Invariant singular points of algebraic curves

1983 ◽  
Vol 34 (6) ◽  
pp. 962-963
Author(s):  
E. I. Shustin
Keyword(s):  
Algebraic Curves ◽  
Singular Points

scholarly journals Singular points of algebraic curves

1990 ◽  
Vol 9 (4) ◽  
pp. 405-421 ◽  
Author(s):  
Takis Sakkalis ◽  
Rida Farouki
Keyword(s):  
Algebraic Curves ◽  
Singular Points

Surface Intersections for Geometric Modeling

1990 ◽  
Vol 112 (1) ◽  
pp. 100-107 ◽  
Author(s):  
N. M. Patrikalakis ◽  
P. V. Prakash
Keyword(s):  
Geometric Modeling ◽  
Algebraic Curves ◽  
Solution Procedure ◽  
Rectangular Domain ◽  
Singular Points ◽  
Algebraic Surfaces ◽  
Surface Patches ◽  
Adaptive Subdivision ◽  
Rational Polynomial ◽  
Spline Surfaces

Evaluation of planar algebraic curves arises in the context of intersections of algebraic surfaces with piecewise continuous rational polynomial parametric surface patches useful in geometric modeling. We address a method of evaluating these curves of intersection that combines the advantageous features of analytic representation of the governing equation of the algebraic curve in the Bernstein basis within a rectangular domain, adaptive subdivision and polyhedral faceting techniques, and the computation of turning and singular points, to provide the basis for a reliable and efficient solution procedure. Using turning and singular points, the intersection problem can be partitioned into subdomains that can be processed independently and which involve intersection segments that can be traced with faceting methods. This partitioning and the tracing of individual segments is carried out using an adaptive subdivision algorithm for Bezier/B-spline surfaces followed by Newton correction of the approximation. The method has been successfully tested in tracing complex algebraic curves and in solving actual intersection problems with diverse features.

Subrings of the first neighbourhood ring: II

1982 ◽  
Vol 92 (1) ◽  
pp. 35-39
Author(s):  
D. Kirby
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Algebraic Curves ◽  
Present Note ◽  
Quotient Ring ◽  
Singular Points ◽  
Local Rings ◽  
Embedding Dimension ◽  
One Dimensional ◽  
Finite Lattices

In (1) and (2) we studied a lattice of extension rings associated with a commutative ring R with identity. When R, M is a one-dimensional Cohen-Macaulay local ring the elements of are just those integral extensions of R contained in the total quotient ring T(R) and such that lengthR(S/R) is finite. Experiments with local rings of singular points on algebraic curves indicate that only the simplest singularities give rise to finite lattices. So the problem arises as to which local rings R give rise to which finite lattices. In later papers this problem will be investigated in detail, at least when R is of low embedding dimension. The purpose of the present note is to establish some general results which indicate the size of the problem.

scholarly journals Computing Singular Points of Projective Plane Algebraic Curves by Homotopy Continuation Methods

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Zhongxuan Luo ◽  
Erbao Feng ◽  
Jielin Zhang
Keyword(s):  
Projective Plane ◽  
Time Complexity ◽  
Algebraic Curve ◽  
Numerical Experiments ◽  
Algebraic Curves ◽  
Singular Points ◽  
Continuation Methods ◽  
Homotopy Continuation ◽  
Univariate Polynomials ◽  
Homotopy Continuation Methods

We present an algorithm that computes the singular points of projective plane algebraic curves and determines their multiplicities and characters. The feasibility of the algorithm is analyzed. We prove that the algorithm has the polynomial time complexity on the degree of the algebraic curve. The algorithm involves the combined applications of homotopy continuation methods and a method of root computation of univariate polynomials. Numerical experiments show that our algorithm is feasible and efficient.

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