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Computation of the topology of real algebraic space curves

2005
*
Journal of symbolic computation
*

An algorithm for

doi:10.1016/j.jsc.2005.01.006
fatcat:zvp5ilcyt5ckjl6n2t5pddaope
*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. ...##
###
On the Complexity of Computing the Topology of Real Algebraic Space Curves
[article]

2019
*
arXiv
*
pre-print

We modify our existing algorithm for

arXiv:1901.10317v1
fatcat:rfmkowmcy5gwvaiojky2wo2gja
*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. ...##
###
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

2013
*
Journal of symbolic computation
*

In this paper, an algorithm to determine

doi:10.1016/j.jsc.2013.06.004
fatcat:wyor4m7oczbt5jw5msw2wtk2w4
*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. ...##
###
Geometric moduli for klein surfaces

1989
*
Rocky Mountain Journal of Mathematics
*

In other words, we actually want to parametrize

doi:10.1216/rmj-1989-19-3-939
fatcat:yvpgufpx4zcmvjxidldt7fpa7q
*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. ...##
###
GENOM3CK

2011
*
ACM SIGSAM Bulletin
*

We notice that

doi:10.1145/1940475.1940519
fatcat:67fbzxrdnvccvnp4vzaolykgv4
*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*. ...##
###
Determining the Topology of Real Algebraic Surfaces
[chapter]

2005
*
Lecture Notes in Computer Science
*

An algorithm is proposed to determine

doi:10.1007/11537908_8
fatcat:244kvej2t5hg5djbk5rojq23hy
*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*. ...##
###
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

2012
*
Journal of symbolic computation
*

We extend

doi:10.1016/j.jsc.2011.09.010
fatcat:elh2gme7dja5lmaedn3m7vkxge
*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. ...##
###
Applications of Level Curves to Some Problems on Algebraic Surfaces
[article]

2007
*
arXiv
*
pre-print

In a previous work

arXiv:0710.3352v1
fatcat:hv6nz6zacba3nbawpklmhxr7nq
*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*. ...##
###
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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