Triangular B-splines Research Articles

Reconstruction of Low Degree B-spline Surfaces with Arbitrary Topology Using Inverse Subdivision Scheme

Multivariate B-spline surfaces over triangular parametric domain have many interesting properties in the construction of smooth free-form surfaces. This paper introduces a novel approach to reconstruct triangular B-splines from a set of data points using inverse subdivision scheme. Our proposed method consists of two major steps. First, a control polyhedron of the triangular B-spline surface is created by applying the inverse subdivision scheme on an initial triangular mesh. Second, all control points of this B-spline surface, as well as knotclouds of its parametric domain are iteratively adjusted locally by a simple geometric fitting algorithm to increase the accuracy of the obtained B-spline. The reconstructed B-spline having the low degree along with arbitrary topology is interpolative to most of the given data points after some fitting steps without solving any linear system. Some concrete experimental examples are also provided to demonstrate the effectiveness of the proposed method. Results show that this approach is simple, fast, flexible and can be successfully applied to a variety of surface shapes.

Reproducing kernel triangular B-spline-based FEM for solving PDEs

We propose a reproducing kernel triangular B-spline-based finite element method (FEM) as an improvement to the conventional triangular B-spline element for solving partial differential equations (PDEs). In the latter, unexpected errors can occur throughout the analysis domain mainly due to the excessive flexibilities in defining the B-spline. The performance therefore becomes unstable and cannot be controlled in a desirable way. To address this issue, the proposed improvement adopts the reproducing kernel approximation in the calculation of B-spline kernel function. Three types of PDE problems are tested to validate the present element and compare against the conventional triangular B-spline. It has been shown that the improved triangular B-spline satisfies the partition of unity condition even for extreme conditions including corners and holes.

Computation and Simulation of 5-Axis Cutter Based on Scattered Point Cloud Data

The complex surface is modeled by triangular surface, B-spline or non-uniform rational B-spline first and then the planing of the cutter is followed in reverse engineering. But this require CAD/CAM software and operator’s experiene. Because of this，the research on direct NC cutter generation from point cloud have become the focus of research in reverse engineering. The normal，curvature，surface vary et al parameters are calculated by principle component analysis method and so on, which define the local feature of the surface hidden in the point cloud. Then，based on local feature，point cloud is simplified. It is proved that two points with minimal orientation-distance on a pair of smooth surfaces are conjugate points. Thus the conjugate problem between the tool surface and the part surface is converted into optimization process of orientation-distance function. The cutter trace and the cutter interference are achieved by local and global optization of the orientation-distance function.

Generation of Knot Net for Calculation of Quadratic Triangular B-spline Surface of Human Head

This paper deals with calculation of the quadratic triangular B-spline surface of the human head for the purpose of its modeling in the standard videocodec MPEG-4 SNHC. In connection with this we propose an algorithm of generation of the knot net and present the results of its application for triangulation of the 3D polygonal model Candide. Then for the model and generated knot net as well as an established distribution of control points we show the results of the calculated quadratic triangular B-spline surface of the human head including its textured version for the texture of the selected avatar.

The Reproducing Kernel DMS-FEM: 3D Shape Functions and Applications to Linear Solid Mechanics

We propose a family of 3D versions of a smooth finite element method (Sunilkumar and Roy 2010), wherein the globally smooth shape functions are derivable through the condition of polynomial reproduction with the tetrahedral B-splines (DMS-splines) or tensor-product forms of triangular B-splines and ID NURBS bases acting as the kernel functions. While the domain decomposition is accomplished through tetrahedral or triangular prism elements, an additional requirement here is an appropriate generation of knotclouds around the element vertices or corners. The possibility of sensitive dependence of numerical solutions to the placements of knotclouds is largely arrested by enforcing the condition of polynomial reproduction whilst deriving the shape functions. Nevertheless, given the higher complexity in forming the knotclouds for tetrahedral elements especially when higher demand is placed on the order of continuity of the shape functions across inter-element boundaries, we presently emphasize an exploration of the triangular prism based formulation in the context of several benchmark problems of interest in linear solid mechanics. In the absence of a more rigorous study on the convergence analyses, the numerical exercise, reported herein, helps establish the method as one of remarkable accuracy and robust performance against numerical ill-conditioning (such as locking of different kinds) vis-a-vis the conventional FEM.

Surface reconstruction using bivariate simplex splines on Delaunay configurations

Recently, a new bivariate simplex spline scheme based on Delaunay configuration has been introduced into the geometric computing community, and it defines a complete spline space that retains many attractive theoretic and computational properties. In this paper, we develop a novel shape modeling framework to reconstruct a closed surface of arbitrary topology based on this new spline scheme. Our framework takes a triangulated set of points, and by solving a linear least-square problem and iteratively refining parameter domains with newly added knots, we can finally obtain a continuous spline surface satisfying the requirement of a user-specified error tolerance. Unlike existing surface reconstruction methods based on triangular B-splines (or DMS splines), in which auxiliary knots must be explicitly added in advance to form a knot sequence for construction of each basis function, our new algorithm completely avoids this less-intuitive and labor-intensive knot generating procedure. We demonstrate the efficacy and effectiveness of our algorithm on real-world, scattered datasets for shape representation and computing.

Manifold splines

Constructing splines whose parametric domain is an arbitrary manifold and effectively computing such splines in real-world applications are of fundamental importance in solid and shape modeling, geometric design, graphics, etc. This paper presents a general theoretical and computational framework, in which spline surfaces defined over planar domains can be systematically extended to manifold domains with arbitrary topology with or without boundaries. We study the affine structure of domain manifolds in depth and prove that the existence of manifold splines is equivalent to the existence of a manifold’s affine atlas. Based on our theoretical breakthrough, we also develop a set of practical algorithms to generalize triangular B-spline surfaces from planar domains to manifold domains. We choose triangular B-splines mainly because of its generality and many of its attractive properties. As a result, our new spline surface defined over any manifold is a piecewise polynomial surface with high parametric continuity without the need for any patching and/or trimming operations. Through our experiments, we hope to demonstrate that our novel manifold splines are both powerful and efficient in modeling arbitrarily complicated geometry and representing continuously varying physical quantities defined over shapes of arbitrary topology.

Automatic Shape Control of Triangular B-Splines of Arbitrary Topology

Triangular B-splines are powerful and flexible in modeling a broader class of geometric objects defined over arbitrary, non-rectangular domains. Despite their great potential and advantages in theory, practical techniques and computational tools with triangular B-splines are less-developed. This is mainly because users have to handle a large number of irregularly distributed control points over arbitrary triangulation. In this paper, an automatic and efficient method is proposed to generate visually pleasing, high-quality triangular B-splines of arbitrary topology. The experimental results on several real datasets show that triangular B-splines are powerful and effective in both theory and practice.

Geometric interpretation of continuity over triangular domains

In this paper, the geometric interpretation of higher order continuity conditions is presented. The concept of duality of “Bézier points” and “extension points” is developed by considering different domain triangulation. Subsequently, the notion of “Continuity around a vertex” is realized and it results in some very interesting and symmetric geometries. These constructs may motivate the geometric interpretation of triangular B-splines or triangular NURBS.

Triangular NURBS and their dynamic generalizations

Triangular B-splines are a new tool for modeling a broad class of objects defined over arbitrary, nonrectangular domains. They provide an elegant and unified representation scheme for all piecewise continuous polynomial surfaces over planar triangulations. To enhance the power of this model, we propose triangular NURBS, the rational generalization of triangular B-splines, with weights as additional degrees of freedom. Fixing the weights to unity reduces triangular NURBS to triangular B-splines. Conventional geometric design with triangular NURBS can be laborious, since the user must manually adjust the many control points and weights. To ameliorate the design process, we develop a new model based on the elegant triangular NURBS geometry and principles of physical dynamics. Our model combines the geometric features of triangular NURBS with the demonstrated conveniences of interaction within a physics-based framework. The dynamic behavior of the model results from the numerical integration of differential equations of motion that govern the temporal evolution of control points and weights in response to applied forces and constraints. This results in physically meaningful hence highly intuitive shape variation. We apply Lagrangian mechanics to formulate the equations of motion of dynamic triangular NURBS and finite element analysis to reduce these equations to efficient numerical algorithms. We demonstrate several applications, including direct manipulation and interactive sculpting through force-based tools, the fitting of unorganized data, and solid rounding with geometric and physical constraints.

Spherical Triangular B-splines with Application to Data Fitting

Triangular B-splines surfaces are a tool for representing arbitrary piecewise polynomial surfaces over planar triangulations, while automatically maintaining continuity properties across patch boun...

A G1 triangular spline surface of arbitrary topological type

A piecewise G 1 spline surface composed of sextic triangular Bézier patches in one-to-one correspondence with the faces of a triangular control mesh is presented. Surfaces of arbitrary topological type are created by approximating any mesh that represents a triangulated 2-manifold. The surface has local support and is affine invariant. A set of shape parameters are available for additional local control over the shape of the surface. If some local regularities are present in the structure of the control mesh, then the corresponding patches may be parametrized as quintic or even quartic Bézier triangles. In the case of a regular triangulation (with appropriate shape parameters), the surface generated by this method is equivalent to a quartic C 2 triangular B-spline.

Modeling with triangular B-splines

Triangular B-splines are a new tool for modeling complex objects with nonrectangular topology. The scheme is based on blending functions and control points, and lets us model piecewise polynomial surfaces of degree n that are C/sup n-1/-continuous throughout. Triangular B-splines permit the construction of smooth surfaces with the lowest degree possible. Because they can represent any piecewise polynomial surface, they provide a unified data format. The new B-spline scheme for modeling complex irregular objects over arbitrary triangulations has many desirable features. Applications like filling polygonal holes or constructing smooth blends demonstrate its potential for dealing with concrete design problems. The method permits real-time editing and rendering. Currently, we are improving the editor to accept simpler user input, optimizing intersection computations and developing new applications.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An introduction to polar forms

Polar forms, which simplify the construction of polynomial and piecewise-polynomial curves and surfaces and lead to new surface representations and algorithms, are reviewed. The polar forms of polynomial curves, Bezier curves, and B-spline are discussed. Tensor product surfaces, the most popular surfaces in computer-aided geometric design, true surfaces, Bezier triangles, B-patches, and a triangular B-spline scheme that combines B-patches and simplex splines are also discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>