scholarly journals Creases and boundary conditions for subdivision curves

2021 ◽  
Author(s):  
J Kosinka ◽  
M Sabin ◽  
Neil Dodgson
Keyword(s):  
Control Points ◽  
Knot Insertion ◽  
Degree Polynomial ◽  
Arbitrary Degree ◽  
B Splines ◽  
Novel Approach ◽  
Low Degree ◽  
Subdivision Curves ◽  
Subspace Selection ◽  
Subdivision Rules

Our goal is to find subdivision rules at creases in arbitrary degree subdivision for piece-wise polynomial curves, but without introducing new control points e.g. by knot insertion. Crease rules are well understood for low degree (cubic and lower) curves. We compare three main approaches: knot insertion, ghost points, and modifying subdivision rules. While knot insertion and ghost points work for arbitrary degrees for B-splines, these methods introduce unnecessary (ghost) control points. The situation is not so simple in modifying subdivision rules. Based on subdivision and subspace selection matrices, a novel approach to finding boundary and sharp subdivision rules that generalises to any degree is presented. Our approach leads to new higher-degree polynomial subdivision schemes with crease control without introducing new control points. © 2014 The Authors. Published by Elsevier Inc.

scholarly journals Creases and boundary conditions for subdivision curves

2020 ◽  
Author(s):  
J Kosinka ◽  
M Sabin ◽  
Neil Dodgson
Keyword(s):  
Control Points ◽  
Knot Insertion ◽  
Degree Polynomial ◽  
Arbitrary Degree ◽  
B Splines ◽  
Novel Approach ◽  
Low Degree ◽  
Subdivision Curves ◽  
Subspace Selection ◽  
Subdivision Rules

Our goal is to find subdivision rules at creases in arbitrary degree subdivision for piece-wise polynomial curves, but without introducing new control points e.g. by knot insertion. Crease rules are well understood for low degree (cubic and lower) curves. We compare three main approaches: knot insertion, ghost points, and modifying subdivision rules. While knot insertion and ghost points work for arbitrary degrees for B-splines, these methods introduce unnecessary (ghost) control points. The situation is not so simple in modifying subdivision rules. Based on subdivision and subspace selection matrices, a novel approach to finding boundary and sharp subdivision rules that generalises to any degree is presented. Our approach leads to new higher-degree polynomial subdivision schemes with crease control without introducing new control points. © 2014 The Authors. Published by Elsevier Inc.

scholarly journals Creases and boundary conditions for subdivision curves

2020 ◽  
Author(s):  
J Kosinka ◽  
M Sabin ◽  
Neil Dodgson
Keyword(s):  
Control Points ◽  
Knot Insertion ◽  
Degree Polynomial ◽  
Arbitrary Degree ◽  
B Splines ◽  
Novel Approach ◽  
Low Degree ◽  
Subdivision Curves ◽  
Subspace Selection ◽  
Subdivision Rules

Our goal is to find subdivision rules at creases in arbitrary degree subdivision for piece-wise polynomial curves, but without introducing new control points e.g. by knot insertion. Crease rules are well understood for low degree (cubic and lower) curves. We compare three main approaches: knot insertion, ghost points, and modifying subdivision rules. While knot insertion and ghost points work for arbitrary degrees for B-splines, these methods introduce unnecessary (ghost) control points. The situation is not so simple in modifying subdivision rules. Based on subdivision and subspace selection matrices, a novel approach to finding boundary and sharp subdivision rules that generalises to any degree is presented. Our approach leads to new higher-degree polynomial subdivision schemes with crease control without introducing new control points. © 2014 The Authors. Published by Elsevier Inc.

scholarly journals Creases and boundary conditions for subdivision curves

2021 ◽  
Author(s):  
J Kosinka ◽  
M Sabin ◽  
Neil Dodgson
Keyword(s):  
Control Points ◽  
Knot Insertion ◽  
Degree Polynomial ◽  
Arbitrary Degree ◽  
B Splines ◽  
Novel Approach ◽  
Low Degree ◽  
Subdivision Curves ◽  
Subspace Selection ◽  
Subdivision Rules

Our goal is to find subdivision rules at creases in arbitrary degree subdivision for piece-wise polynomial curves, but without introducing new control points e.g. by knot insertion. Crease rules are well understood for low degree (cubic and lower) curves. We compare three main approaches: knot insertion, ghost points, and modifying subdivision rules. While knot insertion and ghost points work for arbitrary degrees for B-splines, these methods introduce unnecessary (ghost) control points. The situation is not so simple in modifying subdivision rules. Based on subdivision and subspace selection matrices, a novel approach to finding boundary and sharp subdivision rules that generalises to any degree is presented. Our approach leads to new higher-degree polynomial subdivision schemes with crease control without introducing new control points. © 2014 The Authors. Published by Elsevier Inc.

BEZIER AND B-SPLINE CURVES WITH KNOTS IN THE COMPLEX PLANE

Fractals ◽  
2011 ◽  
Vol 19 (01) ◽  
pp. 67-86 ◽  
Author(s):  
KONSTANTINOS I. TSIANOS ◽  
RON GOLDMAN
Keyword(s):  
Complex Domain ◽  
Polynomial Algorithms ◽  
Control Points ◽  
Knot Insertion ◽  
Real Numbers ◽  
Koch Curve ◽  
Natural Manner ◽  
B Spline ◽  
B Splines ◽  
Spline Curves

We extend some well known algorithms for planar Bezier and B-spline curves, including the de Casteljau subdivision algorithm for Bezier curves and several standard knot insertion procedures (Boehm's algorithm, the Oslo algorithm, and Schaefer's algorithm) for B-splines, from the real numbers to the complex domain. We then show how to apply these polynomial and piecewise polynomial algorithms in a complex variable to generate many well known fractal shapes such as the Sierpinski gasket, the Koch curve, and the C-curve. Thus these fractals also have Bezier and B-spline representations, albeit in the complex domain. These representations allow us to change the shape of a fractal in a natural manner by adjusting their complex Bezier and B-spline control points. We also construct natural parameterizations for these fractal shapes from their Bezier and B-spline representations.

Non-uniform subdivision for B-splines of arbitrary degree

2009 ◽  
Vol 26 (1) ◽  
pp. 75-81 ◽  
Author(s):  
S. Schaefer ◽  
R. Goldman
Keyword(s):  
Arbitrary Degree ◽  
B Splines

A Dynamics-Based Optimal Trajectory Generation for Controlling an Automated Excavator

2010 ◽  
Vol 224 (10) ◽  
pp. 2109-2119 ◽  
Author(s):  
S Yoo ◽  
C-G Park ◽  
S-H You ◽  
B Lim
Keyword(s):  
Optimal Trajectory ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Linear Constraints ◽  
Control Points ◽  
Weighting Functions ◽  
B Splines ◽  
Finite Dimensional ◽  
Motion Optimization ◽  
Save Energy

This article presents a new methodology to generate optimal trajectories in controlling an automated excavator. By parameterizing all the actuator displacements with B-splines of the same order and with the same number of control points, the coupled actuator limits, associated with the maximum pump flowrate, are described as the finite-dimensional set of linear constraints to the motion optimization problem. Several weighting functions are introduced on the generalized actuator torque so that the solution to each optimization problems contains the physical meaning. Numerical results showing that the generated motions of the excavator are fairly smooth and effectively save energy, which can prevent mechanical wearing and possibly save fuel consumption, are presented. A typical operator's manoeuvre from experiments is referred to bring out the standing features of the optimized motion.

scholarly journals Evaluation of low degree polynomial kernel support vector machines for modelling Pore-water pressure responses

2016 ◽  
Vol 59 ◽  
pp. 04003
Author(s):  
Nuraddeen Muhammad Babangida ◽  
Muhammad Raza Ul Mustafa ◽  
Khamaruzaman Wan Yusuf ◽  
Mohamed Hasnain Isa ◽  
Imran Baig
Keyword(s):  
Support Vector Machines ◽  
Pore Water ◽  
Pore Water Pressure ◽  
Water Pressure ◽  
Polynomial Kernel ◽  
Support Vector ◽  
Degree Polynomial ◽  
Low Degree ◽  
Vector Machines ◽  
Kernel Support Vector Machines

Choosing the optimal number of B-spline control points (Part 2: Approximation of surfaces and applications)

2017 ◽  
Vol 11 (1) ◽  
Author(s):  
Corinna Harmening ◽  
Hans Neuner
Keyword(s):  
Data Analysis ◽  
Laser Scanner ◽  
Point Clouds ◽  
Optimal Number ◽  
Deformation Analysis ◽  
Control Points ◽  
Trial And Error ◽  
B Spline ◽  
B Splines ◽  
Spline Surfaces

AbstractFreeform surfaces like B-splines have proven to be a suitable tool to model laser scanner point clouds and to form the basis for an areal data analysis, for example an areal deformation analysis.A variety of parameters determine the B-spline's appearance, the B-spline's complexity being mostly determined by the number of control points. Usually, this parameter type is chosen by intuitive trial-and-error-procedures.In [The present paper continues these investigations. If necessary, the methods proposed in [The application of those methods to B-spline surfaces reveals the datum problem of those surfaces, meaning that location and number of control points of two B-splines surfaces are only comparable if they are based on the same parameterization. First investigations to solve this problem are presented.

Data Modeling Using NURBs Curves and Modified Genetic Algorithms

2010 ◽  
Author(s):  
Christopher Hammond ◽  
Cameron J. Turner
Keyword(s):  
Genetic Algorithm ◽  
Genetic Algorithms ◽  
Trial Data ◽  
Data Sets ◽  
Control Points ◽  
Nurbs Curve ◽  
B Splines ◽  
Modified Genetic Algorithm ◽  
Nurbs Curves ◽  
2D Data

Non-Uniform Rational B-Splines (NURBS) curves have long been used to model 1D and 2D data because they are efficient to calculate, easy to manipulate, and capable of handling discontinuities and drastic changes in the general topology of the data. However, the user must assist in defining the control points, weights, knots and an order for the curve in order to fit the curve to the data. This paper uses a modified Genetic Algorithm (GA) to choose and manipulate these various parameters to produce a NURBS curve that minimizes the error between the data and the curve and also minimizes the time it takes the algorithm to compute the solution. The algorithm is tested on several 1D trial data sets and the results are explained. Then, several general difficulties for this application of the GA are explained and possible methods for overcoming them are discussed.

Optimum 3D Rapping of CAD Models Using Single NURBS

2014 ◽  
Author(s):  
Mohamed A. El-Komy ◽  
Sayed M. Metwalli
Keyword(s):  
Corner Point ◽  
Basis Functions ◽  
Control Points ◽  
Nurbs Curve ◽  
B Splines ◽  
Full Control ◽  
Solid Models ◽  
Cad Models ◽  
3D Solid ◽  
Number Of Segments

Non-Uniform Rational B-Splines (NURBS) can represent curves and surfaces of any degree. Usually in the same curve, however, the degree is unique. The goal of this work is to identify single and exact corner point of lines represented by cubic or other NURBS. The combination of arcs and lines can then be represented by one NURBS with error not to exceed (10−12). The developed procedure can represent any NURBS curve and surface of any degree with full control on all parameters, control points, weights, knot vectors, and number of segments representing the curve or surface, in addition to, the basis functions examination. The optimization identifies the parameters and geometry to insure any required level of accuracy to represent singular corner solid models to allow a single cubic or other NURBS representing the whole solid. It is concluded that the singular corner point can be identified with cubic NURBS. Applications to several 3D solid CAD models are used to verify such a technique.

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