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Implicitization of rational surfaces by means of polynomial interpolation

2002
*
Computer Aided Geometric Design
*

A method for finding the

doi:10.1016/s0167-8396(02)00094-8
fatcat:jdqo22mptfhxxk3nnnhk524gt4
*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. ...##
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Matrix representations by means of interpolation
[article]

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 ...

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

2014
*
arXiv
*
pre-print

Our approach exploits the sparseness

arXiv:1411.2846v1
fatcat:zv4grsg6ijd2jci6y2f7ofitbq
*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 ...##
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Matrix Representations by Means of Interpolation

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 ...

##
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Splines and geometric modeling
[chapter]

2004
*
Handbook of Discrete and Computational Geometry, Second Edition
*

Parametric splines are vectors

doi:10.1201/9781420035315.ch53
fatcat:e7evq2b77ba5lj3vxlcxwxxzda
*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*. ...##
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Using polynomial interpolation for implicitizing algebraic curves

2001
*
Computer Aided Geometric Design
*

The algorithm is based on an efficient computation

doi:10.1016/s0167-8396(01)00033-4
fatcat:ug3ao3kx55enrlciqml65e54si
*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. ...##
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Computing Isophotes on Free-Form Surfaces Based on Support Function Approximation
[chapter]

2009
*
Lecture Notes in Computer Science
*

These include quadratic

doi:10.1007/978-3-642-03596-8_1
fatcat:m7gwnaprbrdq3ppiayycspzlga
*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. ...##
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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. ...##
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PHYSICS BASED GEOMETRIC DESIGN

1996
*
International journal of shape modeling
*

In this paper, we s u r v ey the diversity

doi:10.1142/s0218654396000099
fatcat:ctnoubb7vvaubpbxuiev5dpmpe
*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. ...##
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Algebraic surface design with Hermite interpolation

1992
*
ACM Transactions on Graphics
*

Positional as well as derivative constraints on

doi:10.1145/102377.120081
fatcat:5bgtbvygyffphlaiktn5nxhney
*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. ...##
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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. ...##
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Hermite interpolation by piecewise polynomial surfaces with polynomial area element

2017
*
Computer Aided Geometric Design
*

One can find a few indirect algorithms for the

doi:10.1016/j.cagd.2017.02.003
fatcat:avkd4tykbbcyzh52cenjp7dd6y
*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. ...##
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Implicit representations of high-codimension varieties

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*...

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

2013
*
Computer-Aided Design
*

We revisit

doi:10.1016/j.cad.2012.10.008
fatcat:5vdvsdzwnnf2vnc4efhbs63q4a
*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. ...##
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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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