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 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 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 The number of connected components of certain real algebraic curves

2001 ◽  
Vol 25 (11) ◽  
pp. 693-701
Author(s):  
Seon-Hong Kim
Keyword(s):  
Algebraic Curve ◽  
Algebraic Curves ◽  
Connected Components ◽  
Connected Component ◽  
The Real ◽  
Real Algebraic Curves ◽  
Real Algebraic Curve

For an integern≥2, letp(z)=∏k=1n(z−αk)andq(z)=∏k=1n(z−βk), whereαk,βkare real. We find the number of connected components of the real algebraic curve{(x,y)∈ℝ2:|p(x+iy)|−|q(x+iy)|=0}for someαkandβk. Moreover, in these cases, we show that each connected component contains zeros ofp(z)+q(z), and we investigate the locus of zeros ofp(z)+q(z).

A Note Concerning the Real Inflexions of Real, Plane, Algebraic Curves

1974 ◽  
Vol 17 (3) ◽  
pp. 411-412
Author(s):  
Gareth J. Griffith
Keyword(s):  
Algebraic Curve ◽  
Double Point ◽  
Algebraic Curves ◽  
Real Plane ◽  
The Real ◽  
Plane Algebraic Curve

Theorem. “If a crunode of a real, irreducible, plane, algebraic curve changes into an acnode via the intermediary stage of a real cusp, two real inflexions are introduced in a neighborhood of the double point.”

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