Systems of 𝑘 points in general position and algebraic curves of different orders

1978 ◽  
pp. 15-45
Author(s):  
D. A. Gudkov
Keyword(s):  
General Position ◽  
Algebraic Curves

scholarly journals AN APPROXIMATE ARRANGEMENT ALGORITHM FOR SEMI-ALGEBRAIC CURVES

2007 ◽  
Vol 17 (02) ◽  
pp. 175-198 ◽  
Author(s):  
VICTOR MILENKOVIC ◽  
ELISHA SACKS
Keyword(s):  
Computational Model ◽  
Linear Order ◽  
General Position ◽  
Algebraic Curves ◽  
Vertical Line ◽  
Floating Point ◽  
Plane Curves ◽  
Data Sets ◽  
Running Time

We present an arrangement algorithm for plane curves. The inputs are (1) continuous, compact, x-monotone curves and (2) a module that computes approximate crossing points of these curves. There are no general position requirements. We assume that the crossing module output is ∊ accurate, but allow it to be inconsistent, meaning that three curves are in cyclic y order over an x interval. The curves are swept with a vertical line using the crossing module to compute and process sweep events. When the sweep detects an inconsistency, the algorithm breaks the cycle to obtain a linear order. We prove correctness in a realistic computational model of the crossing module. The number of vertices in the output is V = 2n + N + min (3kn,n2/2) and the running time is O (V log n) for n curves with N crossings and k inconsistencies. The output arrangement is realizable by curves that are O (∊ + kn∊) close to the input curves, except in kn∊ neighborhoods of the curve tails. The accuracy can be guaranteed everywhere by adding tiny horizontal extensions to the segment tails, but without the running time bound. An implementation is described for semi-algebraic curves based on a numerical equation solver. Experiments show that the extensions only slightly increase the running time and have little effect on the error. On challenging data sets, the number of inconsistencies is at most 3N, the output accuracy is close to ∊, and the running time is close to that of the standard, non-robust floating point sweep.

scholarly journals On the disposition of cubic and pair of conics in a real projective plane

2020 ◽  
Vol 22 (1) ◽  
pp. 24-37
Author(s):  
Victoriya A. Gorskaya ◽  
Grigory M. Polotovskiy
Keyword(s):  
Projective Plane ◽  
General Position ◽  
Algebraic Curves ◽  
Hilbert Problem ◽  
Topological Classification ◽  
Real Projective Plane ◽  
16Th Hilbert Problem ◽  
Algebraic Manifolds ◽  
Manifolds With Singularities

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.

GEOMETRIC THEORY OF MULTIDIMENSIONAL INTERPOLATION

2020 ◽  
Vol 2020 (1) ◽  
pp. 9-16
Author(s):  
Evgeniy Konopatskiy
Keyword(s):  
Response Function ◽  
Algebraic Curves ◽  
Geometric Theory ◽  
Analytical Description ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Mathematical Apparatus ◽  
Multidimensional Interpolation ◽  
Pass Through

The paper presents a geometric theory of multidimensional interpolation based on invariants of affine geometry. The analytical description of geometric interpolants is performed within the framework of the mathematical apparatus BN-calculation using algebraic curves that pass through preset points. A geometric interpretation of the interaction of parameters, factors, and the response function is presented, which makes it possible to generalize the geometric theory of multidimensional interpolation in the direction of increasing the dimension of space. The conceptual principles of forming the tree of the geometric interpolant model as a geometric basis for modeling multi-factor processes and phenomena are described.

Rationality revisited: Politicisation through planning rationality against the rationality of power

2021 ◽  
pp. 147309522110011
Author(s):  
Esin Özdemir
Keyword(s):  
Public Interest ◽  
General Position ◽  
Strong Relationship ◽  
Planning Theory ◽  
The Political ◽  
Rational Thinking ◽  
Liberal Democracies ◽  
Substantive Issue ◽  
Rational Consensus ◽  
Definition Of

In this article, I readdress the issue of rationality, which has been so far considered in western liberal democracies and in planning theory as procedural, and more recently as post-political in the post-foundational approach, aiming to show how it can gain a substantive and politicising character. I first discuss the problems and limits of the treatment of rational thinking as well as rational consensus-seeking as merely procedural and post-political. Secondly, utilising the notion of Realrationalität of Flyvbjerg, I discuss how rationality attains a politicising role due to its strong relationship with power. Using the concept of planning rationality aiming at public interest, I present the general position and actions of professional organisations in Turkey, focusing on the Chamber of City Planners, as an example illustrative of my argument. I finally argue that rationality becomes a substantive issue that politicizes planning, when it is put forward as an alternative to authoritarian market logic. In doing so, I adopt the Rancièrian definition of the political, defined as disclosure of a wrong and staging of equality. In conclusion, I first emphasize the importance of avoiding quick rejections of the concepts of rationality and consensus in the framework of planning activity and planning theory and secondly, call for a broader definition of the political; the political that is not confined to conflict but is open to rational thinking and rational consensus.

scholarly journals THE GENERAL POSITION NUMBER OF THE CARTESIAN PRODUCT OF TWO TREES

2020 ◽  
pp. 1-10
Author(s):  
JING TIAN ◽  
KEXIANG XU ◽  
SANDI KLAVŽAR
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
General Position ◽  
Cartesian Product

Abstract The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.

scholarly journals Real plane algebraic curves with prescribed singularities

Topology ◽  
1993 ◽  
Vol 32 (4) ◽  
pp. 845-856 ◽  
Author(s):  
Eugenii Shustin
Keyword(s):  
Algebraic Curves ◽  
Real Plane
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