A Condition Number Analysis of a Line-Surface Intersection Algorithm

Elimination theory was originally developed in the 18th and 19th centuries to find solvability criteria for a system of polynomial equations. We present some new resolvent methods that are suitable to solve important problems in computer aided geometric design (CAGD), including implicitization, inversion, intersection, computation of self intersection points, and detection of unfaithful parametrizations of rational curves and surfaces.

In the last years, the field of non-linear Computational Geometry has produced very relevant algorithmic advances, for instance, by adapting to low-degree curved objects techniques dealing with linear primitives. Moreover the introduction of new algorithms and approaches for dealing efficiently with curves and surfaces has shown a considerable impact in fields such as Computer-Aided Design where final users are continuously demanding more efficiency, accuracy, robustness, etc. New approaches for solving geometric problems, coming from Computer-Aided Geometric Design and involving curves and surfaces with a Computational Geometry flavour is the main focus of this special issue. Contributions were asked in areas such as: arrangements of low degree curves and surfaces; Voronoi diagrams and medial axes involving curved ob jects; offset computations and their topology; distance predicates for curves and surfaces; closed formulae for geometric queries involving curves and surfaces; intersection detection for moving curved ob jects; adapting algebraic or geometrical techniques for solving algorithmic problems involving curves and surfaces. This special issue is a direct consequence of the Second International Conference on Mathematical Aspects of Computer and Information Sciences - MACIS 2007 (see http:// www- spiral.lip6.fr/ MACIS2007) held in Paris in December 2007 and where Computational Geometry and Computer Aided Geometric Design were one of the main topics (among other two, Polynomial System Solving and Algorithms and Complexity). Finally six papers (out of eight papers submitted by the March 2009 deadline) were finally accepted. The two papers by C. D'Andrea et al. and I. Z. Emiris et al. deal with the problem of analyzing the Newton polygon of the implicit equation of a rational curve starting from its parameterization: both papers are devoted to study from different viewpoints the properties of the set of monomials appearing in the implicit equation of the considered curve without computing it. The two papers by E. Berberich et al. introduce a new approach to deal with the problem of determining the shape of an arrangement of curves on several classes of parametric surfaces which is then applied to compute, for example, the Minkowski sum of polytopes or the envelope of surfaces. The two papers by D. Lazard and J. Cheng et al. deal with the problem of computing the topology of a semialgebraic set (i.e. a set in $\mathbb{R}^n$ defined by polynomial equations and inequations): the first one shows how to deal with the general case by using the so called Cylindrical Algebraic Decomposition and the second one introduces an extremely fast algorithm computing the topology of a real algebraic plane curve defined implicitely. We would like finally to thank all the reviewers who definitively helped with their work to improve the papers that appear into this special issue.

We compare approaches to the location of the algebraic curve f(x,y) = 0 in a rectangular region of the plane, based on recursive use of conservative estimates of the range of the function over a rectangle. Previous work showed that performing interval arithmetic in the Bernstein basis is more accurate than using the power basis, and that affine arithmetic in the power basis is better than using interval arithmetic in the Bernstein basis. This paper shows that using affine arithmetic with the Bernstein basis gives no advantage over affine arithmetic with the power basis. It also considers the Bernstein coefficient method based on the convex hull property, which has similar performance to affine arithmetic.

Macaulay and Dixon resultant formulations are proposed for parametrized multivariate polynomial systems represented in Bernstein basis. It is proved that the Macaulay resultant for a polynomial system in Bernstein basis vanishes for the total degree case if and only if the either the polynomial system has a common Bernstein-toric root, a common infinite root, or the leading forms of the polynomial system obtained by replacing every variable x i in the original polynomial system by \(\frac{y_i}{1+y_i}\) have a non-trivial common root. For the Dixon resultant formulation, the rank sub-matrix constructions for the original system and the transformed system are shown to be essentially equivalent. Known results about exactness of Dixon resultants of a sub-class of polynomial systems as discussed in Chtcherba and Kapur in Journal of Symbolic Computation (August, 2003) carry over to polynomial systems represented in the Bernstein basis. Furthermore, in certain cases, when the extraneous factor in a projection operator constructed from the Dixon resultant formulation is precisely known, such results also carry over to projection operators of polynomial systems in the Bernstein basis where extraneous factors are precisely known. Applications of these results in the context of geometry theorem proving, implicitization and intersection of surfaces with curves are discussed. While Macaulay matrices become large when polynomials in Bernstein bases are used for problems in these applications, Dixon matrices are roughly of the same size.KeywordsresultantsMacaulay matrixDixon matrixBernstein basis

In this paper the fundamental concept of repeated linear interpolation and its possible applications in computer-aided geometric design, and start considering basic constructive methods for curves and surfaces. We discuss here a repeated linear interpolation method that we commonly find in computer graphics and geometric modelling. Repeated linear Interpolation means to calculate a polynomial by using several points. For a given sequence of points, this means to estimate a curve that passes through every single point. The purpose of this paper is to construct a polynomial of degree less than or equal to n, by using repeated linear Interpolation.

Robust arithmetic for multivariate Bernstein-form polynomials

Computer-aided geometric design is a subject which involves the representation, specification, design, manipulation, display, and analysis of free-form curves and surfaces. This eclectic field draws on techniques and principles from numerical analysis, approximation theory, computer graphics, interactive computer systems, mechanical and geometric design, and many other areas. Nevertheless, computer-aided geometric design is an endeavor with a character, emphasis, and purpose of its own, combining techniques and principles from other fields with unique approaches that are often somewhat enigmatic and confusing to the novice. Many things sound and seem familiar, yet the directions and motivations are distinct to this discipline. This decade is proving to be an era of intense interest in computer-aided geometric design, an activity attracting people from many sectors including computing and manufacturing. As the discipline more firmly establishes valid principles and standard techniques, it will undoubtedly be of increasing importance in computer-aided manufacturing and a wide range of other applications.

Chapter 15 - Algebraic Methods for Computer Aided Geometric Design

Modified Jacobi–Bernstein basis transformation and its application to multi-degree reduction of Bézier curves

Explicit differentiation matrices in various polynomial bases are presented in this work. The idea is to avoid any change of basis in the process of polynomial differentiation. This article concerns both degree-graded polynomial bases such as orthogonal bases, and non-degree-graded polynomial bases including the Lagrange and Bernstein bases.

Computer-Aided Geometric Design (CAGD) is one of the most important fields in Computer Graphics. Usually, CAGD is handled in traditional programming languages, such as Fortran, Pascal or C. By contrast, this paper supports the idea that Symbolic Computation Systems (SCS) should be used instead. To this aim, the paper shows how some mathematical expressions for BEzier curves and surfaces can be easily translated to the Mathematica programming language. Then, they are used to prove symbolically some mathematical properties related to these geometric entities.

Ruled surfaces are a class of surfaces generated by a moving straight line and are used in various design and modeling applications, including kinematics, robotics, and computer-aided geometric design. On the other hand, the inextensibility condition, which preserves the intrinsic geometry of a surface during evolution, is significant for motions of surfaces where stretching is not allowed. Nevertheless, comprehensive characterizations of the evolutions of partner-ruled surfaces generated by the canonical vector pair of a space curve and the preservation of their intrinsic properties under evolutions remain open for investigation. This study aims to provide systematic research on the inextensible evolutions of partner-ruled surfaces simultaneously generated by a pair from the set of Darboux and Frenet vectors of a unit-speed space curve. For this purpose, the intrinsic and extrinsic invariants of the partner-ruled surfaces are derived, and their inextensibility conditions are determined. The structural classifications of the parameter curves of these types of surfaces are explored. The main contributions are the necessary and sufficient conditions on the curvature and torsion of the generator curve, which state the simultaneous inextensibility of these partner surfaces. Additionally, characterizations for the simultaneous developability and minimality of these partner surfaces are provided, based on Gauss and mean curvatures calculated using partial differential equations. This work provides a classification for this family of surfaces, supported by clear visual examples.

We prove that rational data of bounded input length are uniformly distributed (in the classical sense of H. Weyl, in [42]) with respect to the probability distribution of condition numbers of numerical analysis. We deal both with condition numbers of linear algebra and with condition numbers for systems of multivariate polynomial equations. For instance, we prove that for a randomly chosen n\times n rational matrix M of bit length O(n 4 log n) + log w , the condition number k(M) satisfies k(M) ≤ w n 5/2 with probability at least 1-2w -1 . Similar estimates are established for the condition number μ_ norm of M. Shub and S. Smale when applied to systems of multivariate homogeneous polynomial equations of bounded input length. Finally, we apply these techniques to estimate the probability distribution of the precision (number of bits of the denominator) required to write approximate zeros of systems of multivariate polynomial equations of bounded input length.

Splines were originally studied in approximation theory where the focus is on approximating explicit functions of the form y = f(x) or z = f(x,y). These splines were later adopted by mathematicians and computer sicentists for use in computer-aided geometric design (CAGD) where the emphasis was shifted to parametric curves and surfaces. Initially the continuity conditions for splines developed in approximation theory were retained in CAGD, but it was soon realized that the old constraints were unnecessarily restrictive in this new context and that they could be relaxed without losing the essential property of smoothness. Beta-splines were developed to take advantage of this new freedom by introducing shape parameters into the constraint equations. These parameters could then be manipulated by a designer to change the shape of a curve of surface in an intuitively meaningful and useful way. Another seemingly unrelated context in which shape parameters appear is in blending functions constructed from discrete urn models. The purpose of this article is to begin to unify these two independent approaches to shape parameters, and in the process apply the techniques of urn models to gain some insight into the properties of Beta-splines.

In boundary representation (B-rep) reconstruction for computer aided design (CAD) applications, it is still challenging with existing methods to distinguish primitives in the smoothly blended regions reasonably. Thus, intensive manual post-processing is always required for correcting the primitives and their neighboring relationships to obtain a valid B-rep solid, seriously preventing the efficiency. In this paper, we address these challenges by presenting two novel techniques. The first is to robustly extract primitives by iteratively estimating the probability distribution of the noise to eliminate outliers. The second is to present intersection-aware constraints, like tangency and collinearity constraints, to correctly obtain intersections between primitives, which have not been explored in existing methods to our knowledge. Therefore, we can effectively extract primitives, especially those blended smoothly, and obtain high-quality relationships between them. As a result, a valid B-rep model can be constructed without a lot of manual post-processing on topology correction, while not with existing methods. As a benefit, with our constructed B-rep models, their corresponding meshes can be intuitively and conveniently edited, which is quite useful in CAD applications. Experimental results show that our proposed B-rep construction method outperforms both classical and recent learning-based methods in terms of reconstruction efficiency and accuracy.