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Ron Goldman - One of the best experts on this subject based on the ideXlab platform.
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representing rational curve segments and surface patches using semi algebraic sets
Computer Aided Geometric Design, 2019Co-Authors: Li-yong Shen, Ron Goldman, Sonia Perezdiaz, Yifei FengAbstract:Abstract We provide a framework for representing segments of rational planar curves or patches of rational tensor product surfaces with no singularities using semi-algebraic sets. Given a rational planar curve segment or a rational tensor product surface patch with no singularities, we find the Implicit Equation of the corresponding unbounded curve or surface and then construct an algebraic box defined by some additional Equations and inequalities associated to the Implicit Equation. This algebraic box is proved to include only the given curve segment or surface patch without any extraneous parts of the unbounded curve or surface. We also explain why it is difficult to construct such an algebraic box if the curve segment or surface patch includes some singular points such as self-intersections. In this case, we show how to isolate a neighborhood of these special points from the corresponding curve segment or surface patch and to represent these special points with small curve segments or surface patches. This framework allows us to dispense with expensive approximation methods such as voxels for representing surface patches.
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Detecting When an Implicit Equation or a Rational Parametrization Defines a Conical or Cylindrical Surface, or a Surface of Revolution
IEEE Transactions on Visualization and Computer Graphics, 2017Co-Authors: Juan Gerardo Alcázar, Ron GoldmanAbstract:Given an Implicit polynomial Equation or a rational parametrization, we develop algorithms to determine whether the set of real and complex points defined by the Equation, i.e., the surface defined by the Equation, in the sense of Algebraic Geometry, is a cylindrical surface, a conical surface, or a surface of revolution. The algorithms are directly applicable to, and formulated in terms of, the Implicit Equation or the rational parametrization. When the surface is cylindrical, we show how to compute the direction of its rulings; when the surface is conical, we show how to compute its vertex; and when the surface is a surface of revolution, we show how to compute its axis of rotation directly from the defining Equations.
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Two additional advantages of complex µ-bases for non-ruled real quadric surfaces
Computer Aided Geometric Design, 2016Co-Authors: Ron Goldman, Xuhui WangAbstract:Real µ-bases for non-ruled real quadric surfaces have two potential drawbacks. First, the resultant of the three moving planes corresponding to a real µ-basis represents the Implicit Equation of the quadric surface, but in some cases contains a linear extraneous factor. Second, even when this resultant contains no extraneous factor, this resultant generates only the affine version of the Implicit Equation. In contrast, the resultant of the two moving planes corresponding to a complex µ-basis always generates the homogeneous version of the Implicit Equation for the quadric surface with no extraneous factors. We illustrate these phenomena here with three examples.
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Implicitization Matrices in the Style of Sylvester with the Order of Bezout
2000Co-Authors: Eng-wee Chionh, Ming Zhang, Ron GoldmanAbstract:Abstract : Resultants are the standard tool used to compute the Implicit Equation of a rational curve or surface. Here we present a new way to compute the Implicit Equation of a rational curve by taking the determinant of a matrix having the style of the Sylvester resultant but the size of the Bezout resultant. Thus the new method has the advantages of both resultant schemes, representing the Implicit Equation as the determinant of a matrix with simple linear entries and lots of zeros just like the Sylvester resultant, but with the same small size as the Bezout resultant.
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Implicitizing rational curves by the method of moving algebraic curves
Journal of Symbolic Computation, 1997Co-Authors: Thomas W Sederberg, Ron Goldman, Hang DuAbstract:Abstract A function F(x,y,t) that assigns to each parameter t an algebraic curve F(x,y,t)=0 is called a moving curve. A moving curve F(x,y,t) is said to follow a rational curve x=x(t)/w(t) , y=y(t)/w(t) if F(x(t)/w(t), y(t)/w(t),t) is identically zero. A new technique for finding the Implicit Equation of a rational curve based on the notion of moving conics that follow the curve is investigated. For rational curves of degree 2 n with no base points the method of moving conics generates the Implicit Equation as the determinant of an n × n matrix, where each entry is a quadratic polynomial in x and y , whereas standard resultant methods generate the Implicit Equation as the determinant of a 2 n × 2 n matrix where each entry is a linear polynomial in x and y . Thus Implicitization using moving conics yields more compact representations for the Implicit Equation than standard resultant techniques, and these compressed expressions may lead to faster evaluation algorithms. Moreover whereas resultants fail in the presence of base points, the method of moving conics actually simplifies, because when base points are present some of the moving conics reduce to moving lines.
Eng-wee Chionh - One of the best experts on this subject based on the ideXlab platform.
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Pacific Conference on Computer Graphics and Applications - Implicitizing Bi-Cubic Toric Surfaces by Dixon -Resultant Quotients
2003Co-Authors: Mao-ching Foo, Eng-wee ChionhAbstract:Toric surface patches have two significant geometric properties: they are multi-sided and they are generalizations of both the triangular and rectangular Bezier surface patches. They also have a very nice algebraic property: their Implicit Equations are closely related to the Dixon determinant. In particular, for bi-cubic toric patches without base points, their Implicit Equation can always be obtained very conveniently using the recently discovered Dixon quotients. In this paper, we explain the relevance of monomial corner cutting to toric patches, and how this leads to their efficient Implicitization by the Dixon quotient. Many examples are given to illustrate the sImplicity and power of this approach.
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Implicitization Matrices in the Style of Sylvester with the Order of Bezout
2000Co-Authors: Eng-wee Chionh, Ming Zhang, Ron GoldmanAbstract:Abstract : Resultants are the standard tool used to compute the Implicit Equation of a rational curve or surface. Here we present a new way to compute the Implicit Equation of a rational curve by taking the determinant of a matrix having the style of the Sylvester resultant but the size of the Bezout resultant. Thus the new method has the advantages of both resultant schemes, representing the Implicit Equation as the determinant of a matrix with simple linear entries and lots of zeros just like the Sylvester resultant, but with the same small size as the Bezout resultant.
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On the existence and the coefficients of the Implicit Equation of rational surfaces
CVGIP: Graphical Models and Image Processing, 1994Co-Authors: Eng-wee Chionh, Ron GoldmanAbstract:Abstract The existence of the Implicit Equation of rational surfaces can be proved by three techniques: elimination theory, undetermined coefficients, and the theory of field extensions. The methods of elimination theory and undetermined coefficients also reveal that the Implicit Equation can be written with coefficients from the coefficient field of the parametric polynomials. All three techniques can be implemented as Implicitization algorithms. For each method, the theoretical limitations of the proof and the practical advantages and disadvantages of the algorithm are discussed. Our results are important for two reasons. First, we caution that elimination theory cannot be generalized in a straightforward manner from rational plane curves to rational surfaces to show the existence of the Implicit Equation; thus other rigorous methods are necessary to bypass the vanishing of the resultant in the presence of base points. Second, as an immediate consequence of the coefficient relationship, we see that the Implicit representation involves only rational (or real) coefficients if a parametric representation involves only rational (or real) coefficients. The existence of the Implicit Equation means every rational surface is a subset of an irreducible algebraic surface. The subset relation can be proper and this may cause problems in certain applications in computer aided geometric design. This anomaly is illustrated by an example.
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Using multivariate resultants to find the Implicit Equation of a rational surface
The Visual Computer, 1992Co-Authors: Eng-wee Chionh, Ronald N. GoldmanAbstract:Given a parametrization of a rational surface, the absence of base points is shown to be a necessary and sufficient condition for the auxiliary resultant to be a power of the Implicit polynomial. The method of resultants also reveals other important properties of rational surface representations, including the coefficients of the Implicit Equation, the relationship between the Implicit and parametric degrees, the degree of each coordinate variable of the Implicit Equation, and the number of correspondence of the parametrization.
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Implitcitizing bi-cubi toric surfaces by dixon alpha-resultant quotients
11th Pacific Conference onComputer Graphics and Applications 2003. Proceedings., 1Co-Authors: Mao-ching Foo, Eng-wee ChionhAbstract:Toric surface patches have two significant geometric properties: they are multi-sided and they are generalizations of both the triangular and rectangular Bezier surface patches. They also have a very nice algebraic property: their Implicit Equations are closely related to the Dixon determinant. In particular, for bi-cubic toric patches without base points, their Implicit Equation can always be obtained very conveniently using the recently discovered Dixon quotients. In this paper, we explain the relevance of monomial corner cutting to toric patches, and how this leads to their efficient Implicitization by the Dixon quotient. Many examples are given to illustrate the sImplicity and power of this approach.
A Papachristodoulou - One of the best experts on this subject based on the ideXlab platform.
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Nonlinear Static State Feedback for Saturated Linear Plants via a Polynomial Approach
IEEE Transactions on Automatic Control, 2017Co-Authors: Giorgio Valmorbida, Luca Zaccarian, Sophie Tarbouriech, Isabelle Queinnec, A PapachristodoulouAbstract:The paper revisits the local exponential stabilization and global asymptotic stabilization problems of saturated linear systems using nonlinear control laws. The proposed nonlinear control law has rational dependence on a parameter σ, which is computed by solving an Implicit Equation depending on the state. Constructive solutions are obtained, based on a sum-of-squares formulation of the proposed conditions.
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A polynomial approach to nonlinear state feedback stabilization of saturated linear systems
2014Co-Authors: Giorgio Valmorbida, Luca Zaccarian, Sophie Tarbouriech, Isabelle Queinnec, A PapachristodoulouAbstract:The paper revisits the local exponential stabilization and global asymptotic stabilization problems of saturated linear systems using nonlinear control laws. The proposed approach takes advantage of a rational formulation of the control action in terms of a parameter σ, which is the solution to an Implicit Equation depending on the state, scheduling the control law. Constructive solutions are then proposed by considering sum-of-squares formulation of the conditions, when imposing a polynomial dependence on parameter σ.
Juan Gerardo Alcázar - One of the best experts on this subject based on the ideXlab platform.
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Detecting When an Implicit Equation or a Rational Parametrization Defines a Conical or Cylindrical Surface, or a Surface of Revolution
IEEE Transactions on Visualization and Computer Graphics, 2017Co-Authors: Juan Gerardo Alcázar, Ron GoldmanAbstract:Given an Implicit polynomial Equation or a rational parametrization, we develop algorithms to determine whether the set of real and complex points defined by the Equation, i.e., the surface defined by the Equation, in the sense of Algebraic Geometry, is a cylindrical surface, a conical surface, or a surface of revolution. The algorithms are directly applicable to, and formulated in terms of, the Implicit Equation or the rational parametrization. When the surface is cylindrical, we show how to compute the direction of its rulings; when the surface is conical, we show how to compute its vertex; and when the surface is a surface of revolution, we show how to compute its axis of rotation directly from the defining Equations.
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a delineability based method for computing critical sets of algebraic surfaces
Journal of Symbolic Computation, 2007Co-Authors: Juan Gerardo Alcázar, Josef Schicho, J R SendraAbstract:In this paper, we address the problem of determining a real finite set of z-values where the topology type of the level curves of a (maybe singular) algebraic surface may change. We use as a fundamental and crucial tool McCallum's theorem on analytic delineability of polynomials (see [McCallum, S., 1998. An improved projection operation for cylindrical algebraic decomposition. In: Caviness, B.F., Johnson, J.R. (Eds.), Quantifier Elimination and Cylindrical Algebraic Decomposition. Springer Verlag, pp. 242-268]). Our results allow to algorithmically compute this finite set by analyzing the real roots of a univariate polynomial; namely, the double discriminant of the Implicit Equation of the surface. As a consequence, an application to offsets is shown.
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Computing the form of highest degree of the Implicit Equation of a rational surface
Advances in Applied Mathematics, 1Co-Authors: Juan Gerardo Alcázar, Sonia Pérez-díazAbstract:Abstract A method is presented for computing the form of highest degree of the Implicit Equation of a rational surface, defined by means of a rational parametrization. Determining the form of highest degree is useful to study the asymptotic behavior of the surface, to perform surface recognition, or to study symmetries of surfaces, among other applications. The method is efficient, and works generally better than known algorithms for Implicitizing the whole surface, in the absence of base points blowing up to a curve at infinity. Possibilities to compute the form of highest degree of the Implicit Equation under the presence of such base points are also discussed. We provide timings to compare our method with known methods for computing the whole Implicit Equation of the surface, both in absence and in presence of base points blowing up to a curve at infinity.
Thomas W Sederberg - One of the best experts on this subject based on the ideXlab platform.
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A new Implicit representation of a planar rational curve with high order singularity
Computer Aided Geometric Design, 2002Co-Authors: Falai Chen, Thomas W SederbergAbstract:This paper presents a new representation for the Implicit Equation of a planar rational curve with high order singularity by using moving curves technique. The new representation is the determinant of a matrix which is less than one half the size of the conventional expression based on the Bezout's resultant. An efficient algorithm to compute the new representation is derived, and comparisons for the computational costs between various resultant based methods are made. The results show that the Implicit representation developed in this paper is not only more compact but also much more efficient to compute the Implicit Equation of the rational curve than previous methods.
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the mu basis of a rational ruled surface
Computer Aided Geometric Design, 2001Co-Authors: Falai Chen, Jianmin Zheng, Thomas W SederbergAbstract:Abstract The mu-basis of a planar rational curve is a polynomial ideal basis comprised of two polynomials that greatly facilitates computing the Implicit Equation of the curve. This paper defines a mu-basis for a rational ruled surface, and presents a simple algorithm for computing the mu-basis. The mu-basis consists of two polynomials p(x,y,z,s) and q(x,y,z,s) that are linear in x,y,z and degree μ and m−μ in s respectively, where m is the degree of the Implicit Equation. The Implicit Equation of the surface is then obtained by merely taking the resultant of p and q with respect to s . This Implicitization algorithm is faster and/or more robust than previous methods.
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A Direct Approach to Computing the µ-basis of Planar Rational Curves
Journal of Symbolic Computation, 2001Co-Authors: Jianmin Zheng, Thomas W SederbergAbstract:This paper presents an O(n2) algorithm, based on Grobner basis techniques, to compute the ? -basis of a degree n planar rational curve. The prior method involved solving a set of linear Equations whose complexity by standard numerical methods was O(n3). The ? -basis is useful in computing the Implicit Equation of a parametric curve and can express the Implicit Equation in the form of a determinant that is smaller than that obtained by taking the resultant of the parametric Equations.
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Implicitizing rational curves by the method of moving algebraic curves
Journal of Symbolic Computation, 1997Co-Authors: Thomas W Sederberg, Ron Goldman, Hang DuAbstract:Abstract A function F(x,y,t) that assigns to each parameter t an algebraic curve F(x,y,t)=0 is called a moving curve. A moving curve F(x,y,t) is said to follow a rational curve x=x(t)/w(t) , y=y(t)/w(t) if F(x(t)/w(t), y(t)/w(t),t) is identically zero. A new technique for finding the Implicit Equation of a rational curve based on the notion of moving conics that follow the curve is investigated. For rational curves of degree 2 n with no base points the method of moving conics generates the Implicit Equation as the determinant of an n × n matrix, where each entry is a quadratic polynomial in x and y , whereas standard resultant methods generate the Implicit Equation as the determinant of a 2 n × 2 n matrix where each entry is a linear polynomial in x and y . Thus Implicitization using moving conics yields more compact representations for the Implicit Equation than standard resultant techniques, and these compressed expressions may lead to faster evaluation algorithms. Moreover whereas resultants fail in the presence of base points, the method of moving conics actually simplifies, because when base points are present some of the moving conics reduce to moving lines.
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SIGGRAPH - Implicitization using moving curves and surfaces
Proceedings of the 22nd annual conference on Computer graphics and interactive techniques - SIGGRAPH '95, 1995Co-Authors: Thomas W Sederberg, Falai ChenAbstract:This paper presents a radically new approach to the century old problem of computing the Implicit Equation of a parametric surface. For surfaces without base points, the new method expresses the Implicit Equation in a determinant which is one fourth the size of the conventional expression based on Dixon’s resultant. If base points do exist, previous Implicitization methods either fail or become much more complicated, while the new method actually simplifies.
