scholarly journals An Upper Bound of the Bezout Number for Piecewise Algebraic Curves over a Rectangular Partition

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Feng-Gong Lang ◽  
Xiao-Ping Xu
Keyword(s):  
Upper Bound ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Zero Set ◽  
Bivariate Spline ◽  
Common Intersection ◽  
Piecewise Algebraic Curve ◽  
Piecewise Algebraic Curves ◽  
The Common ◽  
Intersection Points

A piecewise algebraic curve is a curve defined by the zero set of a bivariate spline function. Given two bivariate spline spaces (Δ) over a domainDwith a partition Δ, the Bezout number BN(m,r;n,t;Δ) is defined as the maximum finite number of the common intersection points of two arbitrary piecewise algebraic curves (Δ). In this paper, an upper bound of the Bezout number for piecewise algebraic curves over a rectangular partition is obtained.

scholarly journals Least Squares Fitting of Piecewise Algebraic Curves

2007 ◽  
Vol 2007 ◽  
pp. 1-11 ◽  
Author(s):  
Chun-Gang Zhu ◽  
Ren-Hong Wang
Keyword(s):  
Least Squares ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Scattered Data ◽  
Energy Term ◽  
Bivariate Spline ◽  
Piecewise Algebraic Curve ◽  
Piecewise Algebraic Curves ◽  
The Given

A piecewise algebraic curve is defined as the zero contour of a bivariate spline. In this paper, we present a new method for fittingC1piecewise algebraic curves of degree 2 over type-2 triangulation to the given scattered data. By simultaneously approximating points, associated normals and tangents, and points constraints, the energy term is also considered in the method. Moreover, some examples are presented.

scholarly journals An Algorithm for Isolating the Real Solutions of Piecewise Algebraic Curves

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Jinming Wu ◽  
Xiaolei Zhang
Keyword(s):  
Iterative Algorithm ◽  
Algebraic Curve ◽  
Spline Function ◽  
Algebraic Curves ◽  
The Real ◽  
Real Solutions ◽  
Bivariate Spline ◽  
Piecewise Algebraic Curve ◽  
Piecewise Algebraic Curves

The piecewise algebraic curve, as the set of zeros of a bivariate spline function, is a generalization of the classical algebraic curve. In this paper, an algorithm is presented to compute the real solutions of two piecewise algebraic curves. It is primarily based on the Krawczyk-Moore iterative algorithm and good initial iterative interval searching algorithm. The proposed algorithm is relatively easy to implement.

scholarly journals SIMULTANEOUS ARITHMETIC PROGRESSIONS ON ALGEBRAIC CURVES

2011 ◽  
Vol 07 (04) ◽  
pp. 921-931 ◽  
Author(s):  
RYAN SCHWARTZ ◽  
JÓZSEF SOLYMOSI ◽  
FRANK DE ZEEUW
Keyword(s):  
Lower Bound ◽  
Elliptic Curve ◽  
Arithmetic Progression ◽  
Upper Bound ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Arithmetic Progressions ◽  
Real Algebraic Curve

A simultaneous arithmetic progression (s.a.p.) of length k consists of k points (xi, yσ(i)), where [Formula: see text] and [Formula: see text] are arithmetic progressions and σ is a permutation. Garcia-Selfa and Tornero asked whether there is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over ℚ. We show that 4319 is such a bound for curves over ℝ. This is done by considering translates of the curve in a grid as a graph. A simple upper bound is found for the number of crossings and the "crossing inequality" gives a lower bound. Together these bound the length of an s.a.p. on the curve. We also extend this method to bound the k for which a real algebraic curve can contain k points from a k × k grid. Lastly, these results are extended to complex algebraic curves.

Intersection Operation on a Complex Plane

2021 ◽  
pp. 19-27
Author(s):  
A. Girsh
Keyword(s):  
Complex Plane ◽  
Algebraic Curves ◽  
Intersection Operation ◽  
Intersection Points

Two plane algebraic curves intersect at the actual intersection points of these curves’ graphs. In addition to real intersection points, algebraic curves can also have imaginary intersection points. The total number of curves intersection points is equal to the product of their orders mn. The number of imaginary intersection points can be equal to or part of mn. The position of the actual intersection points is determined by the graphs of the curves, but the imaginary intersection points do not lie on the graphs of these curves, and their position on the plane remains unclear. This work aims to determine the geometry of imaginary intersection points, introduces into consideration the concept of imaginary complement for these algebraic curves in the intersection operation, determines the form of imaginary complements, which intersect at imaginary points. The visualization of imaginary complements clarifies the curves intersection picture, and the position of the imaginary intersection points becomes expected.

Triangulating Topological Spaces

1997 ◽  
Vol 07 (04) ◽  
pp. 365-378 ◽  
Author(s):  
Herbert Edelsbrunner ◽  
Nimish R. Shah
Keyword(s):  
Homotopy Equivalent ◽  
Topological Spaces ◽  
Sufficient Condition ◽  
Voronoi Cells ◽  
Delaunay Complex ◽  
Common Intersection ◽  
Finite Set ◽  
The Common ◽  
Nerve Theorem

Given a subspace [Formula: see text] and a finite set S⊆ℝd, we introduce the Delaunay complex, [Formula: see text], restricted by [Formula: see text]. Its simplices are spanned by subsets T⊆S for which the common intersection of Voronoi cells meets [Formula: see text] in a non-empty set. By the nerve theorem, [Formula: see text] and [Formula: see text] are homotopy equivalent if all such sets are contractible. This paper proves a sufficient condition for [Formula: see text] and [Formula: see text] be homeomorphic.

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