Pair Of Conics Research Articles

On the disposition of cubic and pair of conics in a real projective plane. II

On the disposition of cubic and pair of conics in a real projective plane. II

In Brief

This article introduces several efficient and easy-to-use tools to analyze the intersection curve between two quadrics, on the basis of the study of its projection on a plane (the so-called cutcurve) to perform the corresponding lifting correctly. This approach is based on an efficient way of determining the topology of the cutcurve through only solving one degree eight (at most) univariate equation and several quadratic univariate equations, intersecting two pairs of conics and, when the parameterization of the cutcurve in closed form cannot be determined, computing the real roots of several degree four univariate squarefree polynomials whose number (of real roots) is known in advance.

Tools for analyzing the intersection curve between two quadrics through projection and lifting

This article introduces several efficient and easy-to-use tools to analyze the intersection curve between two quadrics, on the basis of the study of its projection on a plane (the so-called cutcurve) to perform the corresponding lifting correctly. This approach is based on an efficient way of determining the topology of the cutcurve through only solving one degree eight (at most) univariate equation and several quadratic univariate equations, intersecting two pairs of conics and, when the parameterization of the cutcurve in closed form cannot be determined, computing the real roots of several degree four univariate squarefree polynomials whose number (of real roots) is known in advance.

Взаимные задачи с кониками

The problem for construction of straight lines, which are tangent to conics, is among the dual problems for constructing the common elements of two conics. For example, the problem for construction of a chordal straight line (a common chord for two conics) ~ the problem for construction of an intersection point for two conics’ common tangents. In this paper a new property of polar lines has been presented, constructive connection between polar lines and chordal straight lines has been indicated, and a new way for construction of two conics’ common chords has been given, taking into account the computer graphics possibilities. The construction of imaginary tangent lines to conic, traced from conic’s interior point, as well as the construction of common imaginary tangent lines to two conics, of which one lies inside another partially or thoroughly is considered. As you know, dual problems with two conics can be solved by converting them into two circles, followed by a reverse transition from the circles to the original conics. This method of solution provided some clarity in understanding the solution result. The procedure for transition from two conics to two circles then became itself the subject of research. As and when the methods for solving geometric problems is improved, the problems themselves are become more complex. When assuming the participation of imaginary images in complex geometry, it is necessary to abstract more and more. In this case, the perception of the obtained result’s geometric picture is exposed to difficulties. In this regard, the solution methods’ correctness and imaginary images’ visualization are becoming relevant. The paper’s main results have been illustrated by the example of the same pair of conics: a parabola and a circle. Other pairs of affine different conics (ellipse and hyperbola) have been considered in the paper as well in order to demonstrate the general properties of conics, appearing in investigated operations. Has been used a model of complex figures, incorporating two superimposed planes: the Euclidean plane for real figures, and the pseudo-Euclidean plane for imaginary algebraic figures and their imaginary complements.

On the disposition of cubic and pair of conics in a real projective plane

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.

Algebro-Geometric Approach to an Okamoto Transformation, the Painlevé VI and Schlesinger Equations

A new method of constructing algebro-geometric solutions of rank two four-point Schlesinger system is presented. For an elliptic curve represented as a ramified double covering of $$\mathbb {CP}^1$$ , a meromorphic differential is constructed with the following property: The common projection of its two zeros on the base of the covering, regarded as a function of the only moving branch point of the covering, is a solution of a Painleve VI equation. This differential provides an invariant formulation of a classical Okamoto transformation for the Painleve VI equations. The corresponding solution of the rank two Schlesinger system associated with a family of elliptic curves is constructed in terms of this differential. The initial data for construction of the meromorphic differential include a point in the Jacobian of the curve, under the assumption that this point has non-variable coordinates with respect to the lattice of the Jacobian while the branch points vary. It appears that the cases where the coordinates of the point are rational correspond to the Poncelet polygons inscribed and circumscribed in a pair of conics. Thus, this is a generalization of a situation studied by Hitchin, who related algebraic solutions of a Painleve VI equation with the Poncelet polygons.

A novel geometric invariant of a pair of conics and its application to eyeglasses-tracking interface

This paper deals with a three-dimensional (3-D) pose estimation problem of a pair of two conics. We first introduce a novel geometric invariant derived from a pair of conics, and then propose a procedure of determining a set of three invariants when two independent conics are given. As a solution of P2C (perspective-two-conics) problem, this paper also provides a pose estimation algorithm that employs an iterative procedure, which aims to optimize the orientation parameters by involving the shape information of conics. The proposed method is verified through a series of experiments for analyzing pose estimation accuracy and precision, being compared with some general approaches. As an application example, we present an eyeglasses-tracking interface, and discuss its performance through some experiment results.

Optimizing the structural parameters of conical pairs with circular teeth

In the present work, we propose a method of opti� mizing the parameters of a conical pair with round teeth. This method may be used in the design of new gear systems or in the modernization of those that already exist. In both cases, the initial data for the design of the transmission in the system are its gain, its dimensions, and the load transmitted. The only differ� ence is that, in the second case, the pair must fit within a preexisting housing, as a rule; this imposes con� straints on its size. The variables in optimization are the helical angle β and the change in theoretical tooth thickness xτ. Thus, in optimization, the following characteristics remain unchanged: the mechanical properties of the gear materials; the number of teeth z 1 and z 2 in the gears; the width b of the tooth crown; the angles δa and δf of the cones of the tooth vertices and the troughs

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.

Conic reconstruction and correspondence from two views

Conics are widely accepted as one of the most fundamental image features together with points and line segments. The problem of space reconstruction and correspondence of two conics from two views is addressed in this paper. It is shown that there are two independent polynomial conditions on the corresponding pair of conics across two views, given the relative orientation of the two views. These two correspondence conditions are derived algebraically and one of them is shown to be fundamental in establishing the correspondences of conics. A unified closed-form solution is also developed for both projective reconstruction of conics in space from two uncalibrated camera views and metric reconstruction from two calibrated camera views. Experiments are conducted to demonstrate the discriminality of the correspondence conditions and the accuracy and stability of the reconstruction both for simulated and real images.

Poncelet 5-gons and abelian surfaces

We study the relations between Poncelet 5-gons, abelian surfaces with real multiplication and the Hilbert modular surfaceY(5) for the number field\(\mathbb{Q}\left( {\sqrt 5 } \right)\). These objects are linked by the construction of Kummer surfaces as double convers of the projective plane. Constructing a map from the moduli space of Poncelet 5-gons toY(5), we get a new proof for the rationality ofY(5). As a corollary we get a theorem of plane projective geometry (due to Humbert) describing the combinatorial symmetries of Poncelet pairs of conics with a Poncelet 5-gon and a bitangent.