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Rational offset approximation of rational Bézier curves

Min Cheng, Guo-jin Wang
2006 Journal of Zhejiang University: Science A  
The problem of parametric speed approximation of a rational curve is raised in this paper. Offset curves are widely used in various applications.  ...  Based on the Jacobi polynomial approximation theory with endpoints interpolation, an algebraic rational approximation algorithm of offset curve, which preserves the direction of normal, is presented.  ...  Based on the Jacobi polynomials approximation theory, approximation to the parametric speed of rational Bézier curve is given in this paper.  ... 
doi:10.1631/jzus.2006.a1561 fatcat:ttcuyiqqd5cvbb63pabe3ydk3u


1996 International journal of shape modeling  
Since parametric rational curves, except for straight lines, cannot be parametrized by arc length, we developed a numerical method of approximating the arc length parametrization function.  ...  In this way it was possible to obtain a good parametrization of a NURBS curve with respect to arc length. Numerical results show a good behaviour of the proposed method on several test curves.  ...  to obtain a NURBS curve, we are led to determine a (t) approximation by means of a rational function (t).  ... 
doi:10.1142/s0218654396000075 fatcat:gobceqiigneu7oualx6dgmac5i

Rational Hausdorff divisors: A new approach to the approximate parametrization of curves

Sonia L. Rueda, Juana Sendra, J. Rafael Sendra
2014 Journal of Computational and Applied Mathematics  
Approximate parametrization problem for plane curves: Given the implicit equation of a non-rational real plane curve C and a tolerance e > 0, decide whether there exists a rational real plane curve C at  ...  Approximate parametrization problem The approximate parametrization problem can be stated as follows (we state it for plane curves, but it can be similarly stated for space curves, see [25], and for surfaces  ...  ). • Application to the approximate parametrization problem: Hausdorff curves Given a non-rational irreducible curve, the approximate parametrization problem consists in providing a rational curve being  ... 
doi:10.1016/ fatcat:xgb2itfohrf5bm5htvknvamtzi

NURBS approximation of surface/surface intersection curves

Chandrajit L. Bajaj, Guoliang Xu
1994 Advances in Computational Mathematics  
The algebraic surfaces could be either in implicit or rational parametric form.  ...  We use iI. combination of both symbolic and numerical techniques to construct a degree bounded Ci:-continuous, rational B-spline f-approximations oheal algebraic surface-surface intersection curves.  ...  Rational B-spline Approximation of Parametric Space Curves 2 The 2 Problem and the outline of the Algorithm The Rational Approximation Problem Given a real intersection space curve se which is either  ... 
doi:10.1007/bf02519033 fatcat:7a25bxucpvdtxe575tcyjvn3ie

Page 4971 of Mathematical Reviews Vol. , Issue 96h [page]

1996 Mathematical Reviews  
Based on the dual form, we develop geomet- ric algorithms for approximating a given curve with a G? piecewise rational PH curve.  ...  The spline approximation is both C' and G?, also optimal. The approximation method is very efficient as it is based on local Hermite interpolation and subdivision.  ... 

Piecewise Rational Parametrizations of Canal Surfaces [chapter]

Heidi E. I. Dahl
2014 Lecture Notes in Computer Science  
We also describe an algorithm for minimal bi-degree rational parametrizations of patches on canal surfaces, and show how this can be used to parametrize piecewise rational corner and edge blends.  ...  Canal surfaces, as envelopes of one-parameter families of spheres, correspond to curves in Minkowski space.  ...  There are non-rational curves in R 3,1 whose canal surfaces can be parametrized rationally.  ... 
doi:10.1007/978-3-642-54382-1_6 fatcat:oorxlgy3dzbh5ixhui3mu7l64m

Robust display of arbitrary rational parametric surfaces

Chandrajit L. Bajaj, Andrew V. Royappa, Joe D. Warren
1992 Curves and Surfaces in Computer Vision and Graphics III  
A well-known strength of the parametric representation of a curve or surface is the ease by which a piecewiselinear approximation is generated.  ...  Such surfaces have pole curves in their domain, where the denominators of the parameter functions vanish, domain base points that correspond to entire curves on the surface, and other features that cause  ...  approximation of the parametric.  ... 
doi:10.1117/12.131734 fatcat:oj5wjzyk4ffxxdbb5njjgf3d5m

Quadratic approximation to plane parametric curves and its application in approximate implicitization

Ming Li, Xiao-Shan Gao, Shang-Ching Chou
2006 The Visual Computer  
This paper applies rational quadratic B-spline curves to give a global C 1 continuous approximation to a large class of plane parametric curves including rational parametric curves.  ...  Expressing complex curves with simple parametric curve segments is widely used in computer graphics, CAD and so on.  ...  Sederberg et al. considered monoid curves and surfaces to find an approxi- mate implicit equation and an approximate inversion map of a plane rational parametric curve or a rational parametric surface  ... 
doi:10.1007/s00371-006-0075-6 fatcat:dk2zj6vogfggjljhbc4o64jtrq

C 1 Spline Implicitization of Planar Curves [chapter]

Mohamed Shalaby, Bert Jüttler, Josef Schicho
2004 Lecture Notes in Computer Science  
We present a new method for constructing a low degree C 1 implicit spline representation of a given parametric planar curve.  ...  To ensure the low degree condition, quadratic B-splines are used to approximate the given curve via orthogonal projection in Sobolev spaces.  ...  equation and approximate inversion map of a planar rational parametric curve or a rational parametric surface.  ... 
doi:10.1007/978-3-540-24616-9_10 fatcat:4u3yttuzczfllauw7panubicby

An approximate algorithm to parametrize algebraic curves

S. L. Rueda, J. Sendra, J. R. Sendra
2015 ACM Communications in Computer Algebra  
We provide approximate algorithms to solve the approximate parametrization problem for planar and spacial real algebraic curves.  ...  The solution we give to this problem is a global one, under the requirement that the Hausdorff distance between input and output curves is finite.  ...  In particular, we provide algorithms to solve the approximate parametrization problem for curves: Given the implicit equation of a non-rational real curve C and a tolerance ϵ > 0, decide whether there  ... 
doi:10.1145/2815111.2815130 fatcat:4nqvmi4runbith6vf3gqadsqfu

Page 998 of Mathematical Reviews Vol. , Issue 93b [page]

1993 Mathematical Reviews  
We present a simple algorithm to compute the proper parametrization of a polynomial curve and reduce the problem of detecting cusps in a rational curve to that of a polynomial curve.  ...  Zuhe Shen (PRC-NAN) 93b:65024 65D15 41A20 41A50 Varga, Richard S. (1-KNTS-CM); Carpenter, Amos J. (1-BUTL-CS) Some numerical results on best uniform rational approximation of x* on [0, 1]. Numer.  ... 

Approximate Implicitization Using Monoid Curves and Surfaces

Thomas W. Sederberg, Jianmin Zheng, Kris Klimaszewski, Tor Dokken
1999 Graphical Models and Image Processing  
This paper presents an approach to nding an approximate implicit equation and an approximate inversion map of a planar rational parametric curve or a rational parametric surface.  ...  High accuracy of the approximation is achieved with a relatively small numberof low-degree curve segments or surface patches.  ...  Furthermore, since Ps; t; u lies on the curve, it satis es the implicit a rational parametric curve of degree n.  ... 
doi:10.1006/gmip.1999.0497 fatcat:zyrvkpv4ine4nfqbgkt53y3n7y

Page 4236 of Mathematical Reviews Vol. , Issue 95g [page]

1995 Mathematical Reviews  
(F-LILL-NA; Villeneuve d’Ascq) Characterizations of the set of rational parametric curves with rational offsets.  ...  Summary: “The paper presents a very powerful construction for rational curves and surfaces on quadrics.  ... 

Page 2004 of Mathematical Reviews Vol. , Issue 99c [page]

1991 Mathematical Reviews  
One way to describe rational curves/surfaces with rational off- sets is through Laguerre geometry.  ...  Among other results, they prove a theorem on rational parametrizations for envelopes of natural quadrics which yields algorithms for evaluation of rational parametrizations of sur- faces such as canal  ... 

Page 6244 of Mathematical Reviews Vol. , Issue 2003h [page]

2003 Mathematical Reviews  
Summary: “Universal parametrizations make it possible to find all rational parametrizations and all rational curves on an implicit surface.  ...  The rational curves and patches can be described with optimal degree as images under such a universal parametrization. Hence, the rational curves on the surface can be classified in a systematic way.  ... 
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