Introducing conics without eccentricity

This note provides a self-contained introduction to conics as loci of points equidistant from circles, lines and points, including a study of the loci of points equidistant from two circles, separated, intersecting or touching.     

Efficient solution of rational conics

We present efficient algorithms for solving Legendre equations over (equivalently, for finding rational points on rational conics) and parametrizing all solutions. Unlike existing algorithms, no integer factorization is required, provided that the prime factors of the discriminant are known.     

On the six conics problem

There do not exist six points in the plane and six distinct congruent hyperbolas, each containing exactly five of the points. The methods are elementary. Part of the discussion includes determining how two congruent hyperbolas may intersect in four points. When combined with a result of Seidel and van Vollenhoven for ellipses, the above solves the six conics problem in the negative.     

Inherited conics in Hall planes

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of $\mathrm{PG}(2,q)$ remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre–Korchmáros on Desargues configurations with perspective triangles inscribed in a conic.     

圆锥曲线的三次有理多项式参数化

圆锥曲线重新参数化可以提高曲线参数的均匀性,且增强在拼接点处的光滑性.常用的参数化方法是采用一次有理多项式或二次有理多项式.采用三次有理多项式对圆锥曲线重新参数化,使曲线的次数由二次升到六次.以圆弧为例所得的实验结果袁明,在两段圆弧的公共点处的连续性为C~3,而且三次有理多项式参数化与弧长参数化的弦长偏差相比二次有理多项式参数化减小两个数量级.     

Rays of light trapped by conics

In a previous note [2], some geometric constructions were shown which can be interpreted as a ray of light trapped in a polygon by successive reflections on the sides. The same question is addressed here, replacing polygons by conics or pairs of conics. In some cases, the light is concentrated towards a specific segment. The tools used for the computations are elementary geometry, calculus and, in some cases, Commutative Algebra Software.     

Rational cubic/quartic Said-Ball conics

In CAGD, the Said-Ball representation for a polynomial curve has two advantages over the Bézier representation, since the degrees of Said-Ball basis are distributed in a step type. One advantage is that the recursive algorithm of Said-Ball curve for evaluating a polynomial curve runs twice as fast as the de Casteljau algorithm of Bézier curve. Another is that the operations of degree elevation and reduction for a polynomial curve in Said-Ball form are simpler and faster than in Bézier form. However, Said-Ball curve can not exactly represent conics which are usually used in aircraft and machine element design. To further extend the utilization of Said-Ball curve, this paper deduces the representation theory of rational cubic and quartic Said-Ball conics, according to the necessary and sufficient conditions for conic representation in rational low degree Bézier form and the transformation formula from Bernstein basis to Said-Ball basis. The results include the judging method for whether a rational quartic Said-Ball curve is a conic section and design method for presenting a given conic section in rational quartic Said-Ball form. Many experimental curves are given for confirming that our approaches are correct and effective.     

Short cycles of Poncelet’s conics

Main results of this paper are the following:1. A closed N-gon interscribed between two conics exists if and only if a specially constructed polygon with a smaller number of sides (n) is closed. To verify the closure of this n-gon, we need to find a periodic solution of a dynamical system of order n. The proof is based on the connection of Poncelet’s curves and matrices that admit unitary bordering [4,9,10,16]. Application of this criterion makes sense when n?N, in particular when n≈log₂N (see Table 4 where n=m₁). So for example we may say that a polygon with 2049 sides interscribed between two circles is closed if and only if some specially constructed 11-gon is closed.2. A closed N-gon interscribed between two confocal ellipses (the billiard case) exists if and only if an N-gon interscribed between two special nested circles is closed.     

3-Webs generated by confocal conics and circles

We consider families of confocal conics and two pencils of Apollonian circles having the same foci. We will show that these families of curves generate trivial 3-webs and find the exact formulas describing them.     

800 conics on a smooth quartic surface

We construct an example of a smooth spatial quartic surface that contains 800 irreducible conics.     

Characterization of complete exterior sets of conics

Let be a set of exterior points of a nondegenerate conic inPG(2,q) with the property that the line joining any 2 points in misses the conic. Ifq1 (mod 4) then consists of the exterior points on a passant, ifq3 (mod 4) then other examples exist (at least forq=7, 11, ..., 31).Support from the Dutch organization for scientific Research (NWO) is gratefully acknowledged     

A Note on Degenerate and Spectrally Degenerate Graphs

A graph G is called spectrally d‐degenerate if the largest eigenvalue of each subgraph of it with maximum degree D is at most . We prove that for every constant M there is a graph with minimum degree M, which is spectrally 50‐degenerate. This settles a problem of Dvo?ák and Mohar (Spectrally degenerate graphs: Hereditary case, arXiv: 1010.3367).