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Theorems and corresponding algorithms are presented which produce a rational Bezier curve of a specified are length subject to certain constraints. Extraneous inflection points are avoided. The problem is reduced to expressing the...
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Theorems and corresponding algorithms are presented which produce a rational Bezier curve of a specified are length subject to certain constraints. Extraneous inflection points are avoided. The problem is reduced to expressing the are length as a function of a single variable. A general theorem from a previous paper of the authors is used which gives conditions under which the are length function is convex or strictly convex. An algorithm to automatically choose the initial parameters for the secant method will produce a solution to this problem with performance comparable to the Newton-Raphson method. Theory and algorithms for rational parametric curves are presented. It is shown that in certain cases rational parametric curves of degree three can be used while polynomials of bounded degree cannot. [References: 8]
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Given a set S of elements in a number field k, we discuss the existence of planar algebraic curves over k which possess rational points whose x-coordinates are exactly the elements of S. If the size vertical bar S vertical bar of ...
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Given a set S of elements in a number field k, we discuss the existence of planar algebraic curves over k which possess rational points whose x-coordinates are exactly the elements of S. If the size vertical bar S vertical bar of S is either 4, 5, or 6, we exhibit infinite families of (twisted) Edwards curves and (general) Huff curves for which the elements of S are realized as the x-coordinates of rational points on these curves. This generalizes earlier work on progressions of certain types on some algebraic curves.
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Bezier curves are an essential tool for curve design. Due to their properties, common operations such as translation, rotation, or scaling can be applied to the curve by simply modifying the control polygon of the curve. More flex...
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Bezier curves are an essential tool for curve design. Due to their properties, common operations such as translation, rotation, or scaling can be applied to the curve by simply modifying the control polygon of the curve. More flexibility, and thus more diverse types of curves, can be achieved by associating a weight with each control point, that is, by considering rational Bezier curves. As shown by Ramanantoanina and Hormann (2021), additional and more direct control over the curve shape can be achieved by exploiting the correspondence between the rational Bezier and the interpolating barycentric form and by exploring the effect of changing the degrees of freedom of the latter (interpolation points, weights, and nodes). In this paper, we explore similar editing possibilities for closed curves, in particular for the rational extension of the periodic Bezier curves that were introduced by Sanchez-Reyes (2009). We show how to convert back and forth between the periodic rational Bezier and the interpolating trigonometric barycentric form, derive a necessary condition to avoid poles of a trigonometric rational interpolant, and devise a general framework to perform degree elevation of periodic rational Bezier curves. We further discuss the editing possibilities given by the trigonometric barycentric form.
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Bezier curves are indispensable for geometric modelling and computer graphics. They have numerous favourable properties and provide the user with intuitive tools for editing the shape of a parametric polynomial curve. Even more co...
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Bezier curves are indispensable for geometric modelling and computer graphics. They have numerous favourable properties and provide the user with intuitive tools for editing the shape of a parametric polynomial curve. Even more control and flexibility can be achieved by associating a shape parameter with each control point and considering rational Bezier curves, which comes with the additional advantage of being able to represent all conic sections exactly. In this paper, we explore the editing possibilities that arise from expressing a rational Bezier curve in barycentric form. In particular, we show how to convert back and forth between the Bezier and the barycentric form, we discuss the effects of modifying the constituents (nodes, interpolation points, weights) of the barycentric form, and we study the connection between point insertion in the barycentric form with degree elevation of the Bezier form. Moreover, we analyse the favourable performance of the barycentric form for evaluating the curve.
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A rational surface is the locus of a rational curve that is moving through space and thereby changing its shape by changing its control points and weights. This intuitive definition can be used to derive hodographs of rational Bez...
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A rational surface is the locus of a rational curve that is moving through space and thereby changing its shape by changing its control points and weights. This intuitive definition can be used to derive hodographs of rational Bezier surfaces and their bounds of magnitude.
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We show that any k-dimensional subvariety with geometric genus zero of a very general degree d hypersurface X (is contained in) P~n is a union of lines and conics if d > 1/5(7n + 2(9 - k)). In particular, any rational curve on a v...
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We show that any k-dimensional subvariety with geometric genus zero of a very general degree d hypersurface X (is contained in) P~n is a union of lines and conics if d > 1/5(7n + 2(9 - k)). In particular, any rational curve on a very general degree d hypersurface X (is contained in) P~n is a line or a conic if d > 1/5(7n + 16).
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We show that the Taylor coefficients of the series q(z) = z exp(G(z)/F(z)) are integers, where F(z) and G(z) + log(z)F(z) are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at ...
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We show that the Taylor coefficients of the series q(z) = z exp(G(z)/F(z)) are integers, where F(z) and G(z) + log(z)F(z) are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at z = 0. We also address the question of finding the largest integer it such that the Taylor coefficients of (z(-1) q(z))(1/u) are still integers. As consequences, we are able to prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi-kau complete intersections in weighted projective spaces, which improve and refine previous results by Lian and Yau and by Zudilin. In particular, we prove the general "integrality" conjecture of Zudilin about these mirror maps.
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We construct an abstract; theory of Gromov Witten invariants of genus 0 for quantum minimal Fano varieties (a minimal class of varieties which is natural from the quantum cohomological viewpoint.). Namely, we. consider the minimal...
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We construct an abstract; theory of Gromov Witten invariants of genus 0 for quantum minimal Fano varieties (a minimal class of varieties which is natural from the quantum cohomological viewpoint.). Namely, we. consider the minimal Gromov Witten ring: a commutative algebra whose generators and relations are of the form used in the Gromov Witten theory of Fano varieties (of unspecified dimension). The Gromov Witten theory of any quantum minimal variety is a homomorphism from this ring to C We prove an abstract reconstruction theorem which says that. this ring is isomorphic to the free commutative ring generated by 'prime two-pointed invariants'. We also find solutions of the differential equation of type DN for a Fano variety of dimension N in terms of the generating series of one-pointed Gromov Witten invariants.
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In this paper we study smooth complex projective polarized varieties (X, H) of dimension n ≥ 2 which admit a covering family V of rational curves of degree 3 with respect to H such that two general points of X may be joined by a ...
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In this paper we study smooth complex projective polarized varieties (X, H) of dimension n ≥ 2 which admit a covering family V of rational curves of degree 3 with respect to H such that two general points of X may be joined by a curve parametrized by V, and such that there is a covering family of rational curves of H-degree one. We prove that the Picard number of these manifolds is at most three, and that, if equality holds, (X, H) has an adjunction theoretic scroll structure over a smooth variety.
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Let K be a characteristic zero field, let a be an algebraic element over K and C a rational curve defined over K given by a parametrization (sic) with coefficients in K( a). We propose an algorithm to solve the following problem, ...
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Let K be a characteristic zero field, let a be an algebraic element over K and C a rational curve defined over K given by a parametrization (sic) with coefficients in K( a). We propose an algorithm to solve the following problem, that is, a parametric version of Hilbert-Hurwitz: To compute a linear fraction u = at+b/ct+d such that (sic) (u) has coefficients over an algebraic extension of K of degree at most two and a conic K-birational to C. Moreover, if the degree of C is odd or a is of odd degree over K, we can compute a parametrization of C with coefficients over K. The problem is solved without implicitization methods nor analyzing the singularities of C.
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