Algorithm to determine the intersection curves between bezier surfaces by the solution of multivariable polynomial system and the differential marching method

The determination of the intersection curve between Bézier Surfaces may be seen as the composition of two separated problems: determining initial points and tracing the intersection curve from these points. The Bézier Surface is represented by a parametric function (polynomial with two variables) that maps a point in the tridimensional space from the bidimensional parametric space. In this article, it is proposed an algorithm to determine the initial points of the intersection curve of Bézier Surfaces, based on the solution of polynomial systems with the Projected Polyhedral Method, followed by a method for tracing the intersection curves (Marching Method with differential equations). In order to allow the use of the Projected Polyhedral Method, the equations of the system must be represented in terms of the Bernstein basis, and towards this goal it is proposed a robust and reliable algorithm to exactly transform a multivariable polynomial in terms of power basis to a polynomial written in terms of Bernstein basis .

Differential geometry of intersection curves in R(4) of three implicit surfaces

We present algorithms for computing the differential geometry properties of intersection Curves of three implicit surfaces in R(4), using the implicit function theorem and generalizing the method of X. Ye and T. Maekawa for 4-dimension. We derive t, n, b(1), b(2) vectors and curvatures (k(1), k(2), k(3)) for transversal intersections of the intersection problem. (C) 2008 Elsevier B.V. All rights reserved.

Topologically correct intersection curves of tori and natural quadrics

This thesis provides efficient and robust algorithms for the computation of the intersection curve between a torus and a simple surface (e.g. a plane, a natural quadric or another torus), based on algebraic and numeric methods. The algebraic part includes the classification of the topological type of the intersection curve and the detection of degenerate situations like embedded conic sections and singularities. Moreover, reference points for each connected intersection curve component are determined. The required computations are realised efficiently by solving quartic polynomials at most and exactly by using exact arithmetic. The numeric part includes algorithms for the tracing of each intersection curve component, starting from the previously computed reference points. Using interval arithmetic, accidental incorrectness like jumping between branches or the skipping of parts are prevented. Furthermore, the environments of singularities are correctly treated. Our algorithms are complete in the sense that any kind of input can be handled including degenerate and singular configurations. They are verified, since the results are topologically correct and approximate the real intersection curve up to any arbitrary given error bound. The algorithms are robust, since no human intervention is required and they are efficient in the way that the treatment of algebraic equations of high degree is avoided.

Topological invariants of isolated complete intersection curve singularities

In this paper we present some formulae for topological invariants of projective complete intersection curves with isolated singularities in terms of the Milnor number, the Euler characteristic and the topological genus. We also present some conditions, involving the Milnor number and the degree of the curve, for the irreducibility of complete intersection curves.

Exact computation of the adjacency graph of an arrangement of quadrics

Präsentiert wird ein vollständiger, exakter und effizienter Algorithmus zur Berechnung des Nachbarschaftsgraphen eines Arrangements von Quadriken (Algebraische Flächen vom Grad 2). Dies ist ein wichtiger Schritt auf dem Weg zur Berechnung des vollen 3D Arrangements. Dabei greifen wir auf eine bereits existierende Implementierung zur Berechnung der exakten Parametrisierung der Schnittkurve von zwei Quadriken zurück. Somit ist es möglich, die exakten Parameterwerte der Schnittpunkte zu bestimmen, diese entlang der Kurven zu sortieren und den Nachbarschaftsgraphen zu berechnen. Wir bezeichnen unsere Implementierung als vollständig, da sie auch die Behandlung aller Sonderfälle wie singulärer oder tangentialer Schnittpunkte einschließt. Sie ist exakt, da immer das mathematisch korrekte Ergebnis berechnet wird. Und schließlich bezeichnen wir unsere Implementierung als effizient, da sie im Vergleich mit dem einzigen bisher implementierten Ansatz gut abschneidet. Implementiert wurde unser Ansatz im Rahmen des Projektes EXACUS. Das zentrale Ziel von EXACUS ist es, einen Prototypen eines zuverlässigen und leistungsfähigen CAD Geometriekerns zu entwickeln. Obwohl wir das Design unserer Bibliothek als prototypisch bezeichnen, legen wir dennoch größten Wert auf Vollständigkeit, Exaktheit, Effizienz, Dokumentation und Wiederverwendbarkeit. Über den eigentlich Beitrag zu EXACUS hinaus, hatte der hier vorgestellte Ansatz durch seine besonderen Anforderungen auch wesentlichen Einfluss auf grundlegende Teile von EXACUS. Im Besonderen hat diese Arbeit zur generischen Unterstützung der Zahlentypen und der Verwendung modularer Methoden innerhalb von EXACUS beigetragen. Im Rahmen der derzeitigen Integration von EXACUS in CGAL wurden diese Teile bereits erfolgreich in ausgereifte CGAL Pakete weiterentwickelt.

Intersections on moduli spaces of curves

We present a new approach to perform calculations with the certain standard classes in cohomology of the moduli spaces of curves. It is based on an important lemma of Ionel relating the intersection theoriy of the moduli space of curves and that of the space of admissible coverings. As particular results, we obtain expressions of Hurwitz numbers in terms of the intersections in the tautological ring, expressions of the simplest intersection numbers in terms of Hurwitz numbers, an algorithm of calculation of certain correlators which are the subject of the Witten conjecture, an improved algorithm for intersections related to the Boussinesq hierarchy, expressions for the Hodge integrals over two-pointed ramification cycles, cut-and-join type equations for a large class of intersection numbers, etc.

Growth Curves For Girls With Turner Syndrome.

The objective of this study was to review the growth curves for Turner syndrome, evaluate the methodological and statistical quality, and suggest potential growth curves for clinical practice guidelines. The search was carried out in the databases Medline and Embase. Of 1006 references identified, 15 were included. Studies constructed curves for weight, height, weight/height, body mass index, head circumference, height velocity, leg length, and sitting height. The sample ranged between 47 and 1,565 (total = 6,273) girls aged 0 to 24 y, born between 1950 and 2006. The number of measures ranged from 580 to 9,011 (total = 28,915). Most studies showed strengths such as sample size, exclusion of the use of growth hormone and androgen, and analysis of confounding variables. However, the growth curves were restricted to height, lack of information about selection bias, limited distributional properties, and smoothing aspects. In conclusion, we observe the need to construct an international growth reference for girls with Turner syndrome, in order to provide support for clinical practice guidelines.

STRESS-STRAIN CURVES FOR STEEL FIBER-REINFORCED CONCRETE IN COMPRESSION

This paper presents a study on the compressive behavior of steel fiber-reinforced concrete. In this study, an analytical model for stress-strain curve for steel fiber-reinforced concrete is derived for concretes with strengths of 40 MPa and 60 MPa at the age of 28 days. Those concretes were reinforced with steel fibers with hooked ends 35 mm long and with aspect ratio of 65. The analytical model was compared with some experimental stress-strain curves and with some models reported in technical literature. Also, the accuracy of the proposed stress-strain curve was evaluated by comparison of the area under stress-strain curve. The results showed good agreement between analytical and experimental data and the benefits of the using of fibers in the compressive behavior of concrete.

A family of implementation-friendly BN elliptic curves

For the last decade, elliptic curve cryptography has gained increasing interest in industry and in the academic community. This is especially due to the high level of security it provides with relatively small keys and to its ability to create very efficient and multifunctional cryptographic schemes by means of bilinear pairings. Pairings require pairing-friendly elliptic curves and among the possible choices, Barreto-Naehrig (BN) curves arguably constitute one of the most versatile families. In this paper, we further expand the potential of the BN curve family. We describe BN curves that are not only computationally very simple to generate, but also specially suitable for efficient implementation on a very broad range of scenarios. We also present implementation results of the optimal ate pairing using such a curve defined over a 254-bit prime field. (C) 2001 Elsevier Inc. All rights reserved.

Nickel adsorption in two Oxisols and an Alfisol as affected by pH, nature of the electrolyte, and ionic strength of soil solution

Background, aim, and scope The retention of potentially toxic metals in highly weathered soils can follow different pathways that variably affect their mobility and availability in the soil-water-plant system. This study aimed to evaluate the effects of pH, nature of electrolyte, and ionic strength of the solution on nickel (Ni) adsorption by two acric Oxisols and a less weathered Alfisol. Materials and methods The effect of pH on Ni adsorption was evaluated in surface and subsurface samples from a clayey textured Anionic `Rhodic` Acrudox ( RA), a sandy-clayey textured Anionic `Xantic` Acrudox (XA), and a heavy clayey textured Rhodic Kandiudalf (RK). All soil samples were equilibrated with the same concentration of Ni solution (5.0 mg L(-1)) and two electrolyte solutions (CaCl(2) or NaCl) with different ionic strengths (IS) (1.0, 0.1 and 0.01 mol L(-1)). The pH of each sample set varied from 3 to 10 in order to obtain sorption envelopes. Results and discussion Ni adsorption increased as the pH increased, reaching its maximum of nearly pH 6. The adsorption was highest in Alfisol, followed by RA and XA. Competition between Ni(2+) and Ca(2+) was higher than that between Ni(2+) and Na(+) in all soil samples, as shown by the higher percentage of Ni adsorption at pH 5. At pH values below the intersection point of the three ionic strength curves (zero point of salt effect), Ni adsorption was generally higher in the more concentrated solution (highest IS), probably due to the neutralization of positive charges of soil colloids by Cl(-) ions and consequent adsorption of Ni(2+). Above this point, Ni adsorption was higher in the more diluted solution (lowest ionic strength), due to the higher negative potential at the colloid surfaces and the lower ionic competition for exchange sites in soil colloids. Conclusions The effect of ionic strength was lower in the Oxisols than in the Alfisol. The main mechanism that controlled Ni adsorption in the soils was the ionic exchange, since the adsorption of ionic species varied according to the variation of pH values. The ionic competition revealed the importance of electrolyte composition and ionic strength on Ni adsorption in soils from the humid tropics. Recommendations and perspectives The presence of NaCl or CaCl(2) in different ionic strengths affects the availability of heavy metals in contaminated soils. Therefore, the study of heavy metal dynamics in highly weathered soils must consider this behavior, especially in soils with large amounts of acric components.

The intersection problem for star designs

We determine those triples (m, II, k) of integers for which there are two m-star designs on the same n-set having exactly k stars in common.

The intersection problem for K_4-e designs

A G-design of order n is a pair (P,B) where P is the vertex set of the complete graph K-n and B is an edge-disjoint decomposition of K-n into copies of the simple graph G. Following design terminology, we call these copies ''blocks''. Here K-4 - e denotes the complete graph K-4 with one edge removed. It is well-known that a K-4 - e design of order n exists if and only if n = 0 or 1 (mod 5), n greater than or equal to 6. The intersection problem here asks for which k is it possible to find two K-4 - e designs (P,B-1) and (P,B-2) of order n, with \B-1 boolean AND B-2\ = k, that is, with precisely k common blocks. Here we completely solve this intersection problem for K-4 - e designs.

The mu-way intersection problem for m-cycle systems

An m-cycle system of order upsilon is a partition of the edge-set of a complete graph of order upsilon into m-cycles. The mu -way intersection problem for m-cycle systems involves taking mu systems, based on the same vertex set, and determining the possible number of cycles which can be common to all mu systems. General results for arbitrary m are obtained, and detailed intersection values for (mu, m) = (3, 4), (4, 5),(4, 6), (4, 7), (8, 8), (8, 9). (For the case (mu, m)= (2, m), see Billington (J. Combin. Des. 1 (1993) 435); for the case (Cc,m)=(3,3), see Milici and Quattrochi (Ars Combin. A 24 (1987) 175. (C) 2001 Elsevier Science B.V. All rights reserved.

On the intersection problem for 1-factorizations and near 1-factorizations of K-v

The number of 1-factors (near 1-factors) that mu 1-factorizations (near 1-factorizations) of the complete graph K-v, v even (v odd), can have in common, is studied. The problem is completely settled for mu = 2 and mu = 3.