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Computation of the topology of real algebraic space curves

Juan Gerardo Alcázar, J. Rafael Sendra
2005 Journal of symbolic computation  
An algorithm for computing the topology of a real algebraic space curve C, implicitly defined as the intersection of two surfaces, is presented.  ...  Given C, the algorithm generates a space graph which is topologically equivalent to the real variety on the Euclidean space.  ...  However, the computation of the topological graph of algebraic space curves has not been treated so extensively.  ... 
doi:10.1016/j.jsc.2005.01.006 fatcat:zvp5ilcyt5ckjl6n2t5pddaope

On the Complexity of Computing the Topology of Real Algebraic Space Curves [article]

Kai Jin, Jin-San Cheng
2019 arXiv   pre-print
We modify our existing algorithm for computing the topology of an algebraic space curve and analyze the bit complexity of the algorithm.  ...  It is Õ (N^20), where N={d,τ}, d, τ are the degree bound and the bit size bound of the coefficients of the defining polynomials of the algebraic space curve.  ...  Acknowledgement The work is partially supported by NKBRPC (2011CB302400), NSFC Grants (11001258, 60821002), SRF for ROCS, SEM.  ... 
arXiv:1901.10317v1 fatcat:rfmkowmcy5gwvaiojky2wo2gja

Page 7247 of Mathematical Reviews Vol. , Issue 95m [page]

1995 Mathematical Reviews  
is a topological model for a real plane projective algebraic curve if T and the curve have the same topological type. The author offers a constructive method to find in RP?  ...  an algebraic curve (possibly singular) with a prescribed topological type in the case where all the singularities of the curve are nondegenerate double points.  ... 

Certified rational parametric approximation of real algebraic space curves with local generic position method

Jin-San Cheng, Kai Jin, Daniel Lazard
2013 Journal of symbolic computation  
In this paper, an algorithm to determine the topology of an algebraic space curve and to compute a certified G 1 rational parametric approximation of the algebraic space curve is given by extending the  ...  We also analyze the complexity of computing the topology of an algebraic space curve. Nontrivial examples are used to show the effectivity of the method.  ...  All the authors would like to thank the anonymous referees, their suggestions improve the paper.  ... 
doi:10.1016/j.jsc.2013.06.004 fatcat:wyor4m7oczbt5jw5msw2wtk2w4

Geometric moduli for klein surfaces

M. Seppala, T. Sorvali
1989 Rocky Mountain Journal of Mathematics  
In other words, we actually want to parametrize the Teichmüller space of real algebraic curves of topological type E.  ...  By a real algebraic curve of the topological type E we then mean a real algebraic curve of the type E* in the sense of Ailing and Greenleaf.  ... 
doi:10.1216/rmj-1989-19-3-939 fatcat:yvpgufpx4zcmvjxidldt7fpa7q

GENOM3CK

Mădălina Hodorog, Bernard Mourrain, Josef Schicho
2011 ACM SIGSAM Bulletin  
We notice that the computation of the genus reduces to the computation of the delta-invariant of each singularity of the plane complex algebraic curve.  ...  The algorithm takes as input parameter a positive real number ∈ R * + , and a domain , with a, b, c ∈ N * , for the x, y and z coordinates of the threedimensional space.  ...  complex algebraic curve in the 4dimensional space.  ... 
doi:10.1145/1940475.1940519 fatcat:67fbzxrdnvccvnp4vzaolykgv4

Determining the Topology of Real Algebraic Surfaces [chapter]

Jin-San Cheng, Xiao-Shan Gao, Ming Li
2005 Lecture Notes in Computer Science  
An algorithm is proposed to determine the topology of an implicit real algebraic surface in R 3 .  ...  The topology is represented by a set of surface patches, and each surface patch is presented by an ordered list of space curve segments.  ...  Fortuna et al presented an algorithm (Fortuna etc. 2003) to determine the topology of non-singular, orientable real algebraic surface in projective space.  ... 
doi:10.1007/11537908_8 fatcat:244kvej2t5hg5djbk5rojq23hy

Page 5384 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
The space of non-singular real algebraic curves of bidegree (4,3) on a hyperboloid consists of a finite number of connected com- ponents.  ...  The subject of the paper is topological description of possible mutual position of two non-singular algebraic curves in the real projective plane.  ... 

Page 3961 of Mathematical Reviews Vol. , Issue 2000f [page]

2000 Mathematical Reviews  
The author revisits an extremal case of Rokhlin’s inequalities concerning the disposition of the ovals of a nonsingular real algebraic curve [S. P. Paris, Math.  ...  computer aided solution of a system of polynomial equations on the coef- ficients of a curve); then this curve is perturbed into the required nonsingular curve.  ... 

Page 3716 of Mathematical Reviews Vol. , Issue 94g [page]

1994 Mathematical Reviews  
The following topics are considered in detail: restrictions on the topology of real algebraic curves; constructions of real algebraic curves; smoothing of singular points.  ...  (RS-SAMA; Samara) Topology of real plane algebraic curves. Real algebraic geometry (Rennes, 1991), 97-109, Lecture Notes in Math., 1524, Springer, Berlin, 1992.  ... 

Page 5563 of Mathematical Reviews Vol. , Issue 2002H [page]

2002 Mathematical Reviews  
The subject of the paper is possible topological types of smooth algebraic surfaces of a given degree in the real projective 3-space. There are finitely many types for each degree.  ...  space curve, (ii) reconstruction of the space curve from two or more of its pro- jections on some planes, (iii) application of algebraic curves to the structure recovery of dynamic scenes from the stream  ... 

Page 2469 of Mathematical Reviews Vol. , Issue 92e [page]

1992 Mathematical Reviews  
The ap- plication of Morse theory to study the topology of real algebraic curves or varieties of higher dimension is classical [see Gudkov, op. cit.]; on the other hand, the existence and construction  ...  Kharlamov (Strasbourg) 92e:14056 14P25 68Q25 Sakkalis, Takis (1-OAKL) The topological configuration of a real algebraic curve. Bull. Austral. Math. Soc. 43 (1991), no. 1, 37-50.  ... 

On the isotopic meshing of an algebraic implicit surface

Daouda Niang Diatta, Bernard Mourrain, Olivier Ruatta
2012 Journal of symbolic computation  
We extend the work in Diatta et al. (2008) , on the topology of non-reduced algebraic space curves, and apply it to the polar curve or apparent contour of the surface S.  ...  We present a new and complete algorithm for computing the topology of an algebraic surface S given by a square free polynomial in Q[X , Y , Z ].  ...  Topology of an implicit space algebraic curve This section is devoted to the description of an algorithm allowing to compute the topology of algebraic space curves with certainty.  ... 
doi:10.1016/j.jsc.2011.09.010 fatcat:elh2gme7dja5lmaedn3m7vkxge

Applications of Level Curves to Some Problems on Algebraic Surfaces [article]

J. G. Alcazar, J. R. Sendra
2007 arXiv   pre-print
In a previous work of the authors, a result to algorithmically compute the topology types of the level curves of an algebraic surface, is given.  ...  From this result, here we derive applications based on level curves to determine some topological features of surfaces (reality, compactness, connectivity) and to the problem of plotting.  ...  in (2) to compute the topology of space algebraic curves.  ... 
arXiv:0710.3352v1 fatcat:hv6nz6zacba3nbawpklmhxr7nq

Page 2847 of Mathematical Reviews Vol. , Issue 97E [page]

1997 Mathematical Reviews  
Thus N, should be x(4/"), which can be computed from knowledge of the cohomology of 14P Real algebraic and real analytic geometry 97e: 14069 Sym*(S).  ...  The paper under review applies this philoso- phy from a purely mathematical point of view to the computation of: (A) Poincaré polynomials for the moduli spaces Mo,, of sta- ble n-pointed curves of genus  ... 
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