Intersection of Offset Surfaces of Parametric Patches

1992 ◽  
Author(s):  
Yu Wang
Keyword(s):  
Parametric Surface ◽  
Intersection Curve ◽  
B Spline ◽  
Numerical Examples ◽  
Surface Patches ◽  
Normal Projection ◽  
Offset Surfaces ◽  
Nurbs Surfaces ◽  
Effectiveness And Efficiency ◽  
Offset Surface

Abstract In this paper, we present a new method to calculate intersection curve of offset surfaces of two parametric surface patches. This method is based on the concept of normal projection and the intersection problem is formulated with a set of tensorial differential equations. The intersection is represented in the parametric spaces of the base surfaces and no offset surface approximation is needed. Numerical examples are presented for non-uniform rational b-spline (NURBS) surfaces to illustrate the effectiveness and efficiency of the proposed approach.

Free-Form Plate Modeling Using Offset Surfaces

1988 ◽  
Vol 110 (3) ◽  
pp. 287-294 ◽  
Author(s):  
N. M. Patrikalakis ◽  
P. V. Prakash
Keyword(s):  
Boundary Representation ◽  
Free Form ◽  
Parametric Surfaces ◽  
B Spline ◽  
Spline Surfaces ◽  
Accuracy Criterion ◽  
Offset Surfaces ◽  
Representation Method ◽  
Class Of Functions ◽  
Offset Surface

This paper addresses the representation of plates within the framework of the Boundary Representation method in a Solid Modeling environment. Plates are defined as the volume bounded by a progenitor surface, its offset surface and other, possibly ruled surfaces for the sides. Offset surfaces of polynomial parametric surfaces cannot be represented exactly within the same class of functions describing the progenitor surface. Therefore, if the offset surface is to be represented in the same form as the progenitor surface, approximation is required. A method of approximation relevant to non-uniform rational parametric B-spline surfaces is described. The method employs the properties of the control polyhedron and a recently developed subdivision algorithm to satisfy a certain accuracy criterion. Representative examples are given which illustrate the efficiency and robustness of the proposed method.

scholarly journals Parametric Surface Modelling for Tea Leaf Point Cloud Based on Non-Uniform Rational Basis Spline Technique

Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1304
Author(s):  
Wenchao Wu ◽  
Yongguang Hu ◽  
Yongzong Lu
Keyword(s):  
Point Cloud ◽  
Principal Component ◽  
Point Clouds ◽  
Parametric Surface ◽  
B Spline ◽  
Surface Modelling ◽  
Modelling Method ◽  
3D Architecture ◽  
Tea Leaf ◽  
Fitting Error

Plant leaf 3D architecture changes during growth and shows sensitive response to environmental stresses. In recent years, acquisition and segmentation methods of leaf point cloud developed rapidly, but 3D modelling leaf point clouds has not gained much attention. In this study, a parametric surface modelling method was proposed for accurately fitting tea leaf point cloud. Firstly, principal component analysis was utilized to adjust posture and position of the point cloud. Then, the point cloud was sliced into multiple sections, and some sections were selected to generate a point set to be fitted (PSF). Finally, the PSF was fitted into non-uniform rational B-spline (NURBS) surface. Two methods were developed to generate the ordered PSF and the unordered PSF, respectively. The PSF was firstly fitted as B-spline surface and then was transformed to NURBS form by minimizing fitting error, which was solved by particle swarm optimization (PSO). The fitting error was specified as weighted sum of the root-mean-square error (RMSE) and the maximum value (MV) of Euclidean distances between fitted surface and a subset of the point cloud. The results showed that the proposed modelling method could be used even if the point cloud is largely simplified (RMSE < 1 mm, MV < 2 mm, without performing PSO). Future studies will model wider range of leaves as well as incomplete point cloud.

Computation of Self-Intersections of Offsets of Be´zier Surface Patches

1997 ◽  
Vol 119 (2) ◽  
pp. 275-283 ◽  
Author(s):  
Takashi Maekawa ◽  
Wonjoon Cho ◽  
Nicholas M. Patrikalakis
Keyword(s):  
Differential Geometry ◽  
Distance Function ◽  
Polynomial Equations ◽  
Intersection Curve ◽  
Parameter Domain ◽  
Surface Patches ◽  
Intersection Curves

Self-intersection of offsets of regular Be´zier surface patches due to local differential geometry and global distance function properties is investigated. The problem of computing starting points for tracing self-intersection curves of offsets is formulated in terms of a system of nonlinear polynomial equations and solved robustly by the interval projected polyhedron algorithm. Trivial solutions are excluded by evaluating the normal bounding pyramids of the surface subpatches mapped from the parameter boxes computed by the polynomial solver with a coarse tolerance. A technique to detect and trace self-intersection curve loops in the parameter domain is also discussed. The method has been successfully tested in tracing complex self-intersection curves of offsets of Be´zier surface patches. Examples illustrate the principal features and robustness characteristics of the method.

scholarly journals A Numerical Solution to Fractional Diffusion Equation for Force-Free Case

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
O. Tasbozan ◽  
A. Esen ◽  
N. M. Yagmurlu ◽  
Y. Ucar
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Diffusion Equation ◽  
Exact Analytical Solution ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Spline Functions ◽  
B Spline ◽  
Numerical Examples ◽  
Free Case

A collocation finite element method for solving fractional diffusion equation for force-free case is considered. In this paper, we develop an approximation method based on collocation finite elements by cubic B-spline functions to solve fractional diffusion equation for force-free case formulated with Riemann-Liouville operator. Some numerical examples of interest are provided to show the accuracy of the method. A comparison between exact analytical solution and a numerical one has been made.

scholarly journals B-splines Algorithms for Solving Fredholm Linear Integro-Differential Equations

2004 ◽  
Vol 1 (2) ◽  
pp. 340-346
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equations ◽  
Spline Function ◽  
Cubic Spline ◽  
Second Order ◽  
Third Order ◽  
B Spline ◽  
Numerical Examples ◽  
B Splines ◽  
The Third ◽  
New Procedures

Algorithms using the second order of B -splines [B (x)] and the third order of B -splines [B,3(x)] are derived to solve 1' , 2nd and 3rd linear Fredholm integro-differential equations (F1DEs). These new procedures have all the useful properties of B -spline function and can be used comparatively greater computational ease and efficiency.The results of these algorithms are compared with the cubic spline function.Two numerical examples are given for conciliated the results of this method.

Superimposed RBF and B-spline parametric surface for reverse engineering applications

2019 ◽  
Vol 27 (1) ◽  
pp. 17-35 ◽  
Author(s):  
Ivo Marinić-Kragić ◽  
Stipe Perišić ◽  
Damir Vučina ◽  
Milan Ćurković
Keyword(s):  
Reverse Engineering ◽  
Parametric Surface ◽  
Engineering Applications ◽  
B Spline

Infinitesimal-Area Radiative Analysis Using Parametric Surface Representation, Through NURBS

2000 ◽  
Vol 123 (2) ◽  
pp. 249-256 ◽  
Author(s):  
K. J. Daun ◽  
K. G. T. Hollands
Keyword(s):  
Integral Equation ◽  
Form Factors ◽  
General Purpose ◽  
Transparent Medium ◽  
Parametric Surface ◽  
Surface Representation ◽  
B Spline ◽  
Industry Standard ◽  
Cad Cam ◽  
User Intervention

The use of form factors in the treatment of radiant enclosures requires the radiosity be approximated as uniform over finite areas, and so when higher accuracy is required, an infinitesimal-area analysis should be applied. This paper describes a generic infinitesimal-area formulation suited in principle for any enclosure containing a transparent medium. The surfaces are first represented parametrically, through “non-uniform rational B-spline” (NURBS) functions, the industry standard in CAD-CAM codes. The kernel of the integral equation is obtained without user intervention, using NURBS algorithms, and then the integral equation is solved numerically. The resulting general-purpose code, which proceeds directly from surface specification to solution, is tested on problems taken from the literature.

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