scholarly journals Bicubic splines and biquartic polynomials

2016 ◽  
Vol 6 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Lukáš Mino ◽  
Imrich Szabó ◽  
Csaba Török
Keyword(s):  
Efficient Approach ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Hermite Spline ◽  
Approximating Polynomials ◽  
Bicubic Splines

AbstractThe paper proposes a new efficient approach to computation of interpolating spline surfaces. The generalization of an unexpected property, noticed while approximating polynomials of degree four by cubic ones, confirmed that a similar interrelation property exists between biquartic and bicubic polynomial surfaces as well. We prove that a 2×2-component C1 -class bicubic Hermite spline will be of class C2 if an equispaced grid is used and the coefficients of the spline components are computed from a corresponding biquartic polynomial. It implies that a 2×2 uniform clamped spline surface can be constructed without solving any equation. The applicability of this biquartic polynomials based approach to reducing dimensionalitywhile computing spline surfaces is demonstrated on an example.

scholarly journals Optimal Approximation of Biquartic Polynomials by Bicubic Splines

2018 ◽  
Vol 173 ◽  
pp. 03012
Author(s):  
Viliam Kačala ◽  
Csaba Török
Keyword(s):  
Optimal Approximation ◽  
First Derivative ◽  
Polynomial Coefficients ◽  
Spline Curves ◽  
Spline Surfaces ◽  
Chebyshev Norm ◽  
Quartic Polynomial ◽  
Hermite Spline ◽  
Open Question ◽  
Bicubic Splines

Recently an unexpected approximation property between polynomials of degree three and four was revealed within the framework of two-part approximation models in 2-norm, Chebyshev norm and Holladay seminorm. Namely, it was proved that if a two-component cubic Hermite spline’s first derivative at the shared knot is computed from the first derivative of a quartic polynomial, then the spline is a clamped spline of classC2and also the best approximant to the polynomial.Although it was known that a 2 × 2 component uniform bicubic Hermite spline is a clamped spline of classC2if the derivatives at the shared knots are given by the first derivatives of a biquartic polynomial, the optimality of such approximation remained an open question.The goal of this paper is to resolve this problem. Unlike the spline curves, in the case of spline surfaces it is insufficient to suppose that the grid should be uniform and the spline derivatives computed from a biquartic polynomial. We show that the biquartic polynomial coefficients have to satisfy some additional constraints to achieve optimal approximation by bicubic splines.

Generation and Modification of Non-Uniform B-Spline Surface Approximations to PDE Surfaces Using the Finite Element Method

1990 ◽  
Author(s):  
Joanna M. Brown ◽  
Malcolm I. G. Bloor ◽  
M. Susan Bloor ◽  
Michael J. Wilson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Spline Approximation ◽  
The Finite Element Method ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Spline Basis ◽  
Element Method

Abstract A PDE surface is generated by solving partial differential equations subject to boundary conditions. To obtain an approximation of the PDE surface in the form of a B-spline surface the finite element method, with the basis formed from B-spline basis functions, can be used to solve the equations. The procedure is simplest when uniform B-splines are used, but it is also feasible, and in some cases desirable, to use non-uniform B-splines. It will also be shown that it is possible, if required, to modify the non-uniform B-spline approximation in a variety of ways, using the properties of B-spline surfaces.

Sequential B-Spline Surface Construction Using Multiresolution Data Clouds

2012 ◽  
Vol 12 (2) ◽  
Author(s):  
Yuan Yuan ◽  
Shiyu Zhou
Keyword(s):  
Control Points ◽  
Finite Sample ◽  
B Spline ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Filtering Technique ◽  
Engineering Practices ◽  
Asymptotical Convergence ◽  
Surface Construction ◽  
Fitting Error

B-spline surfaces are widely used in engineering practices as a flexible and efficient mathematical model for product design, analysis, and assessment. In this paper, we propose a new sequential B-spline surface construction procedure using multiresolution measurements. At each iterative step of the proposed procedure, we first update knots vectors based on bias and variance decomposition of the fitting error and then incorporate new data into the current surface approximation to fit the control points using Kalman filtering technique. The asymptotical convergence property of the proposed procedure is proved under the framework of sieves method. Using numerical case studies, the effectiveness of the method under finite sample is tested and demonstrated.

scholarly journals Hierarchical Genetic Algorithm for B-Spline Surface Approximation of Smooth Explicit Data

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
C. H. Garcia-Capulin ◽  
F. J. Cuevas ◽  
G. Trejo-Caballero ◽  
H. Rostro-Gonzalez
Keyword(s):  
Genetic Algorithm ◽  
Geometric Modeling ◽  
Data Set ◽  
Surface Approximation ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Surface Dimension ◽  
Hierarchical Genetic Algorithm

B-spline surface approximation has been widely used in many applications such as CAD, medical imaging, reverse engineering, and geometric modeling. Given a data set of measures, the surface approximation aims to find a surface that optimally fits the data set. One of the main problems associated with surface approximation by B-splines is the adequate selection of the number and location of the knots, as well as the solution of the system of equations generated by tensor product spline surfaces. In this work, we use a hierarchical genetic algorithm (HGA) to tackle the B-spline surface approximation of smooth explicit data. The proposed approach is based on a novel hierarchical gene structure for the chromosomal representation, which allows us to determine the number and location of the knots for each surface dimension and the B-spline coefficients simultaneously. The method is fully based on genetic algorithms and does not require subjective parameters like smooth factor or knot locations to perform the solution. In order to validate the efficacy of the proposed approach, simulation results from several tests on smooth surfaces and comparison with a successful method have been included.

Direct slicing of T-spline surfaces for additive manufacturing

2018 ◽  
Vol 24 (4) ◽  
pp. 709-721 ◽  
Author(s):  
Jiawei Feng ◽  
Jianzhong Fu ◽  
Zhiwei Lin ◽  
Ce Shang ◽  
Bin Li
Keyword(s):  
Additive Manufacturing ◽  
Computer Aided Design ◽  
Content Type ◽  
Geometrical Properties ◽  
Direct Slicing ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Complex Models ◽  
Aided Design ◽  
Design And Manufacture

Purpose T-spline is the latest powerful modeling tool in the field of computer-aided design. It has all the merits of non-uniform rational B-spline (NURBS) whilst resolving some flaws in it. This work applies T-spline surfaces to additive manufacturing (AM). Most current AM products are based on Stereolithograph models. It is a kind of discrete polyhedron model with huge amounts of data and some inherent defects. T-spline offers a better choice for the design and manufacture of complex models. Design/methodology/approach In this paper, a direct slicing algorithm of T-spline surfaces for AM is proposed. Initially, a T-spline surface is designed in commercial software and saved as a T-spline mesh file. Then, a numerical method is used to directly calculate all the slicing points on the surface. To achieve higher manufacturing efficiency, an adaptive slicing algorithm is applied according to the geometrical properties of the T-spline surface. Findings Experimental results indicate that this algorithm is effective and reliable. The quality of AM can be enhanced at both the designing and slicing stages. Originality/value The T-spline and direct slicing algorithm discussed here will be a powerful supplement to current technologies in AM.

Filling N-Sided Holes With Trimmed B-Spline Surfaces Based on Energy-Minimization Method

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Xiaodong Liu
Keyword(s):  
Energy Minimization ◽  
Computer Aided Design ◽  
Control Points ◽  
Minimization Method ◽  
B Spline ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Key Issues ◽  
Complex Constraints ◽  
Aided Design

Using a trimmed rectangular B-Spline surface to fill an n-sided hole is a much desired operation in computer aided design (CAD), but few papers have addressed this issue. Based on an energy-minimization or variational B-Spline technique, the paper presents the technique of using one single trimmed rectangular B-Spline surface to fill an n-sided hole. The method is efficient and robust, and takes a fraction of a second to fill n-sided holes with high-quality waterproof B-Spline surfaces under complex constraints. As the foundation of filling n-sided holes, the paper also presents the framework and addresses the key issues on variational B-Spline technique. Without any precalculation, the variational B-Spline technique discussed in this paper can solve virtually any B-Spline surface with up to 20,000 control points in real time, which is much more efficient and powerful than previous work in the variational B-Spline field. Moreover, the result is accurate and satisfies CAD systems' high-precision requirements.

AUTOMATED SPLINE SURFACE MODELING AND MATCHING FOR RECOGNITION OF FREE-FORM OBJECTS

2004 ◽  
Vol 04 (01) ◽  
pp. 51-84
Author(s):  
G. MAMIC ◽  
M. BENNAMOUN
Keyword(s):  
Hash Table ◽  
Free Form ◽  
Surface Errors ◽  
Fitting Procedure ◽  
Surface Recognition ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Model Control ◽  
Surface Representations ◽  
High Level

Surface representations are utilized in a multitude of applications such as computer vision, medical imaging and computer graphics. B-spline surfaces have a number of desirable properties for representing surfaces, however, the complexity of knot placement strategies has prevented their widespread use in high-level vision environments. A solution to this problem is formulated within the reversible jump Markov chain Monte Carlo framework, whereby a derived posterior distribution may be sampled to calculate expected values for the number of knots required, their expected positions and a maximum likelihood estimate for the resulting control net of a given surface. Recognition of individual models may then be achieved using a hash table constructed using the principal components of the model control nets. Results of the fitting procedure, in terms of estimated knot vectors and spline surface errors, and the recognition of objects are provided for a set of free-form objects.

A New Method for Deformation of B-Spline Surfaces

2010 ◽  
Vol 139-141 ◽  
pp. 1260-1263
Author(s):  
Xian Guo Cheng ◽  
Wei Jun Liu
Keyword(s):  
Optimization Problem ◽  
Local Deformation ◽  
Geometric Constraints ◽  
Surface Patch ◽  
The Finite Element Method ◽  
B Spline ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Single Force ◽  
External Forces

This paper presents an efficient method for deforming B-spline surfaces, based on the surface energy minimization. Firstly, using an analogy between the B-spline surface patch and the thin-plate element of the finite element method, and applying external forces on the surface with some given geometric constraints, the forces can locate on part of the surface or the surface. Then, the energy of the B-spline surface can change with the change of the forces. Finally, a new B-spline surface is generated by solving an optimization problem of change of the energy. The forces can be a single force, a distributed force and set of isolated force. The method can accomplish easily local deformation and total deformation of the B-spline surface.

Filling N-Sided Holes Using Trimmed B-Spline Surfaces

2012 ◽  
Author(s):  
Xiaodong Liu
Keyword(s):  
Energy Minimization ◽  
Control Points ◽  
Variational Technique ◽  
High Quality ◽  
B Spline ◽  
Cad Systems ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Key Issues ◽  
Complex Constraints

Using one single trimmed B-Spline surface to fill an n-sided hole is a much desired operation in CAD, but few papers have addressed this issue. The paper presents the method of using trimmed B-Spline surfaces to fill n-sided holes based on energy minimization or variational technique. The method is efficient and robust, and takes less than one second to fill n-sided holes with high quality B-Spline surfaces under complex constraints. As the foundation of filling n-sided holes, some key issues on variational B-Spline technique are also discussed. The variational technique discussed is significantly much more efficient and powerful than previous research, and the result is very accurate to satisfy CAD systems’ high-precision requirements. We demonstrate that, without any pre-calculation, the discussed technique is efficient enough to solve a B-Spline surface with up to 20,000 control points in real time while satisfying an arbitrary combination of point and curve constraints.

B2: An Interactive Quality Visualization and Improvement Technique for B-Spline Surfaces

1998 ◽  
Author(s):  
Yifan Chen ◽  
Klaus-Peter Beier
Keyword(s):  
Surface Quality ◽  
Control Points ◽  
Small Surface ◽  
B Spline ◽  
Original Surface ◽  
Interactive Technique ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Surface Irregularities

Abstract A new interactive technique for B-spline surface quality visualization and improvement, called the B2 method, is presented. This method interpolates the control points of a given B-spline surface using a second B-spline surface. If small irregularities exist in the control points of the original surface, they will be magnified through the second B-spline and demonstrated as large distortions in its control points. This facilitates the detection of small surface irregularities. Subsequently, the surface may be improved through direct and interactive adjustment of the second B-spline’s control polyhedron.

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