The projective characterization of genus two plane curves which have one place at infinity

The characterization of algebraic plane curves

Duke Mathematical Journal ◽

10.1215/s0012-7094-47-01466-x ◽

1947 ◽

Vol 14 (4) ◽

pp. 837-853

Author(s):

Th. Motzkin ◽

A. Robinson

Keyword(s):

Plane Curves

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On the approximation of convex, closed plane curves by multifocal ellipses

Journal of Applied Probability ◽

10.1017/s0021900200034483 ◽

1982 ◽

Vol 19 (A) ◽

pp. 89-96

Author(s):

P. Erdös I. Vincze

Keyword(s):

Open Problem ◽

Closed Curve ◽

Plane Curves ◽

Straight Segment ◽

Convex Figure ◽

Closed Plane

The question whether a convex closed curve can be approximated by ellipses having a large number of foci is considered. It is shown that the limiting, convex figure of multifocal ellipses may have only one single straight segment. This happens only in the case, when the foci tend partly to infinity and partly to points of the line through the straight segment. The approximations of certain ‘distance integrals' are treated; the characterization of approximability remains an open problem.

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Splitting types of bundles of logarithmic vector fields along plane curves

International Journal of Mathematics ◽

10.1142/s0129167x18500556 ◽

2018 ◽

Vol 29 (08) ◽

pp. 1850055 ◽

Cited By ~ 8

Author(s):

Takuro Abe ◽

Alexandru Dimca

Keyword(s):

Vector Fields ◽

Plane Curve ◽

Plane Curves ◽

Tjurina Number ◽

Arrangements Of Lines ◽

Splitting Type ◽

Intersection Points

We give a formula relating the total Tjurina number and the generic splitting type of the bundle of logarithmic vector fields associated to a reduced plane curve. By using it, we give a characterization of nearly free curves in terms of splitting types. Several applications to free and nearly free arrangements of lines are also given, in particular a proof of a form of Terao’s Conjecture for arrangements having a line with at most four intersection points.

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Completion of Katz-Qin-Ruan's enumeration of genus-two plane curves

Journal of Algebraic Geometry ◽

10.1090/s1056-3911-03-00353-9 ◽

2004 ◽

Vol 13 (3) ◽

pp. 547-561 ◽

Cited By ~ 2

Author(s):

Aleksey Zinger

Keyword(s):

Plane Curves ◽

Genus Two

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Algebraic characterization of generic strongly semi-regular rational PH plane curves

Journal of Applied Mathematics and Computing ◽

10.1007/bf02935802 ◽

2005 ◽

Vol 19 (1-2) ◽

pp. 241-251

Author(s):

Gwang-Il Kim

Keyword(s):

Plane Curves ◽

Algebraic Characterization

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On curves with circles as their isoptics

Aequationes Mathematicae ◽

10.1007/s00010-021-00828-4 ◽

2021 ◽

Author(s):

Waldemar Cieślak ◽

Witold Mozgawa

Keyword(s):

Geometric Characterization ◽

Plane Curves ◽

Support Functions ◽

Convex Plane ◽

The Family

AbstractIn the present paper we describe the family of all closed convex plane curves of class $$C^1$$ C 1 which have circles as their isoptics. In the first part of the paper we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the paper contains considerations on curves which have circles as their isoptics and we show the form of support functions of all considered curves.

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On the approximation of convex, closed plane curves by multifocal ellipses

Journal of Applied Probability ◽

10.2307/3213552 ◽

1982 ◽

Vol 19 (A) ◽

pp. 89-96 ◽

Cited By ~ 6

Author(s):

P. Erdös I. Vincze

Keyword(s):

Open Problem ◽

Closed Curve ◽

Plane Curves ◽

Straight Segment ◽

Convex Figure ◽

Closed Plane

The question whether a convex closed curve can be approximated by ellipses having a large number of foci is considered. It is shown that the limiting, convex figure of multifocal ellipses may have only one single straight segment. This happens only in the case, when the foci tend partly to infinity and partly to points of the line through the straight segment. The approximations of certain ‘distance integrals' are treated; the characterization of approximability remains an open problem.

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