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Implicitization of rational surfaces by means of polynomial interpolation

A Marco
2002 Computer Aided Geometric Design  
A method for finding the implicit equation of a surface given by rational parametric equations is presented.  ...  The method is based on an efficient computation of the resultant by means of classical multivariate polynomial interpolation.  ...  Acknowledgements This research has been partially supported by Research Grant BFM 2000-1253 from the Spanish Ministerio de Ciencia y Tecnología.  ... 
doi:10.1016/s0167-8396(02)00094-8 fatcat:jdqo22mptfhxxk3nnnhk524gt4

Matrix representations by means of interpolation [article]

Ioannis Z. Emiris, Christos Konaxis, Ilias S. Kotsireas, Clement Laroche
2017 arXiv   pre-print
By means of Chow forms, we propose a new, practical, randomized algorithm that always produces correct output but possibly with a non-minimal number of surfaces.  ...  We examine implicit representations of parametric or point cloud models, based on interpolation matrices, which are not sensitive to base points.  ...  Acknowledgments IE, CK, CL belong to team AROMATH, joint between INRIA Sophia-Antipolis and NKUA; their work is partially supported as part of Project ARCADES that has received funding from the European  ... 
arXiv:1606.00789v2 fatcat:ktmr5elqdfb4hicpaplgkkwmxy

Sparse implicitization by interpolation: Geometric computations using matrix representations [article]

Ioannis Emiris, Tatjana Kalinka, Christos Konaxis
2014 arXiv   pre-print
Our approach exploits the sparseness of the given parametric equations and of the implicit polynomial.  ...  Based on the computation of a superset of the implicit support, implicitization of a parametrically given hyper-surface is reduced to computing the nullspace of a numeric matrix.  ...  This research has been co-financed by the European Union (European Social Fund -ESF) and Greek national funds through the Operational Program "Education and Lifelong Learning" of the National Strategic  ... 
arXiv:1411.2846v1 fatcat:zv4grsg6ijd2jci6y2f7ofitbq

Matrix Representations by Means of Interpolation

Ioannis Z. Emiris, Christos Konaxis, Ilias S. Kotsireas, Clément Laroche
2017 Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation - ISSAC '17  
By means of Chow forms, we propose a new, practical, randomized algorithm that always produces correct output but possibly with a non-minimal number of surfaces.  ...  We examine implicit representations of parametric or point cloud models, based on interpolation matrices, which are not sensitive to base points.  ...  Acknowledgments IE, CK, CL belong to team AROMATH, joint between INRIA Sophia-Antipolis and NKUA; their work is partially supported as part of Project ARCADES that has received funding from the European  ... 
doi:10.1145/3087604.3087629 dblp:conf/issac/EmirisKKL17 fatcat:as5w65a6ozagzoeaqe7ddkgzzu

Splines and geometric modeling [chapter]

Chandrajit Bajaj
2004 Handbook of Discrete and Computational Geometry, Second Edition  
Parametric splines are vectors of multivariate polynomial or rational functions while implicit splines are zero contours of multivariate polynomials.  ...  Second, for the same polynomial of degree n, implicit algebraic surfaces have more degrees of freedom n+3n+2n+1 6 ,1 compared with 2n+2n+1,1 degrees of freedom for rational parametric surfaces.  ... 
doi:10.1201/9781420035315.ch53 fatcat:e7evq2b77ba5lj3vxlcxwxxzda

Using polynomial interpolation for implicitizing algebraic curves

A. Marco, J.J. Martı́nez
2001 Computer Aided Geometric Design  
The algorithm is based on an efficient computation of the resultant by means of classical bivariate polynomial interpolation.  ...  A simple algorithm for finding the implicit equation of a rational plane algebraic curve given by its parametric equations is presented.  ...  Acknowledgements This research has been partially supported by Research Grant BFM 2000-1253 from the Spanish Ministerio de Ciencia y Tecnología.  ... 
doi:10.1016/s0167-8396(01)00033-4 fatcat:ug3ao3kx55enrlciqml65e54si

Computing Isophotes on Free-Form Surfaces Based on Support Function Approximation [chapter]

M. Aigner, L. Gonzalez-Vega, B. Jüttler, M. L. Sampoli
2009 Lecture Notes in Computer Science  
These include quadratic polynomial surfaces, ruled surfaces with direction vectors of low degree and polynomial translational surfaces of bidegree (3,2).  ...  The support function of a free-form-surface is closely related to the implicit equation of the dual surface, and the process of computing both the dual surface and the support function can be seen as dual  ...  One may easily express both u and v by rational functions of n. The dual implicit equation takes the form (4) with d = 1. The support function of the surface is an odd rational function.  ... 
doi:10.1007/978-3-642-03596-8_1 fatcat:m7gwnaprbrdq3ppiayycspzlga

Page 2018 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews  
[Martinez, José-Javier] (E-ALCA; Alcala de Henares) Implicitization of rational surfaces by means of polynomial interpolation. (English summary) Comput. Aided Geom. Design 19 (2002), no. 5, 327-344.  ...  Summary: “A method for finding the implicit equation of a surface given by rational parametric equations is presented.  ... 

PHYSICS BASED GEOMETRIC DESIGN

HONG QIN
1996 International journal of shape modeling  
In this paper, we s u r v ey the diversity of shape representations ranging from the primitive polynomial to the sophisticated Non-Uniform Rational B-Spline (NURBS).  ...  are generally awkward when designers are confronted by complex, real-world objects.  ...  Every rational parametric curve/surface can be represented by an implicit algebraic equation, but not vice versa.  ... 
doi:10.1142/s0218654396000099 fatcat:ctnoubb7vvaubpbxuiev5dpmpe

Algebraic surface design with Hermite interpolation

Chanderjit L. Bajaj, Insung Ihm
1992 ACM Transactions on Graphics  
Positional as well as derivative constraints on interpolating implicitly defined algebraic surfaces are translated into a homogeneous linear system, where the unknowns are the coefficients of the polynomial  ...  A heuristic approach to interactive shape control of implicit algebraic surfaces is also given, and various open problems in algebraic surface design are discussed.  ...  Kolb's ray tracer adding capabilities for rendering arbitrary implicit surfaces, and helped us in generating and displaying some of the pictures on HP9000/370SRX.  ... 
doi:10.1145/102377.120081 fatcat:5bgtbvygyffphlaiktn5nxhney

Page 6151 of Mathematical Reviews Vol. , Issue 91K [page]

1991 Mathematical Reviews  
The author presents an algorithm for the interpolation of a “mesh of points” in 3-space by a C! surface.  ...  Summary: “In a square, we prove stability results for the lifting of polynomial traces on the four edges by a global polynomial function, in standard and weighted Sobolev spaces.  ... 

Hermite interpolation by piecewise polynomial surfaces with polynomial area element

Michal Bizzarri, Miroslav Lávička, Zbyněk Šír, Jan Vršek
2017 Computer Aided Geometric Design  
One can find a few indirect algorithms for the interpolations with PN surfaces, where 'indirect' means that the resulting surfaces become rational PN only after a suitable reparameterization -we recall  ...  Clearly, the area element, and thus also the surface area, of polynomial PN surfaces is polynomial but the surface area of rational PN surfaces is again not rational, in general.  ...  Acknowledgments The authors Michal Bizzarri, Miroslav Lávička and Jan Vršek were supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports.  ... 
doi:10.1016/j.cagd.2017.02.003 fatcat:avkd4tykbbcyzh52cenjp7dd6y

Implicit representations of high-codimension varieties

Ioannis Z. Emiris, Christos Konaxis, Clément Laroche
2019 Computer Aided Geometric Design  
Implicitization usually focuses on plane curves and (hyper)surfaces, in other words, varieties of codimension 1.  ...  Our first contribution is a direct generalization of an implicitization method based on interpolation matrices for objects of high codimension given parametrically or as point clouds.  ...  The monomial set is determined quite tightly for parametric models, by means of the sparse resultant of the parametric polynomials, thus exploiting the sparseness of the parametric and implicit polynomials  ... 
doi:10.1016/j.cagd.2019.07.003 fatcat:wpu5m42k4rckdj77kclvlgyqpu

Sparse implicitization by interpolation: Characterizing non-exactness and an application to computing discriminants

Ioannis Z. Emiris, Tatjana Kalinka, Christos Konaxis, Thang Luu Ba
2013 Computer-Aided Design  
We revisit implicitization by interpolation in order to examine its properties in the context of sparse elimination theory.  ...  The approach is applicable to polynomial and rational parameterizations of curves and (hyper)surfaces of any dimension, including the case of parameterizations with base points.  ...  Our main motivation is in changing the representation of geometric (hyper)surfaces given parametrically by polynomial, rational, or trigonometric parameterizations.  ... 
doi:10.1016/j.cad.2012.10.008 fatcat:5vdvsdzwnnf2vnc4efhbs63q4a

Page 1237 of Mathematical Reviews Vol. , Issue 2001B [page]

2001 Mathematical Reviews  
In this way, existing methods for implicit curves and surfaces can always be used to solve CAGD problems independently of the representation chosen for the curves and surfaces involved.  ...  This paper is a new contribution to the construction of G'- piecewise quadric surfaces, first introduced by W. A.  ... 
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