multivariate geometric aggregation to lightweightly capture and adaptively aggregate multivariate geometric features, providing a comprehensive depiction of 3D geometries. DSE, designed for real-world autonomous driving scenarios, enhances the semantic perception of point clouds, particularly for distant points. Our method demonstrates flexibility by seamlessly integrating with a classification/segmentation head or embedding into off-the-shelf 3D object detection networks, achieving notable performance improvements at a minimal cost. Extensive experiments on object-level datasets, including ModelNet40, ScanObjectNN, ShapeNetPart, and the scene-level dataset KITTI, demonstrate the superior performance of PointeNet over state-of-the-art methods in point cloud analysis. Notably, PointeNet outperforms PointMLP with significantly fewer parameters on ModelNet40, ScanObjectNN, and ShapeNetPart, and achieves a substantial improvement of over 2% in $3DA{P}_{R40}$ for PointRCNN on KITTI with a minimal parameter cost of 1.4 million. Code is publicly available at https://github.com/lipeng-gu/PointeNet.

Computer-Aided Design (CAD) software remains a pivotal tool in modern engineering and manufacturing, driving the design of a diverse range of products. In this work, we introduce VQ-CAD, the first CAD generation model based on Denoising Diffusion Probabilistic Models. This model utilizes a vector quantized diffusion model, employing multiple hierarchical codebooks generated through VQ-VAE. This integration not only offers a novel perspective on CAD model generation but also achieves state-of-the-art performance in 3D CAD model creation in a fully automatic fashion. Our model is able to recognize and incorporate implicit design constraints by simply forgoing traditional data augmentation. Furthermore, by melding our approach with CLIP, we significantly simplify the existing design process, directly generate CAD command sequences from initial design concepts represented by text or sketches, capture design intentions, and ensure designs adhere to implicit constraints.

Although significant advances have been made in the field of multi-view 3D reconstruction using implicit neural field-based methods, existing reconstruction methods overlook the estimation of the material information (e.g. the base color, albedo, roughness, and metallic) during the learning process. In this paper, we propose a novel differentiable rendering framework, named as material NueS (M-NeuS), for simultaneously achieving precise surface reconstruction and competitive material estimation. For surface reconstruction, we perform multi-view geometry optimization by proposing an enhanced-low-to-high frequency encoding registration strategy (EFERS) and a second-order interpolated signed distance function (SI-SDF) for precise details and outline reconstruction. For material estimation, inspired by the NeuS, we first propose a volume-rendering-based material estimation strategy (VMES) to estimate the base color, albedo, roughness, and metallic accurately. And then, different from most material estimation methods that need ground-truth geometric priors, we use the geometry information reconstructed in the surface reconstruction stage and the directions of incidence from different viewpoints to model a neural light field, which can extract the lighting information from image observations. Next, the extracted lighting and the estimated base color, albedo, roughness, and metallic are optimized by the physics-based rendering equation. Extensive experiments demonstrate that our M-NeuS can not only reconstruct more precise geometry surface than existing state-of-the-art (SOTA) reconstruction methods but also can estimate competitive material information: the base color, albedo, roughness, and metallic.

One of the main challenges of indoor scene synthesis is preserving the functionality of synthesized scenes to create practical and usable indoor environments. Function groups exhibit the capability of balancing the global structure and local scenes of an indoor space. In this paper, we propose a function-centric indoor scene synthesis framework, named FuncScene. Our key idea is to use function groups as an intermedium to connect the local scenes and the global structure, thus achieving a coarse-to-fine indoor scene synthesis while maintaining the functionality and practicality of synthesized scenes. Indoor scenes are synthesized by first generating function groups using generative models and then instantiating by searching and matching the specific function groups from a dataset. The proposed framework also makes it easier to achieve multi-level generation control of scene synthesis, which was challenging for previous works. Extensive experiments on various indoor scene synthesis tasks demonstrate the validity of our method. Qualitative and quantitative evaluations show the proposed framework outperforms the existing state-of-the-art.

The generalization of barycentric coordinates to arbitrary simple polygons with more than three vertices has been a subject of study for a long time. Among the different constructions proposed, mean value coordinates have emerged as a popular choice, particularly due to their suitability for the non-convex setting. Since their introduction, they have found applications in numerous fields, and several equivalent formulas for their evaluation have been presented in the literature. However, so far, there has been no study regarding their numerical stability. In this paper, we aim to investigate the numerical stability of the algorithms that compute mean value coordinates. We show that all the known methods exhibit instability in some regions of the domain. To address this problem, we introduce a new formula for computing mean value coordinates, explain how to implement it, and formally prove that our new algorithm provides a stable evaluation of mean value coordinates. We validate our results through numerical experiments.