Smooth B-Spline Curves Extension with Ordered Points Constraint

2011 ◽  
Vol 311-313 ◽  
pp. 1439-1445 ◽  
Author(s):  
Jin Xu
Keyword(s):  
Experimental Results ◽  
Maximum Deviation ◽  
Curve Segment ◽  
B Spline ◽  
Original Curve ◽  
Spline Curves ◽  
Minimum Number ◽  
The Given ◽  
Curve Extension ◽  
Extension Process

An algorithm for extending B-spline curves with a sequence of ordered points constraint is presented based on the curve unclamping algorithm. The ordered points are divided into two categories: interpolation points and approximation points. The number of interpolation points increases gradually during the curve extension process. The most important feature of this algorithm is the ability to optimize the knots of the extended curve segment according to the ordered points. Thus, with minimum number of interpolation points, the maximum deviation of the extended curve segment from the ordered points is less than the given tolerance. The extended curve segment connects to the original curve with maximum continuity intrinsically. Several experimental results have shown the validity and applicability of the proposed algorithm.

Parametric Generation of Planing Hulls with NURBS Surfaces

2013 ◽  
Vol 57 (04) ◽  
pp. 241-261
Author(s):  
Francisco L. Perez-Arribas ◽  
Erno Peter-Cosma
Keyword(s):  
Maximum Deviation ◽  
Geometric Features ◽  
Surface Geometry ◽  
Orthogonal Projections ◽  
Parametric Generation ◽  
B Spline ◽  
Spline Curves ◽  
Hull Design ◽  
Data Points ◽  
Control Curves

This article presents a mathematical method for producing hard-chine ship hulls based on a set of numerical parameters that are directly related to the geometric features of the hull and uniquely define a hull form for this type of ship. The term planing hull is used generically to describe the majority of hard-chine boats being built today. This article is focused on unstepped, single-chine hulls. B-spline curves and surfaces were combined with constraints on the significant ship curves to produce the final hull design. The hard-chine hull geometry was modeled by decomposing the surface geometry into boundary curves, which were defined by design constraints or parameters. In planing hull design, these control curves are the center, chine, and sheer lines as well as their geometric features including position, slope, and, in the case of the chine, enclosed area and centroid. These geometric parameters have physical, hydrodynamic, and stability implications from the design point of view. The proposed method uses two-dimensional orthogonal projections of the control curves and then produces three-dimensional (3-D) definitions using B-spline fitting of the 3-D data points. The fitting considers maximum deviation from the curve to the data points and is based on an original selection of the parameterization. A net of B-spline curves (stations) is then created to match the previously defined 3-D boundaries. A final set of lofting surfaces of the previous B-spline curves produces the hull surface.

scholarly journals Inverse-Problem-Based Accuracy Control for Arbitrary-Resolution Fairing of Quasiuniform Cubic B-Spline Curves

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Xiaogang Ji ◽  
Jie Xue ◽  
Yan Yang ◽  
Xueming He
Keyword(s):  
Inverse Problem ◽  
Approximation Algorithm ◽  
Linear Hypothesis ◽  
Accuracy Control ◽  
B Spline ◽  
Accuracy Requirement ◽  
Dyadic Wavelet ◽  
Spline Curves ◽  
Minimum Number ◽  
Fairing Process

In the process of curves and surfaces fairing with multiresolution analysis, fairing accuracy will be determined by final fairing scale. On the basis of Dyadic wavelet fairing algorithm (DWFA), arbitrary resolution wavelet fairing algorithm (ARWFA), and corresponding software, accuracy control of multiresolution fairing was studied for the uncertainty of fairing scale. Firstly, using the idea of inverse problem for reference, linear hypothesis was adopted to predict the corresponding wavelet scale for any given fairing error. Although linear hypothesis has error, it can be eliminated by multiple iterations. So faired curves can be determined by a minimum number of control vertexes and have the best faring effect under the requirement of accuracy. Secondly, in consideration of efficiency loss caused by iterative algorithm, inverse calculation of fairing scale was presented based on the least squares fitting. With the increase of order of curves, inverse calculation accuracy becomes higher and higher. Verification results show that inverse calculation scale can meet the accuracy requirement when fitting curve is sextic. In the whole fairing process, because there is no approximation algorithm such as interpolation and approximation, faired curves can be reconstructed again exactly. This algorithm meets the idea and essence of wavelet analysis well.

T-B Spline Curves with a Shape Parameter

2012 ◽  
Vol 241-244 ◽  
pp. 2144-2148
Author(s):  
Li Juan Chen ◽  
Ming Zhu Li
Keyword(s):  
Shape Parameter ◽  
Simple Structure ◽  
Control Points ◽  
Curve Segment ◽  
Closed Curves ◽  
B Spline ◽  
Spline Curve ◽  
Spline Curves

A T-B spline curves with a shape parameter λ is presented in this paper, which has simple structure and can be used to design curves. Analogous to the four B-spline curves, each curve segment is generated by five consecutive control points. For equidistant knots, the curves are C^2 continuous, but when the shape parameter λ equals to 0 , the curves are C^3 continuous. Moreover, this spline curve can be used to construct open and closed curves and can express ellipses conveniently.

scholarly journals Approximating curve by a single segment of B-Spline or Bézier curve directly in CAD environment

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mariusz Sobolak ◽  
Piotr Połowniak ◽  
Adam Marciniec ◽  
Patrycja Ewa Jagiełowicz
Keyword(s):  
Simulated Annealing ◽  
Optimization Methods ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Points ◽  
Curve Segment ◽  
B Spline ◽  
Single Segment ◽  
Original Procedure ◽  
The Given

AbstractThe paper presents the method of approximating curves with a single segment of the B-Spline and Bézier curves. The method for determining a single curve segment using the optimization methods in the CATIA environment is shown. The algorithms of simulated annealing and design of experiment are used for optimization. For the same purpose, a new original procedure for determining the distance between the given curves using explicit parameters in the CATIA environment was also used. This approximation of the cyclic curves results in the curve oscillation as shown in the examples. The results show that the approximation method with Bézier curve using control points as “free” points can be applied to obtain the best results of approximation.

Matching Minimum Strain Energy Curves to B-Spline Curves for Thick Layer Manufacturing

2002 ◽  
Author(s):  
Adrie Kooijman ◽  
Joris S. M. Vergeest
Keyword(s):  
Strain Energy ◽  
Cutting Tool ◽  
Thick Layer ◽  
B Spline ◽  
Spline Curves ◽  
Layer Manufacturing ◽  
Point Density ◽  
Key Issues ◽  
Minimum Strain Energy ◽  
The Given

One of the key issues of thick layer manufacturing is matching the shape of the flexible cutting blade to the local surface curvature of the model to be created. In this paper we explore a method to find the best matching minimum strain energy (MSE) curve for a given B-spline curve. For this purpose we developed software to a) generate a dataset containing MSE curves for a range of settings of the cutting tool and b) find the best matching curve from this MSE dataset to the given target curve. Both the MSE and the target curves are represented as point sets, the target curves having a considerable higher point density than the curves in the MSE dataset. The best matching MSE curve is defined as the curve with the minimum directed Hausdorff distance to the target curve. It is found that despite the relative low density of the dataset, for several practical domains of target curve shape, a satisfying match can be found. Numerical results concerning the matching accuracy are presented.

Cubic TP B-Spline Curves with a Shape Parameter

2013 ◽  
Vol 11 ◽  
pp. 59-72 ◽  
Author(s):  
Mridula Dube ◽  
Reenu Sharma
Keyword(s):  
Tensor Product ◽  
Shape Parameter ◽  
Control Points ◽  
Shape Parameters ◽  
Special Shape ◽  
B Spline ◽  
Control Polygon ◽  
Spline Curves ◽  
Trigonometric Spline ◽  
The Given

In this paper a new kind of splines, called cubic trigonometric polynomial B-spline (cubic TP B-spline) curves with a shape parameter, are constructed over the space spanned by As each piece of the curve is generated by three consecutive control points, they posses many properties of the quadratic B-spline curves. These trigonometric curves with a non-uniform knot vector are C1 and G2 continuous. They are C2 continuous when choosing special shape parameter for non-uniform knot vector. These curves are closer to the control polygon than the quadratic B-spline curves when choosing special shape parameters. With the increase of the shape parameter, the trigonometric spline curves approximate to the control polygon. The given curves posses many properties of the quadratic B-spline curves. The generation of tensor product surfaces by these new splines is straightforward.

PIECEWISE QUARTIC TRIGONOMETRIC POLYNOMIAL B-SPLINE CURVES WITH TWO SHAPE PARAMETERS

2012 ◽  
Vol 12 (04) ◽  
pp. 1250028
Author(s):  
MRIDULA DUBE ◽  
REENU SHARMA
Keyword(s):  
Trigonometric Polynomial ◽  
Shape Parameters ◽  
B Spline ◽  
Spline Curve ◽  
Control Polygon ◽  
Spline Curves ◽  
Spline Surfaces ◽  
Polynomial Curve ◽  
Trigonometric Spline ◽  
The Given

Analogous to the quartic B-splines curve, a piecewise quartic trigonometric polynomial B-spline curve with two shape parameters is presented in this paper. Each curve segment is generated by three consecutive control points. The given curve posses many properties of the B-spline curve. These curves are closer to the control polygon than the different other curves considered in this paper, for different values of shape parameters for each curve. With the increase of the value of shape parameters, the curve approach to the control polygon. For nonuniform and uniform knot vector the given curves have C0, G3; C1, G3; C1, G7; and C3 continuity for different choice of shape parameters. A quartic trigonometric Bézier curves are also introduced as a special case of the given trigonometric spline curves. A comparison of quartic trigonometric polynomial curve is made with different other curves. In the last, quartic trigonometric spline surfaces with two shape parameters are constructed. They have most properties of the corresponding curves.

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