Backward Errors and Condition Numbers of Regular and Singular Points on Algebraic Curves

Author(s):  
Joab R. Winkler
1983 ◽  
Vol 34 (6) ◽  
pp. 962-963
Author(s):  
E. I. Shustin

1990 ◽  
Vol 9 (4) ◽  
pp. 405-421 ◽  
Author(s):  
Takis Sakkalis ◽  
Rida Farouki

1990 ◽  
Vol 112 (1) ◽  
pp. 100-107 ◽  
Author(s):  
N. M. Patrikalakis ◽  
P. V. Prakash

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.


Author(s):  
D. Kirby

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.


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