scholarly journals Implicitizing Rational Curves by the Method of Moving Algebraic Curves

1997 ◽  
Vol 23 (2-3) ◽  
pp. 153-175 ◽  
Author(s):  
TOM SEDERBERG ◽  
RON GOLDMAN ◽  
HANG DU
Keyword(s):  
Algebraic Curves ◽  
Rational Curves

scholarly journals Approximate Implicitization of Parametric Curves Using Cubic Algebraic Splines

2009 ◽  
Vol 2009 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaolei Zhang ◽  
Jinming Wu
Keyword(s):  
Algebraic Curves ◽  
Approximation Error ◽  
Rational Curves ◽  
Parametric Curves ◽  
Approximate Implicitization

This paper presents an algorithm to solve the approximate implicitization of planar parametric curves using cubic algebraic splines. It applies piecewise cubic algebraic curves to give a globalG2continuity approximation to planar parametric curves. Approximation error on approximate implicitization of rational curves is given. Several examples are provided to prove that the proposed method is flexible and efficient.

scholarly journals ALGEBRAIC GEOMETRY AND HOFSTADTER TYPE MODEL

2002 ◽  
Vol 16 (14n15) ◽  
pp. 2097-2106
Author(s):  
SHAO-SHIUNG LIN ◽  
SHI-SHYR ROAN
Keyword(s):  
Algebraic Geometry ◽  
Physical Model ◽  
Algebraic Curves ◽  
Spectral Curve ◽  
Rational Curves ◽  
Type Model ◽  
Bethe Equation ◽  
Explicit Solutions ◽  
Polynomial Roots ◽  
High Genus

In this report, we study the algebraic geometry aspect of Hofstadter type models through the algebraic Bethe equation. In the diagonalization problem of certain Hofstadter type Hamiltonians, the Bethe equation is constructed by using the Baxter vectors on a high genus spectral curve. When the spectral variables lie on rational curves, we obtain the complete and explicit solutions of the polynomial Bethe equation; the relation with the Bethe ansatz of polynomial roots is discussed. Certain algebraic geometry properties of Bethe equation on the high genus algebraic curves are discussed in cooperation with the consideration of the physical model.

scholarly journals Correspondence theorems via tropicalizations of moduli spaces

2016 ◽  
Vol 18 (03) ◽  
pp. 1550043 ◽  
Author(s):  
Andreas Gross
Keyword(s):  
Moduli Spaces ◽  
Toric Variety ◽  
Algebraic Curves ◽  
Rational Curves ◽  
Tropical Curves ◽  
Algebraic Tori ◽  
Pass Through ◽  
General Correspondence ◽  
Correspondence Theorem ◽  
Generic Points

We show that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli spaces. These embeddings are shown to respect the evaluation morphisms in the sense that evaluation commutes with tropicalization. With this particular setting in mind, we prove a general correspondence theorem for enumerative problems which are defined via “evaluation maps” in both the algebraic and tropical world. Applying this to our motivational example, we show that the tropicalizations of the curves in a given toric variety which intersect the boundary divisors in their interior and with prescribed multiplicities, and pass through an appropriate number of generic points are precisely the tropical curves in the corresponding tropical toric variety satisfying the analogous condition. Moreover, the intersection-theoretically defined multiplicities of the tropical curves are equal to the numbers of algebraic curves tropicalizing to them.

Construction of Belonging to Surfaces One-Dimensional Contours by Mapping Them to a Plane

2018 ◽  
Vol 6 (1) ◽  
pp. 3-9 ◽  
Author(s):  
Геннадий Иванов ◽  
Gennadiy Ivanov
Keyword(s):  
Quadratic Forms ◽  
Information Technologies ◽  
Algebraic Curves ◽  
Complex Geometry ◽  
Rational Curves ◽  
Cremona Transformations ◽  
One Dimensional ◽  
Curves And Surfaces ◽  
Theoretical Support ◽  
The 19Th Century

As is known, differential geometry studies the properties of curve lines (tangent, curvature, torsion), surfaces (bending, first and second basic quadratic forms) and their families in small, that is, in the neighborhood of the point by means of differential calculus. Algebraic geometry studies properties of algebraic curves, surfaces, and algebraic varieties in general [1; 17]: order, class, genre, existence of singular points and lines, curves and surfaces family intersections (sheaves, bundles, congruences, complexes and their characteristics). Rational curves and surfaces occupy a special place among them: • their design by bi-rational (Cremona) transformations [10; 21]; • investigation of their properties by mapping to lines and planes [9; 21; 22]; • construction of smooth contours from arcs of rational curves belonging to surfaces [10]. It seems that the main results obtained in this direction by mathematicians in the second half of the 19th century by structural and geometric methods should be the theoretical support for the design of technical forms that meet a number of pre-set requirements using modern computational tools and information technologies. It is obvious that application of Cremona transformations’ powerful apparatus is useful when designing, for example, pipes of complex geometry according to set of streamlines, thin-walled shells for a given mesh manifold of curvature lines etc. Apparently, this stage should precede computer graphics’ calculation procedures. However, in Russian publications on applied (engineering) geometry, only a little attention is paid to the study of surfaces in general. The author knows nothing about the use of this approach for solving of these applied problems. In this regard, the aims of this paper are: • illustration of method for mapping a surface to a plane to study its properties in general by the example of construction a flat model for a hyperboloid of one sheet; • constructive approach to the construction of smooth one-dimensional contours on rational surfaces.

scholarly journals The $2$-Hessian and sextactic points on plane algebraic curves

2019 ◽  
Vol 125 (1) ◽  
pp. 13-38
Author(s):  
Paul Aleksander Maugesten ◽  
Torgunn Karoline Moe
Keyword(s):  
Algebraic Curve ◽  
Algebraic Curves ◽  
Rational Curves ◽  
Veronese Embedding ◽  
Plane Algebraic Curve ◽  
Special Case

In an article from 1865, Arthur Cayley claims that given a plane algebraic curve there exists an associated $2$-Hessian curve that intersects it in its sextactic points. In this paper we fix an error in Cayley's calculations and provide the correct defining polynomial for the $2$-Hessian. In addition, we present a formula for the number of sextactic points on cuspidal curves and tie this formula to the $2$-Hessian. Lastly, we consider the special case of rational curves, where the sextactic points appear as zeros of the Wronski determinant of the 2nd Veronese embedding of the curve.

GEOMETRIC THEORY OF MULTIDIMENSIONAL INTERPOLATION

2020 ◽  
Vol 2020 (1) ◽  
pp. 9-16
Author(s):  
Evgeniy Konopatskiy
Keyword(s):  
Response Function ◽  
Algebraic Curves ◽  
Geometric Theory ◽  
Analytical Description ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Mathematical Apparatus ◽  
Multidimensional Interpolation ◽  
Pass Through

The paper presents a geometric theory of multidimensional interpolation based on invariants of affine geometry. The analytical description of geometric interpolants is performed within the framework of the mathematical apparatus BN-calculation using algebraic curves that pass through preset points. A geometric interpretation of the interaction of parameters, factors, and the response function is presented, which makes it possible to generalize the geometric theory of multidimensional interpolation in the direction of increasing the dimension of space. The conceptual principles of forming the tree of the geometric interpolant model as a geometric basis for modeling multi-factor processes and phenomena are described.

scholarly journals Non Kählerian surfaces with a cycle of rational curves

2021 ◽  
Vol 8 (1) ◽  
pp. 208-222
Author(s):  
Georges Dloussky
Keyword(s):  
Deformation Theory ◽  
Complex Surface ◽  
Rational Curves ◽  
Intersection Form ◽  
Connected Component ◽  
Hirzebruch Surface ◽  
Compact Complex Surface ◽  
A Chain

Abstract Let S be a compact complex surface in class VII0 + containing a cycle of rational curves C = ∑Dj . Let D = C + A be the maximal connected divisor containing C. If there is another connected component of curves C ′ then C ′ is a cycle of rational curves, A = 0 and S is a Inoue-Hirzebruch surface. If there is only one connected component D then each connected component Ai of A is a chain of rational curves which intersects a curve Dj of the cycle and for each curve Dj of the cycle there at most one chain which meets Dj . In other words, we do not prove the existence of curves other those of the cycle C, but if some other curves exist the maximal divisor looks like the maximal divisor of a Kato surface with perhaps missing curves. The proof of this topological result is an application of Donaldson theorem on trivialization of the intersection form and of deformation theory. We apply this result to show that a twisted logarithmic 1-form has a trivial vanishing divisor.

Sign in / Sign up

Export Citation Format

Share Document