曲面实体造型中曲线和曲面交点的求解

求交是曲面实体造型系统中影响拼合算法效率和稳定性的重要因素，而求交算法又是和曲面的几何表示密切相关的。ＮＵＲＢＳ虽然能统一表示所有曲面，但却给二次曲面的求交带来了不必要的复杂性。二次曲面经常在机械零件的设计中被用来描述轴、孔、槽等几何特征，因此它们的求交算法应具有高精度、高效率和高稳定性。为此，对一种实用的二次曲面表示方法——几何法进行了深入研究后，给出了构成二次曲面轮廓的几种二次曲线和空间四次曲线与二次曲面交点的求法。     

A Framework for Synchronized Editing of Multiple Curve Representations

Editing curves and surfaces is difficult in part because their mathematical representations rarely correspond to most people's idea of a curve or surface. The implementation (and hence, behavior) of most manipulation tools is intertwined with a particular curve or surface representation; this can make reimplementing the tool with a different representation problematic. A system using a single representation must therefore either limit the types of tools available or convert existing tools to work on the system's representation.
In this paper we present a framework for editing curves or surfaces which supports multiple representations and ensures that they stay synchronized. As a proof of concept, we have created a curve editor which contains several tools each of which manipulate one of three different curve representations: polylines, NURBs, and multi-resolution B-splines.     

Parametric representation of a surface pencil with a common spatial geodesic

In this paper, we study the problem of constructing a family of surfaces from a given spatial geodesic curve. We derive a parametric representation for a surface pencil whose members share the same geodesic curve as an isoparametric curve. By utilizing the Frenet trihedron frame along the given geodesic, we express the surface pencil as a linear combination of the components of this local coordinate frame, and derive the necessary and sufficient conditions for the coefficients to satisfy both the geodesic and the isoparametric requirements. We illustrate and verify the method by finding exact surface pencil formulations for some simple surfaces, such as surfaces of revolution and ruled surfaces. Finally, we demonstrate the use of this method in a garment design application.     

Surface approximation via sparse representation and parameterization optimization

Surface approximation with smooth functions suffers the problems of choosing the basis functions and representing non-smooth features. In this work, we introduce a sparse representation for surfaces with a set of redundant basis functions, which efficiently overcomes the overfitting artifacts. Moreover, we propose an approach of parameterization transformation, which makes the possibility to represent non-smooth features by the composition of a smooth function and a non-smooth domain optimization. We couple the sparse representation and the parameterization transformation in a global optimization to respect sharp features with smooth polynomial basis functions. Our approach is capable for approximating a wide range of surfaces with different level of sharp features. Experimental results have shown the feasibility and applicability of our proposed method in various applications.     

NURBS曲线曲面的显式矩阵表示及其算法

从 B样条的差商定义出发 ,提出差商展开系数的概念 ,通过差商展开系数显式解析表示式的导出 ,得到任意次 NU RBS曲线曲面系数矩阵的显式解析表示式 ,并给出了求差商展开系数和 NURBS曲线曲面系数矩阵的数值算法 .文中给出的方法适用于一切 NU RBS曲线曲面 ,包括有理和非有理的 Bézier、均匀和非均匀的 B样条曲线曲面 .相应的数值算法计算简单 ,易于实现 .差商展开系数解析表示式为 NU RBS曲线曲面的表示、转换和节点插入、升阶等基本运算以及与差商相关的问题的研究提供了一个统一的构造性工具和应用方法 .     

Real‐Time Fluid Effects on Surfaces using the Closest Point Method

The Closest Point Method (CPM) is a method for numerically solving partial differential equations (PDEs) on arbitrary surfaces, independent of the existence of a surface parametrization. The CPM uses a closest point representation of the surface, to solve the unmodified Cartesian version of a surface PDE in a 3D volume embedding, using simple and well‐understood techniques. In this paper, we present the numerical solution of the wave equation and the incompressible Navier‐Stokes equations on surfaces via the CPM, and we demonstrate surface appearance and shape variations in real‐time using this method. To fully exploit the potential of the CPM, we present a novel GPU realization of the entire CPM pipeline. We propose a surface‐embedding adaptive 3D spatial grid for efficient representation of the surface, and present a high‐performance approach using CUDA for converting surfaces given by triangulations into this representation. For real‐time performance, CUDA is also used for the numerical procedures of the CPM. For rendering the surface (and the PDE solution) directly from the closest point representation without the need to reconstruct a triangulated surface, we present a GPU ray‐casting method that works on the adaptive 3D grid.     

Neural Representation of Open Surfaces

Neural implicit surfaces have emerged as an effective, learnable representation for shapes of arbitrary topology. However, representing open surfaces remains a challenge. Different methods, such as unsigned distance fields (UDF), have been proposed to tackle this issue, but a general solution remains elusive. The generalized winding number (GWN), which is often used to distinguish interior points from exterior points of 3D shapes, is arguably the most promising approach. The GWN changes smoothly in regions where there is a hole in the surface, but it is discontinuous at points on the surface. Effectively, this means that it can be used in lieu of an implicit surface representation while providing information about holes, but, unfortunately, it does not provide information about the distance to the surface necessary for e.g. ray tracing, and special care must be taken when implementing surface reconstruction. Therefore, we introduce the semi-signed distance field (SSDF) representation which comprises both the GWN and the surface distance. We compare the GWN and SSDF representations for the applications of surface reconstruction, interpolation, reconstruction from partial data, and latent vector analysis using two very different data sets. We find that both the GWN and SSDF are well suited for neural representation of open surfaces.     

基于C1连续的NURBS边界Gregory曲面片实现曲面拼接

该文通过使用C1连续的NUR BS边界Gregory(N BG)曲面片进行插值,实现曲面拼接,使得在曲面连接处达到G1连续。实现了插值曲面的高阶连续,解决了用平滑的NUR BS曲面对曲线网格区域进行插值这一难以实现的问题,使用户在设计曲线网格时,只需考虑形状设计而不必关心曲线的类型及插值到曲线网格区域的曲面方程。     

Sharp Features on Multiresolution Subdivision Surfaces

In this paper we describe a method for creating sharp features and trim regions on multiresolution subdivision surfaces along a set of user-defined curves. Operations such as engraving, embossing, and trimming are important in many surface modeling applications. Their implementation, however, is nontrivial due to computational, topological, and smoothness constraints that the underlying surface has to satisfy. The novelty of our work lies in the ability to create sharp features anywhere on a surface and in the fact that the resulting representation remains within the multiresolution subdivision framework. Preserving the original representation has the advantage that other operations applicable to multiresolution subdivision surfaces can subsequently be applied to the edited model. We also introduce an extended set of subdivision rules for Catmull–Clark surfaces that allows the creation of creases along diagonals of control mesh faces.     

Multi-scale geometric detail enhancement for time-varying surfaces

This paper presents a new multi-scale geometric detail enhancement approach for time-varying surfaces. We first develop an adaptive spatio-temporal bilateral filter, which produces temporally-coherent and feature-preserving multi-scale representation for the time-varying surfaces. We then extract the geometric details from the time-varying surfaces, and enhance geometric details by exaggerating detail information at each scale across the time-varying surfaces. Our approach can process mesh sequences with consistent connections or point sequences with unconstructed point set. In addition, as applications, based on the developed multi-scale surface representation and detail enhancement operators, we present geometric detail transfer, space–time morphing, and local regions detail enhancement for the time-varying surface.     

On bending invariant signatures for surfaces

Isometric surfaces share the same geometric structure, also known as the "first fundamental form." For example, all possible bendings of a given surface that includes all length preserving deformations without tearing or stretching the surface are considered to be isometric. We present a method to construct a bending invariant signature for such surfaces. This invariant representation is an embedding of the geometric structure of the surface in a small dimensional Euclidean space in which geodesic distances are approximated by Euclidean ones. The bending invariant representation is constructed by first measuring the intergeodesic distances between uniformly distributed points on the surface. Next, a multidimensional scaling technique is applied to extract coordinates in a finite dimensional Euclidean space in which geodesic distances are replaced by Euclidean ones. Applying this transform to various surfaces with similar geodesic structures (first fundamental form) maps them into similar signature surfaces. We thereby translate the problem of matching nonrigid objects in various postures into a simpler problem of matching rigid objects. As an example, we show a simple surface classification method that uses our bending invariant signatures.     

Tunnel-free voxelisation of rational Bézier surfaces

To synthesize natural and artificial objects into a hybrid graphics scene represented by a set of voxels, voxelisation of geometric models is necessary. Rational parametric surfaces have been widely used in the representation of free-form surfaces. Voxelisation of these surfaces is therefore of great importance in the development of a voxel-based modeling system. A key issue is to develop a tunnel-free voxelisation algorithm for these continuous surfaces. In this paper, we propose such an algorithm for a rational Bézier surface. We derive the bound of the parametric steps to ensure that the voxelised rational Bézier surface is, by our algorithm, 6-tunnel-free, and we give the mathematical proof of this property. For efficient computation, we employ the forward difference technique in homogeneous form in the implementation of the algorithm. For more general applications, we show that voxelisation of a NURBS surface can be realised by first converting it into a piecewise rational Bézier surface and then voxelising each of the rational Bézier surfaces. We indicate the advantages carrying out this procedure.     

Smoke surfaces: an interactive flow visualization technique inspired by real-world flow experiments

Smoke rendering is a standard technique for flow visualization. Most approaches are based on a volumetric, particle based, or image based representation of the smoke. This paper introduces an alternative representation of smoke structures: as semi-transparent streak surfaces. In order to make streak surface integration fast enough for interactive applications, we avoid expensive adaptive retriangulations by coupling the opacity of the triangles to their shapes. This way, the surface shows a smoke-like look even in rather turbulent areas. Furthermore, we show modifications of the approach to mimic smoke nozzles, wool tufts, and time surfaces. The technique is applied to a number of test data sets.     

基于几何法表示的曲面实体的分割算法

分割是实体造型的重要步骤,而边界问题是影响分割算法效率和稳定性的主要因素。工程中常用的二次曲面有自封闭的特性,在相交时交线会出现退化和自交等情况,给拓扑表示和帝体重建带来困难,另外,边界重合也是实体分割时经常要遇到的问题,该文在二次曲面几何法表示的基础上,针对二次曲面相交时交线的特性,设计了合理的分割策略,提出了有效的分割算法,并对边界重合等问题做了很好的处理,同时用实例验证了本文算法的有效性。     

Qualitative and quantitative comparisons of B-spline offset surface approximation methods

Surface offset is one of the most useful operations in Computer Aided Geometric Design (CAGD). However, an implementation of this operator is not trivial primarily because the offset surface, in general, does not have the same representation as the original surface. Hence, it is difficult or impossible to represent an exact offset in a system with limited surface forms. For this reason, some CAGD surface offset operators produce results that are, at times, unsatisfactory. In this article, we discuss surface offset approximation methods in B-spline environment: both the original surfaces and their offsets are B-spline surfaces. This article summarizes research contained in the first named author's PhD thesis.     

Learning to Predict 3D Surfaces of Sculptures from Single and Multiple Views

International Journal of Computer Vision - The objective of this work is to reconstruct the 3D surfaces of sculptures from one or more images using a view-dependent representation. To this end, we...     

Intrinsic Decompositions for Image Editing

Intrinsic images are a mid‐level representation of an image that decompose the image into reflectance and illumination layers. The reflectance layer captures the color/texture of surfaces in the scene, while the illumination layer captures shading effects caused by interactions between scene illumination and surface geometry. Intrinsic images have a long history in computer vision and recently in computer graphics, and have been shown to be a useful representation for tasks ranging from scene understanding and reconstruction to image editing. In this report, we review and evaluate past work on this problem. Specifically, we discuss each work in terms of the priors they impose on the intrinsic image problem. We introduce a new synthetic ground‐truth dataset that we use to evaluate the validity of these priors and the performance of the methods. Finally, we evaluate the performance of the different methods in the context of image‐editing applications.     

三角B样条上可展曲面的设计与形状调节

提出了计算机辅助设计可展曲面的新方法,利用该文的方法,可展曲面可用具有三角B样条基函数的控制平面来设计,这种设计方法具有现存曲线设计方法的特征。同时,通过引入形状控制参数,使生成的可展曲面在较大的范围内可进行调节和控制,增加了造型的自由度。该文的设计方法直接、简单、有效,不仅能克服传统方法在可展曲面设计方法上的缺陷,而且能方便地解决工程中经常遇到的可展曲面的形状难以调节和控制的问题。     

Generalized Helicoids for Modeling Hair Geometry

In computer graphics, modeling the geometry of hair and hair‐like patterns such as grass and fur remains a significant challenge. Hair strands can exist in an extensive variety of arrangements and the choice of an appropriate representation for tasks such as hair synthesis, fitting, editing, or reconstruction from samples, is non‐trivial. To support such applications we present a novel mathematical representation of hair based on a class of minimal surfaces called generalized helicoids. This representation allows us to characterize the geometry of a single hair strand, as well as of those in its vicinity, by three intuitive curvature parameters and an elevation angle. We introduce algorithms for fitting piecewise generalized helicoids to unparameterized hair strands, and for interpolating hair between these fits. We showcase several applications of this representation including the synthesis of different hair geometries, wisp generation, hair interpolation from samples and hair‐style parametrization and reconstruction from real hair data.     

Representing higher-order singularities in vector fields on piecewise linear surfaces

Accurately representing higher-order singularities of vector fields defined on piecewise linear surfaces is a non-trivial problem. In this work, we introduce a concise yet complete interpolation scheme of vector fields on arbitrary triangulated surfaces. The scheme enables arbitrary singularities to be represented at vertices. The representation can be considered as a facet-based "encoding" of vector fields on piecewise linear surfaces. The vector field is described in polar coordinates over each facet, with a facet edge being chosen as the reference to define the angle. An integer called the period jump is associated to each edge of the triangulation to remove the ambiguity when interpolating the direction of the vector field between two facets that share an edge. To interpolate the vector field, we first linearly interpolate the angle of rotation of the vectors along the edges of the facet graph. Then. we use a variant of Nielson's side-vertex scheme to interpolate the vector field over the entire surface. With our representation, we remove the bound imposed on the complexity of singularities that a vertex can represent by its connectivity. This bound is a limitation generally exists in vertex-based linear schemes. Furthermore, using our data structure, the index of a vertex of a vector field can be combinatorily determined. We show the simplicity of the interpolation scheme with a GPU-accelerated algorithm for a LIC-based visualization of the so-defined vector fields, operating in image space. We demonstrate the algorithm applied to various vector fields on curved surfaces.