SELF-INTERSECTION LOCAL TIME OF ADDITIVE LEVY PROCESS

This article discusses the problem of existence of jointly continuous self-intersection local time for an additive levy process. Here, 'local time' is understood in the sense of occupation density, and by an additive Levy process the authors mean a process X = {X(t),t∈ R+N} which has the decomposition X = Xi X2 … XN, each Xl has the lower index αl, α= min{α1,…, αN}. Let Z = (Xt2 - Xt1, …, Xtr - Xtr-1). They prove that if Nrα > d(r-1), then a jointly continuous local time of Z, i.e. the self-intersection local time of X, can be obtained.

Trimming self-intersections in swept volume solid modeling

Swept volume solid modeling has been applied to many areas such as NC machining simulation and verification, robot workspace analysis, collision detection, and CAD. But self-intersections continue to be a challenging problem in the boundary representation of swept volume solids. A novel algorithm is presented in this paper to trim self-intersection regions in swept volume solids modeling. This trimming algorithm consists of two major steps: (1) roughly detecting self-intersection regions by checking intersections or overlapping of the envelop profiles; (2) splitting the whole envelop surfaces of the swept volume solid into separate non-self-intersecting patches to trim global self-intersections, and to trim local self-intersections, dividing local self-intersecting regions into patches and replacing self-intersecting patches with non-self-intersecting ones. Examples show that our algorithm is efficient and robust.

A method of dealing polygon’s self-intersection contour in SLA

The contour of the slices of SLA parts is composed of a great deal of small lines. When offsetting the contour to compensate for the radius of laser spot, many self-intersection contours come into being, which decrease the precision of formed parts. A new lemma to judge the local self-intersection contour and the global self-intersection contour separately is put forward, according to which self-intersection contour can be removed reliably. Meanwhile, a new beam offsetting algorithm for SLA parts is described, which brings about good resuits in the practical manufacturing process.

Self-intersection local times and collision local times of bifractional Brownian motions

In this paper, we consider the local time and the self-intersection local time for a bifractional Brownian motion, and the collision local time for two independent bifractional Brownian motions. We mainly prove the existence and smoothness of the self-intersection local time and the collision local time, through the strong local nondeterminism of bifractional Brownian motion, L2 convergence and Chaos expansion.

Smoothness of local times and self-intersection local times of Gaussian random fields

This paper is concerned with the smoothness （in the sense of Meyer- Watanabe） of the local times of Gaussian random fields. Sufficient and necessary conditions for the existence and smoothness of the local times, collision local times, and self-intersection local times are established for a large class of Gaussian random fields, including fractional Brownian motions, fractional Brownian sheets and solutions of stochastic heat equations driven by space-time Gaussian noise.

Deforming generalized cylinders without self-intersection by means of a parametric center curve

Large-scale deformations of a tubular object, or generalized cylinder, are often defined by a target shape for its center curve, typically using a parametric target curve. This task is non-trivial for free-form deformations or direct manipulation methods because it is hard to manually control the centerline by adjusting control points. Most skeleton-based methods are no better, again due to the small number of manually adjusted control points. In this paper, we propose a method to deform a generalized cylinder based on its skeleton composed of a centerline and orthogonal cross sections. Although we are not the first to use such a skeleton, we propose a novel skeletonization method that tries to minimize the number of intersections between neighboring cross sections by means of a relative curvature condition to detect intersections. The mesh deformation is first defined geometrically by deforming the centerline and mapping the cross sections. Rotation minimizing frames are used during mapping to control twisting. Secondly, given displacements on the cross sections, the deformation is decomposed into finely subdivided regions. We limit distortion at these vertices by minimizing an elastic thin shell bending energy,in linear time. Our method can handle complicated generalized cylinders such as the human colon.

An Algorithm for Voxelizing Non-manifold Triangle Geometry

Taking advantage of the characteristic of distance field and octree, we brought up a method which can discrete any triangle-geometry into voxel. As many triangles used for cutting simulation are unoriented, non-manifold or self-intersecting, which leads to ambiguity in mathematical terms. The algorithm firstly computes sign distance field and use the threshold value to reconstruct the surface, which is very close to the original mode. At last, the reconstructed surface is voxelized. Also we can produce voxelized model which is suitable for cutting simulation.

Automatic Cage Building with Quadric Error Metrics

Modern computer graphics applications usually require high resolution object models for realistic rendering. However, it is expensive and difficult to deform such models in real time. In order to reduce the computational cost during deformations, a dense model is often manipulated through a simplified structure, called cage, which envelops the model. However, cages are usually built interactively by users, which is tedious and time-consuming. In this paper, we introduce a novel method that can build cages automatically for both 2D polygons and 3D triangular meshes. The method consists of two steps： 1） simplifying the input model with quadric error metrics and quadratic programming to build a coarse cage; 2） removing the self-intersections of the coarse cage with Delaunay partitions. With this new method, a user can build a cage to envelop an input model either entirely or partially with the approximate vertex number the user specifies. Experimental results show that, compared to other cage building methods with the same number of vertex, cages built by our method are more similar to the input models. Thus, the dense models can be manipulated with higher accuracy through our cages.

Homology classes that can be represented by embedded spheres in rational surfaces

The problem of embedding spheres in rational surfaces CP 2#n $\overline {CP} $ 2 is studied. For homology classes u = (b 1 + k, b 2,...,b n) with positive self-intersection numbers, a necessary and sufficient condition to detect its representability is given when k ≤ 5.

Three-dimensional deformation in curl vector field

Deformation is an important research topic in graphics.There are two key issues in mesh deformation:(1) selfintersection and(2) volume preserving.In this paper,we present a new method to construct a vector field for volume-preserving mesh deformation of free-form objects.Volume-preserving is an inherent feature of a curl vector field.Since the field lines of the curl vector field will never intersect with each other,a mesh deformed under a curl vector field can avoid self-intersection between field lines.Designing the vector field based on curl is useful in preserving graphic features and preventing self-intersection.Our proposed algorithm introduces distance field into vector field construction;as a result,the shape of the curl vector field is closely related to the object shape.We define the construction of the curl vector field for translation and rotation and provide some special effects such as twisting and bending.Taking into account the information of the object,this approach can provide easy and intuitive construction for free-form objects.Experimental results show that the approach works effectively in real-time animation.