Diagonal curve is one of the most important shape measurements of tensor-product Bézier surfaces. An approach to construct Bézier surfaces with energy-minimizing from two prescribed diagonal curves ...is presented in this paper. Firstly, a general second order functional energy is formulated with several parameters. This functional includes many common functionals as special cases, such as the Dirichlet energy, the biharmonic functional and the quasi-harmonic functional etc. Secondly, the necessary and sufficient conditions for tensor-product Bézier surface with minimal energy from prescribed diagonal curves are derived. Besides prescribing the diagonal curves, other related problems are considered, those where boundary curves are also prescribed. Finally, the effectiveness of the proposed method is illustrated by several surface modeling examples.
Modeling of free-form curves and surfaces is vital in the manufacturing industry and engineering. Curves and surfaces that have flexible shapes and adjustable lengths and sizes are a necessity to ...fulfill the needs of the manufacturing industry. Hence, researchers develop numerous aesthetic Bézier curves and surfaces to have such flexibility and adjustability. In this paper, the generalized Riemann–Liouville fractional Bézier curves and surfaces are proposed in the modeling of complex surfaces. The generalized Riemann–Liouville fractional Bézier curves and surfaces have two outstanding parameters: shape and fractional parameters. Shape parameters are utilized to change the shape of the curves and surfaces without altering the control points hence, adding flexibility in controlling the shape. While fractional parameters are utilized in curves and surfaces’ length and size adjustability. By adjusting the sizes of surfaces via fractional parameters, surfaces with optimal sizes can be modeled. In this paper, five types of engineering surfaces will be modeled, namely ruled surface, swept surface, swung surface, rotation surface, and coons patch. The geometric effects of the implementation of shape and fractional parameters to these engineering surfaces will also be analyzed, thus proving the generalized Riemann–Liouville fractional Bézier surfaces is an excellent tool in designing complex surfaces.
Diagonal curve is one of the most important shape measurements of tensor-product Bézier surfaces. An approach to construct Bézier surfaces with energy-minimizing diagonal curves from four input ...boundary curves is presented in this paper. Firstly, the expression for the diagonal energy is formulated. Secondly, the necessary and sufficient conditions for tensor-product Bézier surface with minimal diagonal energy are derived. Both stretch energy and bending energy of the diagonal curves are studied. Finally, from the derived conditions, a construction algorithm for Bézier surfaces with minimal diagonal energy by using Lagrange-multiplier method is presented. The effectiveness of the proposed method is illustrated by several surface modeling examples.
The affine space of all tensor product Bézier patches of degree n×n with prescribed main diagonal curves is determined. First, the pair of Bézier curves which can be diagonals of a Bézier patch is ...characterized. Besides prescribing the diagonal curves, other related problems are considered, those where boundary curves or tangent planes along boundary curves are also prescribed.
•An interactive and integrative approach to truss design is proposed by developing a Grasshopper component for simultaneous optimization of geometry and topology of trusses.•Numerical difficulty due ...to melting nodes can be successfully avoided using force density as design variable, and various optimal geometry and topology can be obtained.•Tensor product Bezier surface is introduced as a design surface to control optimal shapes.•Efficiency and accuracy of the proposed method are demonstrated through two numerical examples of semi-cylindrical and semi-spherical models.
This paper presents a new efficient tool for simultaneous optimization of topology and geometry of truss structures. Force density method is applied to formulate optimization problem to minimize compliance under constraint on total structural volume, and objective and constraint functions are expressed as explicit functions of force density only. This method does not need constraints on nodal locations to avoid coalescent nodes, and enables to generate optimal solutions with a variety in topology and geometry. Furthermore, for the purpose of controlling optimal shapes, tensor product Bézier surface is introduced as a design surface. The optimization problem is solved using sensitivity coefficients and the optimizer is compiled as a component compatible with Grasshopper, an algorithmic modeling plug-in for Rhinoceros, which is a popular 3D modeling software. Efficiency and accuracy of the proposed method are demonstrated through two numerical examples of semi-cylindrical and semi-spherical models.
The surface of partial differential equation (PDE surface) is a surface that satisfies the PDE with boundary conditions, which can be applied in surface modeling and construction. In this paper, the ...construction of tensor product Bézier surfaces of triharmonic equation from different boundary conditions is presented. The internal control points of the resulting triharmonic Bézier surface can be obtained uniquely by the given boundary condition. Some representative examples show the effectiveness of the presented method.
Recently, a functional named quasi-area functional was proposed in 14 to approximate the area functional in the Plateau-Bézier problem. The quasi-area functional is constructed by a balanced sum ...among the quasi-harmonic functional, Dirichlet functional and a functional which measures isothermality. It improves greatly the approximation efficiency of existing methods. Therefore in this paper, based on the quasi-area functional, we introduce two methods to study the triangular Plateau-Bézier problem. On the one hand, we minimize directly the functional among all the triangular Bézier surfaces with prescribed border determined by the exterior control points. On the other hand, we compute and analyze the Euler-Lagrange equation of the quasi-area functional. Both methods are illustrated by some representative examples to show the effectiveness.
In this paper, a new generalization of Bézier curves with one shape parameter is constructed. It is based on the Lupaş q-analogue of Bernstein operator, which is the first generalized Bernstein ...operator based on the q-calculus. The new curves have some properties similar to classical Bézier curves. Moreover, we establish degree evaluation and de Casteljau algorithms for the generalization. Furthermore, we construct the corresponding tensor product surfaces over the rectangular domain, and study the properties of the surfaces, as well as the degree evaluation and de Casteljau algorithms. Compared with q-Bézier curves and surfaces based on Phillips q-Bernstein polynomials, our generalizations show more flexibility in choosing the value of q and superiority in shape control of curves and surfaces. The shape parameters provide more convenience for the curve and surface modeling.
An extension of the Bézier model Yan, Lanlan; Liang, Jiongfeng
Applied mathematics and computation,
11/2011, Letnik:
218, Številka:
6
Journal Article
Recenzirano
In this paper, we first construct a new kind of basis functions by a recursive approach. Based on these basis functions, we define the Bézier-like curve and rectangular Bézier-like surface. Then we ...extend the new basis functions to the triangular domain, and define the Bernstein–Bézier-like surface over the triangular domain. The new curve and surfaces have most properties of the corresponding classical Bézier curve and surfaces. Moreover, the shape parameter can adjust the shape of the new curve and surfaces without changing the control points. Along with the increase of the shape parameter, the new curve and surfaces approach the control polygon or control net. In addition, the evaluation algorithm for the new curve and triangular surface are provided.
Accurate estimation of the fractional abundances of intimately mixed materials from spectral reflectances is generally hard due to a highly nonlinear relationship between the measured spectrum and ...the composition of the material. Changes in the acquisition and the illumination conditions cause variability in the spectral reflectance, further complicating the spectral unmixing procedure. In this work, we propose a methodology for unmixing intimate mixtures that can tackle both nonlinearity and spectral variability. A supervised approach is proposed that characterizes the nonlinear data manifolds by high-dimensional Bézier surfaces. To deal with spectral variability, a manifold transformation procedure is designed. To generate Bézier surfaces, training samples are required that are uniformly distributed throughout the data manifold. For this, we recently generated a hyperspectral dataset of intimate mineral powder mixtures by homogeneously mixing five different clay powders (Kaolin, Roof clay, Red clay, mixed clay, and Calcium hydroxide) in laboratory settings. In total 330 samples (325 mixtures and five pure materials) were prepared. The ground fractional abundances of these mixtures uniformly cover the 5-D probability simplex. The spectral reflectances of these samples were acquired by multiple sensors with a large variation in sensor types, platforms, and acquisition conditions. Experiments are conducted both on simulated and real intimate mineral powder mixtures. Comparison with a number of unsupervised unmixing methods demonstrates the potential of the proposed approach.
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