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

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.

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

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.

2. Applications of the Theorem that Two Closed Plane Curves intersect an even number of times

1878 ◽  
Vol 9 ◽  
pp. 237-246 ◽  
Author(s):  
Tait
Keyword(s):  
Special Class ◽  
Double Point ◽  
Closed Curve ◽  
British Association ◽  
Plane Curves ◽  
Double Points ◽  
The Wire ◽  
Closed Plane

The theorem itself may be considered obvious, and is easily applied, as I showed at the late meeting of the British Association, to prove that in passing from any one double point of a plane closed curve continuously along the curve to the same point again, an even number of intersections must be passed through. Hence, if we suppose the curve to be constructed of cord or wire, and restrict the crossings to double points, we may arrange them throughout so that, in following the wire continuously, it goes alternately over and under each branch it meets. When this is done it is obviously as completely knotted as its scheme (defined below) will admit of, and except in a special class of cases cannot have the number of crossings reduced by any possible deformation.

scholarly journals Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

2021 ◽  
Author(s):  
Bin Liu ◽  
Jouni Rättyä ◽  
Fanglei Wu
Keyword(s):  
Bergman Space ◽  
Open Problem ◽  
Lebesgue Space ◽  
Composition Operators ◽  
Bergman Spaces ◽  
Weighted Bergman Space ◽  
Carleson Measures ◽  
Doubling Weights ◽  
The Way

AbstractBounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.

On an area-preserving inverse curvature flow of convex closed plane curves

2021 ◽  
Vol 280 (8) ◽  
pp. 108931
Author(s):  
Laiyuan Gao ◽  
Shengliang Pan ◽  
Dong-Ho Tsai
Keyword(s):  
Curvature Flow ◽  
Plane Curves ◽  
Area Preserving ◽  
Closed Plane

The characterization of algebraic plane curves

1947 ◽  
Vol 14 (4) ◽  
pp. 837-853
Author(s):  
Th. Motzkin ◽  
A. Robinson
Keyword(s):  
Plane Curves

Parity conditions for realizability of Gauss diagrams

2019 ◽  
Vol 28 (01) ◽  
pp. 1950015
Author(s):  
Oleg N. Biryukov
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Plane Curves ◽  
Double Points ◽  
Gauss Diagram ◽  
Necessary And Sufficient ◽  
Closed Plane

We consider a problem of realizability of Gauss diagrams by closed plane curves where the plane curves have only double points of transversal self-intersection. We formulate the necessary and sufficient conditions for realizability. These conditions are based only on the parity of double and triple intersections of the chords in the Gauss diagram.

PLANE CURVES IN AN IMMERSED GRAPH IN ℝ2

2013 ◽  
Vol 22 (02) ◽  
pp. 1350003
Author(s):  
MARISA SAKAMOTO ◽  
KOUKI TANIYAMA
Keyword(s):  
Complete Graph ◽  
Closed Curve ◽  
Chord Diagram ◽  
Plane Curves ◽  
Generic Immersion ◽  
The Given

For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane-closed curve whose chord diagram contains the given chord diagram as a sub-chord diagram. For any generic immersion of the complete graph on six vertices to the plane, the sum of averaged invariants of all Hamiltonian plane curves in it is congruent to one quarter modulo one-half.

scholarly journals Contracting convex immersed closed plane curves with slow speed of curvature

2012 ◽  
Vol 364 (11) ◽  
pp. 5735-5763 ◽  
Author(s):  
Yu-Chu Lin ◽  
Chi-Cheung Poon ◽  
Dong-Ho Tsai
Keyword(s):  
Plane Curves ◽  
Slow Speed ◽  
Closed Plane
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