Intersection points algorithm for piecewise algebraic curves based on Groebner bases

2008 ◽  
Vol 29 (1-2) ◽  
pp. 357-366 ◽  
Author(s):  
Feng-Gong Lang ◽  
Ren-Hong Wang
Keyword(s):  
Algebraic Curves ◽  
Groebner Bases ◽  
Piecewise Algebraic Curves ◽  
Intersection Points

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.

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.

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.

Intersection Operation on a Complex Plane

2021 ◽  
Vol 9 (1) ◽  
pp. 20-28
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.

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