A Unified Approach to Conic Visibility

{In this paper we present linear time algorithms for computing the shortest path tree from a point and the weak visibility polygon of an arc inside a triangulated curved polygon. We also present a linear time algorithm for computing the planar subdivision (in the parametric space) of the set of rays emanating from a fixed arc, such that each face of the subdivision corresponds to rays hitting the same arc of the polygon. Although these results, which involve nontrivial generalizations of known results for rectilinear polygons, may have some interest in its own right, the main result of this paper is a linear time algorithm for computing the conic (circular, elliptic, parabolic, and hyperbolic) visibility polygon of a point inside a simple polygon. The main advantage of our technique over previous results on circular visibility is that it provides a simple, unified approach to conic visibility. Finally, we present a linear time algorithm for computing the planar subdivision, in the parametric space, of two-parametric families of conic rays emanating from a fixed point, such that each face of the subdivision corresponds to conic rays hitting the same edge of the polygon. All these algorithms are asymptotically optimal.} Received August 21, 1997; revised December 27, 1998.     

Efficient direct rendering of deforming surfaces via shared subdivision trees

In this paper, we present a subdivision-based approach to rasterize implicit surfaces embedded in volumetric Bézier patches undergoing a nonlinear deformation. Subdividing a given patch into simpler patches to perform the surface rasterization task is numerically robust, and allows guaranteeing visual accuracy even in the presence of geometric degeneracies. However, due to its memory requirements and slow convergence rates, subdivision is challenging to be used in an interactive environment. Unlike previous methods employing subdivision, our approach is based on the idea where for a given patch only one subdivision tree is maintained and shared among pixels. Furthermore, as the geometry of the object changes from frame to frame, a flexible data structure is proposed to manage the geometrically varying Bézier patches. The resulting algorithm is general and maps well to parallel computing platforms such as CUDA. We demonstrate on a variety of representative graphics and visualization examples that our GPU scheme scales well and achieves up to real-time performance on consumer-level graphics cards by guaranteeing visual accuracy.     

一种平面点集Voronoi 图的细分算法

Voronoi 图是计算几何中的重要概念之一,在计算机图形学、计算几何、 计算机辅助几何设计、有限元网格划分、机器人轨迹控制、模式识别、气象学和地质学研究 中得到广泛应用。借助于四叉树和区间算术,提出了一种新的构造平面点集Voronoi 图的细 分算法, 并且和经典的增量算法、栅格扩张法进行了比较, 结果显示新细分算法更为有效。 最重要的是细分算法原理简单,很容易编程实现。     

Real-time ray casting of algebraic B-spline surfaces

Piecewise algebraic B-spline surfaces (ABS surfaces) are capable of modeling globally smooth shapes of arbitrary topology. These can be potentially applied in geometric modeling, scientific visualization, computer animation and mathematical illustration. However, real-time ray casting the surface is still an obstacle for interactive applications, due to the large amount of numerical root findings of nonlinear polynomial systems that are required. In this paper, we present a GPU-based real-time ray casting method for ABS surfaces. To explore the powerful parallel computing capacity of contemporary GPUs, we adopt iterative numerical root-finding algorithms, e.g., the Newton-Raphson and regula falsi algorithms, rather than recursive ones. To facilitate convergence of the Newton-Raphson or regula falsi algorithm, their initial guesses are determined through rasterization of the isotopic isosurface, and the isosurface is generated based on regular criteria for surface domain subdivision. Meanwhile, polar surfaces are adopted to identify single roots or to isolate different roots, i.e., ray and surface intersections. As an important geometric feature, the silhouette curve is elaborately computed to floating-point accuracy, which can be applied in further anti-aliasing processes. The experimental results show that the proposed method can render thousands of piecewise algebraic surface patches of degrees 6-9 in real time.     

Novel Techniques for Robust Voxelization and Visualization of Implicit Surfaces

Voxelization is the transformation of geometric surfaces into voxels. Up to date this process has been done essentially using incremental algorithms. Incremental algorithms have the reputation of being efficient but they lack an important property: robustness. The voxelized representation should envelop its continuous model. However, without robust methods this cannot be guaranteed. This article describes novel techniques of robust voxelization and visualization of implicit surfaces. First of all our recursive subdivision voxelization algorithm is reviewed. This algorithm was initially inspired by Duff's image space subdivision method. Then, we explain the algorithm to voxelize implicit surfaces defined in spherical or cylindrical coordinates. Next, we show a new technique to produce infinite replications of implicit objects and their voxelization method. Afterward, we comment on the parallelization of our voxelization procedure. Finally we present our voxel visualization algorithm based on point display. Our voxelization algorithms can be used with any data structure, thanks to the fact that a voxel is only stored once the last subdivision level is reached. We emphasize the use of the octree, though, because it is a convenient way to store the discrete model hierarchically. In a hierarchy the discrete model refinement is simple and possible from any previous voxelized scene thanks to the fact that the voxelization algorithms are robust.     

A digital mock-up visualization system capable of processing giga-scale CAD models

In the automotive, aerospace and naval industries, digital mock-up tools are always used for assembly examination, layout examination and interference checking. Generally, a digital mock-up is assembled from giga-scale CAD models. Because of limitations of computer hardware resources, a digital mock-up represented by traditional planar facets is too large to load into a computer’s memory for rendering. This article proposes a new geometric compression representation to represent given CAD models with curved triangular patches. Based on the compression representation, our digital mock-up visualization system can import several giga-scale CAD models into a computer’s memory simultaneously. A high performance rendering strategy to display the curved triangular patches is also presented. In the rendering strategy, a dynamic subdivision algorithm is introduced which is different from conventional LOD techniques in order to reduce memory consumption. In addition, an algorithm to convert CAD models to the curved triangular patches is introduced.     

组合曲面参数线五坐标加工刀具轨迹的计算

提出了组合曲面间拓扑关系的建立方法．通过对曲面相邻边界及相邻角点拓扑信息查询，完成刀具路径的合理组织；针对目前在给定加工精度时确定参数增量算法存在的不足，提出基于等参数线的走刀步长追踪法，并对曲率半径趋于无穷大的情况及直纹面加工的情况进行单独处理，保证了算法的稳定性和有效性．在此基础上，系统地阐述了组合曲面加工中刀触点、刀位点的计算以及刀具轨迹的合理化组织。     

两空间三角形的退化关系研究

通过对两空间三角形关系的分类,讨论了几何退化对几何计算的稳健性的影响力。解决一个问题的第一步是描述这个问题,空间几何退化的完整表述是稳健几何计算算法的设计、改进以及测试的重要基础和保障。首次对空间三角形对的退化进行了深入的研究、全面的梳理。基于投影降维原理,抽取繁杂的空间两三角形关系的规律,分离出完整的空间三角形对的退化样本模型。基本策略是建立计算坐标系,通过投影降维,将空间三角形对的位置关系变成一个固定,只有一个变化的平面位置关系。以相离、接触、相交、内含的线索改变另一三角形的位置和大小,分类出空间三角形对的位置关系,检索出两者的所有退化状态。该方法可以推广到其他三维几何间的退化状态分类和几何计算算法的稳健性设计中。     

An improved formalism for quantum computation based on geometric algebra—case study: Grover’s search algorithm

The Grover search algorithm is one of the two key algorithms in the field of quantum computing, and hence it is desirable to represent it in the simplest and most intuitive formalism possible. We show firstly, that Clifford’s geometric algebra, provides a significantly simpler representation than the conventional bra-ket notation, and secondly, that the basis defined by the states of maximum and minimum weight in the Grover search space, allows a simple visualization of the Grover search analogous to the precession of a spin- ${\frac{1}{2}}$ particle. Using this formalism we efficiently solve the exact search problem, as well as easily representing more general search situations. We do not claim the development of an improved algorithm, but show in a tutorial paper that geometric algebra provides extremely compact and elegant expressions with improved clarity for the Grover search algorithm. Being a key algorithm in quantum computing and one of the most studied, it forms an ideal basis for a tutorial on how to elucidate quantum operations in terms of geometric algebra—this is then of interest in extending the applicability of geometric algebra to more complicated problems in fields of quantum computing, quantum decision theory, and quantum information.     

基于控制顶点扰动的平面Offset曲线的NURBS逼近

平面曲线的ｏｆｆｓｅｔ曲线具有丰富的几何结构,它在曲面造型、ＮＣ加工等领域具有广泛应用,但除直线、圆弧或速端曲线等少数几种曲线外,有理多项式参数曲线的ｏｆｆｓｅｔ曲线不能保证仍是有理多项式曲线形式。因此,实际应用中常用逼近方法表示ｏｆｆｓｅｔ曲造型系统中数据结构和几何算法的统一表示。作者针对平面ＮＵＲＢＳ曲线的特点,提出两种逼近表示方法,一种是基于曲线分割的控制顶点动法,另一种是整体控制顶点偏移法     

Hollowing objects with uniform wall thickness

In rapid prototyping, we can significantly reduce the building time and expensive RP material consumption by building a hollowed prototype instead of a solid model. Presented in the paper is a new method to hollow out solid objects with uniform wall thickness to speed up the part building processes in rapid prototyping systems. To achieve uniform wall thickness, it is necessary to generate internal contours by slicing the offset model of an STL model. Since computing an offset model of an STL model involves many difficulties, this paper proposes a new algorithm that computes internal contours without computing the offset model. The proposed algorithm is efficient and relatively easy to implement, because it employs well-known 2D geometric algorithms, such as planar curve offsetting and tracing innermost curves. Various examples have been tested with implementation of the algorithm, and some examples are presented for illustration.     

Visualization of geometric algorithms

Investigates the visualization of geometric algorithms. We discuss how limiting the domain makes it possible to create a system that enables others to use it easily. Knowledge about the domain can be very helpful in building a system which automates large parts of the user's task. A system can be designed to isolate the user from any concern about how graphics is done. The application need only specify “what” happens and need not be concerned with “how” to make it happen on the screen. We develop a conceptual model and a framework for experimenting with it. We also present a system, GASP (Geometric Animation System, Princeton), which implements this model. GASP allows quick generation of 3D geometric algorithm visualizations, even for highly complex algorithms. It also provides a visual debugging facility for geometric computing. We show the utility of GASP by presenting a variety of examples     

Contour Inferences for Image Understanding

We present a new approach to the algorithmic study of planar curves, with applications to estimations of contours in images. We construct spaces of curves satisfying constraints suited to specific problems, exploit their geometric structure to quantify properties of contours, and solve optimization and inference problems. Applications include new algorithms for computing planar elasticae, with enhanced performance and speed, and geometric algorithms for the estimation of contours of partially occluded objects in images.     

A Projective Framework for Polyhedral Mesh Modelling

We present a novel framework for polyhedral mesh editing with face‐based projective maps that preserves planarity by definition. Such meshes are essential in the field of architectural design and rationalization. By using homogeneous coordinates to describe vertices, we can parametrize the entire shape space of planar‐preserving deformations with bilinear equations. The generality of this space allows for polyhedral geometric processing methods to be conducted with ease. We demonstrate its usefulness in planar‐quadrilateral mesh subdivision, a resulting multi‐resolution editing algorithm, and novel shape‐space exploration with prescribed transformations. Furthermore, we show that our shape space is a discretization of a continuous space of conjugate‐preserving projective transformation fields on surfaces. Our shape space directly addresses planar‐quad meshes, on which we put a focus, and we further show that our framework naturally extends to meshes with faces of more than four vertices as well.     

A novel approach to ontology classification

Ontology classification–the computation of the subsumption hierarchies for classes and properties–is a core reasoning service provided by all OWL reasoners known to us. A popular algorithm for computing the class hierarchy is the so-called Enhanced Traversal (ET) algorithm. In this paper, we present a new classification algorithm that attempts to address certain shortcomings of ET and improve its performance. Apart from classification of classes, we also consider object and data property classification. Using several simple examples, we show that the algorithms commonly used to implement these tasks are incomplete even for relatively weak ontology languages. Furthermore, we show that property classification can be reduced to class classification, which allows us to classify properties using our optimised algorithm. We implemented all our algorithms in the OWL reasoner HermiT. The results of our performance evaluation show significant performance improvements on several well-known ontologies.     

A New Class of Guided C2 Subdivision Surfaces Combining Good Shape with Nested Refinement

Converting quadrilateral meshes to smooth manifolds, guided subdivision offers a way to combine the good highlight line distribution of recent G‐spline constructions with the refinability of subdivision surfaces. This avoids the complex refinement of G‐spline constructions and the poor shape of standard subdivision. Guided subdivision can then be used both to generate the surface and hierarchically compute functions on the surface. Specifically, we present a C² subdivision algorithm of polynomial degree bi‐6 and a curvature bounded algorithm of degree bi‐5. We prove that the common eigenstructure of this class of subdivision algorithms is determined by their guide and demonstrate that their eigenspectrum (speed of contraction) can be adjusted without harming the shape. For practical implementation, a finite number of subdivision steps can be completed by a high‐quality cap. Near irregular points this allows leveraging standard polynomial tools both for rendering of the surface and for approximately integrating functions on the surface.     

Spatial Patches - A Primitive for 3D Model Representation

The commonly used solution for real-life 3D model representation is polygonal spatially consistent geometry, with texture, and, optionally, bump or displacement maps attached. Although the idea of displacement mapping is well known, there are just a few approaches to its efficient implementation. In this paper we develop a technique that allows for efficient representation and rendering of 3D models by getting a new angle on the displacement mapping concept. We introduce a new primitive that is defined as the range image of a small part of the model's surface; therefore, it is called a spatial patch. The whole model is just a collection of patches with no connectivity information between them. Such a representation can be directly acquired by 3D scanning machinery, and stored in a compact uniform form. It also allows for efficient visualization, which is the major focus of this paper. Thus, we present the logical structure of a rendering unit based on conventional z-buffering, and discuss the involved algorithms in detail. These algorithms benefit from modern features of computing units for which we believe the proposed technique can be used in a wide range of applications dealing with real-life 3D data.     

沿三维直线的非单位体素遍历的多步整数算法

提出一种只用整数运算的沿三维直线的体素遍历算法,适用的体素空间可以分割成非单位的和非正方体的.首先研究了二维平面中的体素直线遍历算法,然后提出一种以二维平面中的遍历算法为基础的沿三维直线的体素遍历算法.该算法是一个多步整数遍历算法,每一步可以遍历最多3个体素,且所用的判断公式非常精炼,不仅计算量很小而且没有累计误差.与现有的体素遍历算法进行比较的结果表明,该算法不仅没有累计误差,而且执行速度也是最快的.     

Faster Phong Shading via Angular Interpolation

One of the most successful algorithms that brought realism to the world of 3D image generation is Phong shading. It is an algorithm for smooth shading meshes of planar polygons used to represent curved surfaces. The level of realism and depth perception that can be obtained by Phong shading is attractive for 3D CAD applications and related areas. However, per pixel computation costs which were too high and/or artifacts, introduced by some of the more efficient evaluation methods and apparent only when displaying moving objects, are major factors mat blocked the common usage of Phong shading in highly interactive applications.
In this paper we present angular interpolation for Phong shading planar polygons. Angular interpolation was a method especially designed to meet requirements as imposed by special purpose hardware we developed¹, but turned out to be generally applicable. The angular interpolation method appears to be very efficient and reduces artifacts when displaying moving objects. Ideally a shading algorithm imposes no need for subdivision of patches as presented by the solid modelling system. Shading calculation via angular interpolation yields such an ideal algorithm. We will describe two alternative evaluation methods that trade off evaluation cost against level of accuracy. They both can handle light source and view point at arbitrary distances, but differ in level of accuracy. As a consequence these alternative evaluation methods do impose restrictions on the topology of patches and light sources. However, generally, the limitations imposed by these alternative shading methods are much more liberal than the limitations on patch size imposed by the geometry.
The most economic evaluation method we present can incrementally compute the colour intensity along a scanline by two additions per pixel. The methods presented are generally applicable and can easily be implemented in hardware.