Parametric representation of a surface pencil with a common line of curvature

Line of curvature on a surface plays an important role in practical applications. A curve on a surface is a line of curvature if its tangents are always in the direction of the principal curvature. By utilizing the Frenet frame, the surface pencil can be expressed as a linear combination of the components of the local frame. With this parametric representation, we derive the necessary and sufficient condition for the given curve to be the line of curvature on the surface. Moreover, the necessary and sufficient condition for the given curve to satisfy the line of curvature and the geodesic requirements is also analyzed.     

Parametric representation of a surface pencil with a common asymptotic curve

In this paper, we study the problem of finding a surface pencil from a given spatial asymptotic curve. We obtain the parametric representation for a surface pencil whose members have the same curve as a given asymptotic curve. Using the Frenet frame of the given asymptotic curve, we present the surface as a linear combination of this frame and analyse the necessary and sufficient condition for that curve to be asymptotic. We illustrate this method by presenting some examples.     

Constrained design of polynomial surfaces from geodesic curves

In a recent work, Wang et al. [Wang G, Tang K, Tai CH. Parametric representation of a surface pencil with common spatial geodesic. Computer-Aided Design 2004;36(5): 447–59] discuss a constrained design problem appearing in the textile and shoe industry for garment design. Given a model and size, the characteristic curve called girth is usually fixed, and preferably should be a geodesic for manufacturing reasons. The designer must preserve this girth, being allowed to modify other areas according to aesthetic criteria. We present a practical method to construct polynomial surfaces from a polynomial geodesic or a family of geodesics, by prescribing tangent ribbons. Differently from previous procedures, we identify the existing degrees of freedom in terms of control points, and our method yields parametric polynomial surfaces that can be incorporated into commercial CAD programs. The extension to rational geodesics is also outlined.     

G 2 interpolation and blending on surfaces

We introduce a method for curvature-continuous (G ²) interpolation of an arbitrary sequence of points on a surface (implicit or parametric) with prescribed tangent and geodesic curvature at every point. The method can also be used forG ² blending of curves on surfaces. The interpolation/blending curve is the intersection curve of the given surface with a functional spline (implicit) surface. For the construction of blending curves, we derive the necessary formulas for the curvature of the surfaces. The intermediate results areG ² interpolation/blending methods in IR².     

过测地线的优化曲面设计

给定一条曲线,构造以其为测地线的曲面,这是服装鞋帽类产品的设计/制造业中的一个现实课题.已有研究结果是构造出以给定曲线为公共测地线的曲面束,然后用拟合数据点的方法来确定最终曲面.这种通用方法受到曲面参数及其表示方法的影响,且没有对曲面的光顺程度加以考虑.从服装材料的特性和设计思想出发,提出一种利用能量优化来确定最终曲面的新方法.通过改变曲面表示形式和引入能量函数,方便而有效地确定了过给定测地线的一张优化曲面.给出了在插值拟合等约束条件下的相应算法.实例表明,所给算法很好地模拟了成衣的光顺设计与加工,在计算机辅助设计/制造(CAD/CAM)中富有应用价值.     

保测地曲率的曲面曲线设计

给出一种在给定光滑曲面上进行曲线设计的算法．由于曲面上曲线的曲率向量可分解为测地曲率向量与法曲率向量的和,故可以通过设计平面源曲线,并将这条源曲线保测地曲率对应到给定曲面上,得到一条测地曲率和平面源曲线的曲率相同的目标曲线．取给定曲面上一点为初始点,逐次迭代跟踪得到整条目标曲线．该算法既可以用来进行曲面上的图案设计,也可用于求解光滑曲面上的测地线。     

Low-degree approximation of high-degree B-spline surfaces

In this paper, a method for approximate conversion of high degree Bezier and B-spline surfaces to lower degree representations is presented to facilitate the exchange of surface geometry between different geometric modeling systems. Building on previous work on curve approximation, the method uses adaptive sampling to compute approximation error and lofting of isoparametric curves to produce the approximating surface. In addition, a bound for the approximation accuracy is computed using convex hulls.     

Computing isophotos of surface of revolution and canal surface

Isophote of a surface consists of a loci of surface points whose normal vectors form a constant angle with a given fixed vector. It also serves as a silhouette curve when the constant angle is given as π/2. We present efficient and robust algorithms to compute isophotes of a surface of revolution and a canal surface. For the two kinds of surfaces, each point on the isophote is derived by a closed-form solution. To find each connected component in the isophote, we utilize the feature of surface normals. Both surfaces are decomposed into a set of circles, where the surface normal vectors at points on each circle construct a cone. The vectors which form a constant angle with given fixed vector construct another cone. We compute the parametric range of the connected component of the isophote by computing the parametric values of the surface which derive the tangential intersection of these two cones.     

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.     

Freeform surface flattening based on fitting a woven mesh model

This paper presents a robust and efficient surface flattening approach based on fitting a woven-like mesh model on a 3D freeform surface. The fitting algorithm is based on tendon node mapping (TNM) and diagonal node mapping (DNM), where TNM determines the position of a new node on the surface along the warp or weft direction and DNM locates a node along the diagonal direction. During the 3D fitting process, strain energy of the woven model is released by a diffusion process that minimizes the deformation between the resultant 2D pattern and the given surface. Nodes mapping and movement in the proposed approach are based on the discrete geodesic curve generation algorithm, so no parametric surface or pre-parameterization is required. After fitting the woven model onto the given surface, a continuous planar coordinate mapping is established between the 3D surface and its counterpart in the plane, based on the idea of geodesic interpolation of the mappings of the nodes in the woven model. The proposed approach accommodates surfaces with darts, which are commonly utilized in clothing industry to reduce the stretch of surface forming and flattening. Both isotropic and anisotropic materials are supported.     

Line art rendering via a coverage of isoparametric curves

A line art nonphotorealistic rendering scheme of scenes composed of freeform surfaces is presented. A freeform surface coverage is constructed using a set of isoparametric curves. The density of the isoparametric curves is set to be a function of the illumination of the surface determined using a simple shading model, or of regions of special importance such as silhouettes. The outcome is one way of achieving an aesthetic and attractive line art rendering that employs isoparametric curve based drawings that is suitable for printing publication     

Reduce the stretch in surface flattening by finding cutting paths to the surface boundary

This paper presents a method for finding cutting paths on a 3D triangular mesh surface to reduce the stretch in the flattened surface. The cutting paths link the surface boundary and the nodes where the Gaussian curvature is high, and their total length is minimized. First, a linear algorithm for computing an approximate boundary geodesic distance map is introduced; the map encapsulates the undirected geodesic distance from every triangular node to the surface boundary approximately. This is followed by determining the undirected shortest paths passing through all the nodes where the Gaussian curvature is larger than a threshold. The cutting paths walk along the triangular edges of the given surface. Compared with other similar approaches, our method reaches a faster speed, and can deal with surfaces with widely distributed curvatures.     

Mesh analysis using geodesic mean-shift

In this paper, we introduce a versatile and robust method for analyzing the feature space associated with a given mesh surface. The method is based on the mean-shift operator, which was shown to be successful in image and video processing. Its strength lies in the fact that it works in a single joint space of geometry and attributes called the feature-space. The mean-shift procedure works as a gradient ascend finding maxima of an estimated probability density function in feature-space. Our method for using the mean-shift technique on surfaces solves several difficulties. First, meshes as opposed to images do not present a regular and uniform sampling of domain. Second, on surface meshes the shifting procedure must be constrained to stay on the surface and preserve geodesic distances. We define a special local geodesic parameterization scheme, and use it to generalize the mean-shift procedure to unstructured surface meshes. Our method can support piecewise linear attribute definitions as well as piecewise constant attributes.     

Automatic reconstruction of B-spline surfaces with constrained boundaries

The aim of this study is to present an automatic surface reconstruction method that can take practical restrictions on scanned points into consideration and efficiently and reliably output a group of G¹ surfaces. The proposed method is mainly composed of three phases: quadrangle frame generation, point and curve networks planning, and surface patches reconstruction. In the first phase, the original triangle mesh is reduced and converted into a quadrangle mesh, the edges of which serve as the frame of the surfaces. In the second phase, the boundary data of the surfaces are prepared. These include a network of serial points, frame curves and surface normals which are also expressed as curves. In the final phase, surface initialization, harmonization mapping and surface warping are presented to yield the desired surfaces. The main advantage of the proposed method is that it can relax the pre-processing of a scanned triangle mesh, and hence, increase the efficiency and quality of the surface reconstruction. Several examples of various types of air bags are presented to demonstrate the feasibility of the proposed method.     

Generation of Discrete Bicubic G1 B-Spline Ship Hullform Surfaces from a Given Curve Network Using Virtual Iso-Parametric Curves

We propose a method that automatically generates discrete bicubic G1 continuous B-spline surfaces that interpolate the curve network of a ship hullform. First, the curves in the network are classified into two types: boundary curves and "reference curves". The boundary curves correspond to a set of rectangular (or triangular) topological type that can be represented with tensor-product (or degenerate) B-spline surface patches. Next, in the interior of the patches, surface fitting points and cross boundary derivatives are estimated from the reference curves by constructing "virtual" isoparametric curves. Finally, a discrete G1 continuous B-spline surface is generated by a surface fitting algorithm. Several smooth ship hullform surfaces generated from curve networks corresponding to actual ship hullforms demonstrate the quality of the method.     

Geodesic-Controlled Developable Surfaces for Modeling Paper Bending

We present a novel and effective method for modeling a developable surface to simulate paper bending in interactive and animation applications. The method exploits the representation of a developable surface as the envelope of rectifying planes of a curve in 3D, which is therefore necessarily a geodesic on the surface. We manipulate the geodesic to provide intuitive shape control for modeling paper bending. Our method ensures a natural continuous isometric deformation from a piece of bent paper to its flat state without any stretching. Test examples show that the new scheme is fast, accurate, and easy to use, thus providing an effective approach to interactive paper bending. We also show how to handle non-convex piecewise smooth developable surfaces.     

Constructing PDE-based surfaces bounded by geodesics or lines of curvature

In order to explore a new approach to construct surfaces bounded by geodesics or lines of curvature, a method of surface modeling based on fourth-order partial differential equations (PDEs) is presented. Compared with the free-form surface modeling based on finding control points, PDE-based surface modeling has the following three advantages. First, the corresponding biharmonic surface can naturally be derived under some degenerative conditions; second, the parameters in the PDE implicate some physical meaning, such as elasticity or rigidity; third, there are only a few parameters that need to be evaluated, and hence the computation is simple. In addition, this paper constructs two adjacent surfaces with C¹ continuity whose common boundary is the same given curve as well as respective geodesic (or line of curvature). Examples show that this method to construct PDE-based surfaces bounded by geodesics or lines of curvature is easy and effective.     

Surface construction by fitting unorganized curves

We present a novel technique to construct a B-spline surface from unorganized curves in 3D space. Unlike the lofting or skinning methods, where the family of curves to be lofted form isoparametric lines of the resulting surface, our method relaxes this restriction and allows a set of curves to take arbitrary orientation and possibly intersect each other. We employ the concept of a curve on a surface which is used in obtaining the arc element of a curve on the surface in differential geometry. This novel technique is useful in surface construction or creation where surfaces are created based on sketches. The surface creation has applications in aesthetic shape design, reverse engineering, computer graphics, and computer animation. This technique is also useful in surface reconstruction where surfaces are constructed from stripes which are made of a series of points lined up in a sequential manner.     

Elastic geodesic paths in shape space of parameterized surfaces

This paper presents a novel Riemannian framework for shape analysis of parameterized surfaces. In particular, it provides efficient algorithms for computing geodesic paths which, in turn, are important for comparing, matching, and deforming surfaces. The novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape-preserving transformations of surfaces. The basic idea is to formulate a space of embedded surfaces (surfaces seen as embeddings of a unit sphere in IR3) and impose a Riemannian metric on it in such a way that the reparameterization group acts on this space by isometries. Under this framework, we solve two optimization problems. One, given any two surfaces at arbitrary rotations and parameterizations, we use a path-straightening approach to find a geodesic path between them under the chosen metric. Second, by modifying a technique presented in [25], we solve for the optimal rotation and parameterization (registration) between surfaces. Their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces. We illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces.     

Geodesic distance-weighted shape vector image diffusion

This paper presents a novel and efficient surface matching and visualization framework through the geodesic distance-weighted shape vector image diffusion. Based on conformal geometry, our approach can uniquely map a 3D surface to a canonical rectangular domain and encode the shape characteristics (e.g., mean curvatures and conformal factors) of the surface in the 2D domain to construct a geodesic distance-weighted shape vector image, where the distances between sampling pixels are not uniform but the actual geodesic distances on the manifold. Through the novel geodesic distance-weighted shape vector image diffusion presented in this paper, we can create a multiscale diffusion space, in which the cross-scale extrema can be detected as the robust geometric features for the matching and registration of surfaces. Therefore, statistical analysis and visualization of surface properties across subjects become readily available. The experiments on scanned surface models show that our method is very robust for feature extraction and surface matching even under noise and resolution change. We have also applied the framework on the real 3D human neocortical surfaces, and demonstrated the excellent performance of our approach in statistical analysis and integrated visualization of the multimodality volumetric data over the shape vector image.