Combined Subdivision Research Articles

Variations in 2D and 3D Models by a New Family of Subdivision Schemes and Algorithms for its Analysis

A new family of combined subdivision schemes with one tension parameter is proposed by the interpolatory and approximating subdivision schemes. The displacement vectors between the points of interpolatory and approximating subdivision schemes provide the flexibility in designing the limit curves and surfaces. Therefore, the limit curves generated by the proposed subdivision schemes variate in between or around the approximating and interpolatory curves. We also design few analytical algorithms to study the properties of the proposed schemes theoretically. The efficiency of these algorithms is analyzed by calculating their time complexity. The graphical representations and graphical properties of the proposed schemes are also analyzed.

Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials

Abstract In this paper, we propose a family of non-stationary combined ternary ( 2 m + 3 ) \left(2m+3) -point subdivision schemes, which possesses the property of generating/reproducing high-order exponential polynomials. This scheme is obtained by adding variable parameters on the generalized ternary subdivision scheme of order 4. For such a scheme, we investigate its support and exponential polynomial generation/reproduction and get that it can generate/reproduce certain exponential polynomials with suitable choices of the parameters and reach 2 m + 3 2m+3 approximation order. Moreover, we discuss its smoothness and show that it can produce C 2 m + 2 {C}^{2m+2} limit curves. Several numerical examples are given to show the performance of the schemes.

A New Paradigm to Design a Class of Combined Ternary Subdivision Schemes

Subdivision schemes play a vital role in curve modeling. The curves produced by the class of 2 n + 2 -point ternary scheme (Deslauriers and Dubuc (1989)) interpolate the given data while the curves produced by a class of 2 n + 2 -point ternary B-spline schemes approximate the given data. In this research, we merge these two classes to introduce a consolidated and unified class of combined subdivision schemes with two shape control parameters in order to grow versatility for overseeing valuable necessities. However, the proposed class of subdivision schemes gives optimal smoothness in the final shapes, yet we can increase its smoothness by using a proposed general formula in form of its Laurent polynomial. The theoretical analysis of the class of subdivision schemes is done by using various mathematical tools and using their coding in the Maple environment. The graphical analysis of the class of schemes is done in the Maple environment by writing the codes based on the recursive mathematical expressions of the class of subdivision schemes.

Parametric hull shape variations by Reduced Order Model based geometric transformation

The representation and the variation of hull shapes are two challenging problems in naval architecture due to the complexity of the geometry and to the need to ensure the fairness of the surfaces. Conventional CAD techniques are widely used to accurately describe the hull shape. However, they are rather complex to be easily used to generate hull shape variations due to the great number of variables involved. We propose a study to highlight the pro and cons of the application of space reduction techniques, usually referred to as Reduced Order Models (ROMs), to create a parametric model for both global and local hull shape variations. A geometric transformation relying on the Proper Orthogonal Decomposition (POD) method is applied on top of a combined subdivision surface — Free Form Deformation (FFD) approach conceived for modeling and variation of hull shapes. The analysis focuses on highlighting the geometric meaning of the new POD basis functions and on how this model affects the variability of the explored design space. Two studies are developed in order to show possible applications of this technique. In the first one a new set of selection criteria for the POD modes based on geometric considerations are proposed to use it on uncorrelated geometric domains. In the latter the large-dimensional space of the so-called Design Velocities (DV), related to the shape sensitivity, is reduced by using the POD approach to create a new transformation of the hull shape.

Six-Point Subdivision Schemes with Cubic Precision

This paper presents 6-point subdivision schemes with cubic precision. We first derive a relation between the 4-point interpolatory subdivision and the quintic B-spline refinement. By using the relation, we further propose the counterparts of cubic and quintic B-spline refinements based on 6-point interpolatory subdivision schemes. It is proved that the new family of 6-point combined subdivision schemes has higher smoothness and better polynomial reproduction property than the B-spline counterparts. It is also showed that, both having cubic precision, the well-known Hormann-Sabin’s family increase the degree of polynomial generation and smoothness in exchange of the increase of the support width, while the new family can keep the support width unchanged and maintain higher degree of polynomial generation and smoothness.

A non-stationary combined subdivision scheme generating exponential polynomials

In this paper, a non-stationary combined subdivision scheme is presented, which can unify several existing non-stationary approximating and interpolatory subdivision schemes. This scheme is obtained by generalizing the connection between the approximating and interpolatory schemes in the stationary case, first formalized by Maillot & Stam using a push-back operator, to the non-stationary case. For such a combined scheme, we investigate its Cl convergence and exponential polynomial generation/reproduction property and get that it can reach C4 degree of smoothness and generate/reproduce certain exponential polynomials with suitable choices of the parameters. Besides, we give a more generalized combined scheme for the purpose of generating and reproducing more general exponential polynomials. The performance of our new schemes is illustrated by several numerical examples.

A combined approximating and interpolating subdivision scheme with C2 continuity

In this paper a combined approximating and interpolating subdivision scheme is presented. The relationship between approximating subdivision and interpolating subdivision is derived by directly performing operations on geometric rules. The behavior of the limit curve produced by our combined subdivision scheme is analyzed by the Laurent polynomial and attains C 2 degree of smoothness. Furthermore, a non-uniform combined subdivision with shape control parameters is introduced, which allows a different tension value for every edge of the original control polygon.

A Quick Edit Method of Curve Network’s Interpolated Surface Based on Combined Subdivision

A simple and effective combined subdivision surface’s modification method is proposed, which is realized by editing its curve network locally. After discussing the overlapping principles of Catmull-Clark subdivision and combined subdivision, a quick update method of the modified surface is given by building the difference curve network from the original curve network and the edited curve network and overlapping the mesh from the difference curve network to the mesh from the original curve network. This method is closely related with the features of combined subdivision, so it is simple and high efficiency. It is useful in the concept design of complex product.

Study on the Asymmetric Sharp Feature Construct Method in Combined Subdivision Surface

Compound curve network is built to defined asymmetric sharp features in combined subdivision surface. The basic idea is to make the construction of sharp features separate from the definition of global surface and build compound curve networks consisting of base curve network and feature curve network. The curve’s subdivision rules and correct operator applying on common vertexes directly connected with curve vertexes are redefined to generate asymmetric sharp feature, and subdivision times is selected as sharp feature factor to control the sharp degree of features. When different sharp feature factors are defined at the two sides of interpolated curves, asymmetric features are generated on them. Curves of base curve network and curves of feature curve network take part in the subdivision of the interpolated surface in turn. Examples show that our method is effective and can get high quality interpolated surface.

Subdivision surface modeling system based on arbitrary topological curves network and combined subdivision

Arbitrary topological curve network has no restriction in topology structure, so it has more powerful representing ability in defining complex surfaces. A complex surface modeling system is presented based on arbitrary topological curve network and the improved combined subdivision method, its functions including creating and editing curve network, and generating and modifying curve network’s interpolated surface. This modeling system can be used to the process of products’ concept design, and its applications is also significant to the development of subdivision method.

Remeshing into normal meshes with boundaries using subdivision

In this paper, we present a remeshing algorithm using the recursive subdivision of irregular meshes with boundaries. A mesh remeshed by subdivision has several advantages. It has a topological regularity, which enables it to be used as a multiresolution model and to represent an original model with less data. Topological regularity is essential for the multiresolutional analysis of the given meshes and makes additional topological information unnecessary. Moreover, we use a normal mesh to reduce the geometric data size requirements at each resolution level of the regularized meshes. The normal mesh uses one scalar value, i.e. normal offset as wavelet, to represent a vertex position, while the other remeshing schemes use one three-dimensional vector at each vertex. The normal offset is a normal distance from a base face, which is the simplified original mesh. Since the normal offset cannot be properly used for the boundaries of a mesh, we use a combined subdivision scheme that resolves the problem of the proposed normal offset method at the boundaries. Finally, we show examples that demonstrate the effectiveness of the proposed scheme in terms of reducing the data requirements of mesh models.

曲線集合からの細分割曲面生成手法

In CAID (Computer Aided Industrial Design) systems, a free form surface design generally starts with defining characteristic curves in 3D space. Based on these characteristic curves, basic curved surfaces are then generated so as to interpolate them. It is required that these curves be connected in order to compute interpolating surface patches. However, it is desirable to give the designers more freedom to allow them to define the characteristic curves without considering their connectivity. In this research, we propose a method called "Curve Wrapping" to generate the connectivity from a set of unconnected characteristic curves by using "a graph searching mesh" technique. And using this connectivity, the curves are adapted to form a graph so that subdivision surfaces are generated using a subdivision surface fitting method and the "combined subdivision" method. A prototype system was developed to evaluate the approach using some examples.

Normal Meshes for Multiresolution Analysis of Irregular Meshes with Boundaries.

In this paper we present a remeshing algorithm for irregular meshes with boundaries. The irregular meshes are approximated by regular meshes that have topological regularity, which is essential for the multiresolution analysis of the given irregular meshes. Normal meshes are used to reduce the necessary data size at each resolution level of the regularized meshes. The normal mesh uses one scalar value, i. e., normal offset value that is calculated by the regular rule of a uniform subdivision and normal sampling, while other remeshing schemes use one 3D vector at each vertex. Since the normal offset cannot be adopted for the boundaries of meshes, we use a combined subdivision scheme that resolves the problem of the proposed normal offset method at the boundaries. Lastly, an example showing the effectiveness of the proposed scheme to reduce the data size of a mesh model is included.

Trimming for subdivision surfaces

Trimming is an important primitive operation in geometric modeling. It is also the root of many numerical and topological problems in modern NURBS based CAGD systems. In this paper we introduce a new method for trimming subdivision surfaces. It is based on the use of combined subdivision schemes to guarantee exact interpolation of trim curves. The latter ensures, for example, that if two surfaces share a trim curve, they will meet exactly at the trim curve. In contrast to traditional approaches to trimming (e.g., for NURBS) we construct a new control mesh with each trim operation. This causes a perturbation of the surface near the trim region, which we control through the use of multiresolution details. These are computed rapidly and at low cost with the help of a novel set of quasi-interpolation operators. We demonstrate our algorithm with a number of examples.

Combined subdivision schemes for the design of surfaces satisfying boundary conditions

We present a new kind of subdivision schemes, which we call combined subdivision schemes. Such schemes are used to refine a given control net, taking into consideration boundary conditions on the limit surface. Several schemes are presented that generate limit surfaces interpolating given boundary curves and cross boundary derivatives. The interpolated boundary curves are arbitrary parametric curves, and are not restricted to subdivision curves or to spline curves.