Conics In Space Research Articles

Dimension of the space of conics on a Fano hypersurface

R. Beheshti showed that for a smooth Fano hypersurface X of degree ⩽8 over the complex number field C, the dimension of the space of lines lying in X is equal to the expected dimension. We study the space of conics on X. In this case, if X contains some linear subvariety, then the dimension of the space can be larger than the expected dimension. In this paper, we show that for a smooth Fano hypersurface X of degree ⩽6 over C, and for an irreducible component R of the space of conics lying in X, if the 2-plane spanned by a general conic of R is not contained in X, then the dimension of R is equal to the expected dimension.

An example of the geometry of a 5th-order ODE: The metric on the space of conics in [formula omitted

As an application of the method of [4], we find the metric and connection on the space of conics in CP2 determined as the solution space of the ODE (1). These calculations underpin the twistor construction of the Radon transform on conics in CP2 described in [5]. Two further examples of the method are provided.

Mori's Program for the Moduli Space of Conics in Grassmannian

We complete Mori's program for Kontsevich's moduli space of degree $2$ stable maps to the Grassmannian of lines. We describe all birational models in terms of moduli spaces (of curves and sheaves), incidence varieties, and Kirwan's partial desingularization.

On the Space of Conics on Complete Intersections

We get sharp degree bound for generic smoothness and connectedness of the space of lines and conics in low degree complete intersections which generalizes the old work about Fano scheme of lines on hypersurfaces. As a consequence, we prove that for a Fano complete intersection $$X$$ with index $$\ge 2$$ , the $$1$$ -Griffiths group generated by algebraic $$1$$ -cycles homologous to $$0$$ modulo algebraic equivalence is trivial, which is a conjecture for general rationally connected varieties.

Fitting Quadratic Curves to Data Points

Fitting quadratic curves (a.k.a. conic sections, or conics) to data points (digitized images) is a fundamental task in image processing and computer vision. This problem reduces to minimization of a certain function over the parameter space of conics. Here we undertake a thorough investigation of that space and the properties of the objective function on it. We determine under what conditions that function is continuous and differentiable. We identify its discontinuities and other singularities and determined what effect those have on the performance of minimization algorithms. Our analysis shows that algebraic parameters of conics are more suitable for minimization procedures than more popular geometric parameters, for a number of reasons. First, the space of parameters is naturally compact, thus their estimated values cannot grow indefinitely causing divergence. Second, with algebraic parameters minimization procedures can move freely and smoothly between conics of different types allowing shortcuts and faster convergence. Third, with algebraic parameters one avoids known issues occurring when the fitting conic becomes a circle. To support our conclusions we prove a dozen of mathematical theorems and provide a plenty of illustrations.

Spaces of rational curves on complete intersections

AbstractWe prove that the space of smooth rational curves of degree $e$ on a general complete intersection of multidegree $(d_1, \ldots , d_m)$ in $\mathbb {P}^n$ is irreducible of the expected dimension if $\sum _{i=1}^m d_i \lt (2n+m+1)/3$ and $n$ is sufficiently large. This generalizes a result of Harris, Roth and Starr [Rational curves on hypersurfaces of low degree, J. Reine Angew. Math. 571 (2004), 73–106], and is achieved by proving that the space of conics passing through any point of a general complete intersection has constant dimension if $\sum _{i=1}^m d_i$ is small compared to $n$.

Path splines with conic envelopes

We look at path splines as constructed by joining tangent continuously segments of envelopes of 1-parameter families of conics. Each envelope is an algebraic curve of degree four or less. A path spline interpolates a sequence of point/tangent pairs and for each segment there is an additional shape handle which allows for the dilation or contraction of that segment of the path spline without altering the rest of it. The mathematical construction of the spline reduces to defining a Bézier curve in higher dimensional space: the space of conics. The cubic hypersurface corresponding to the degenerate conics plays a role in the construction.

A natural smooth compactification of the space of elliptic curves in projective space

The space of smooth genus- 0 0 curves in projective space has a natural smooth compactification: the moduli space of stable maps, which may be seen as the generalization of the classical space of complete conics. In arbitrary genus, no such natural smooth model is expected, as the space satisfies “Murphy’s Law”. In genus 1 1 , however, the situation remains beautiful. We give a natural smooth compactification of the space of elliptic curves in projective space, and describe some of its properties. This space is a blowup of the space of stable maps. It can be interpreted as a result of blowing up the most singular locus first, then the next most singular, and so on, but with a twist—these loci are often entire components of the moduli space. We give a number of applications in enumerative geometry and Gromov-Witten theory. For example, this space is used by the second author to prove physicists’ predictions for genus-1 Gromov-Witten invariants of a quintic threefold. The proof that this construction indeed gives a desingularization will appear in a subsequent paper.

Tangent Conics at Quartic Surfaces and Conics in Quartic Double Solids

For a quartic double solid we study the parameter space of conics (i.e. of smooth rational curves C ⊂ Z such that C · φ*O(1) = 2). This parameter space is naturally fibred (with disconnected fibres) over Pˇ3. We study the monodromy of the fibres and determine this way the irreducible components of the parameter space.

Duality for Curves in Affine Plane Geometry

The dual of a plane curve in equiaffine geometry is defined as a curve in the space of conics. The image and inverse of this duality are described. It is shown that the duality transforms equiaffine vertices into cusps. As an application, an analogue of Kneser's lemma for osculating conics is proved.

Wonderful varieties of rank two

LetG be a complex connected reductive group. Well known wonderfulG-varieties are those of rank zero, namely the generalized flag varietiesG/P, those of rank one, classified in [A], and certain complete symmetric varieties described in [DP] such as the famous space of complete conics. Recently, there is a renewed interest in wonderful varieties of rank two since they were shown to hold a keystone position in the theory of spherical varieties, see [L], [BP], and [K].

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.

Algebraic and geometric invariant of a pair of non-coplanar conics in space

It is easy to construct the geometric invariant of a pair of non-coplanar conics in space. It is the crossratio of the 4 intersection points of the two conics with the common line of the two conic planes. In this paper, the algebraic invariant of a pair of non-coplanar conics is derived from the invariant algebra of a pair of quaternary quadratic forms by using the dual representation of space conics. Then, the relationship between the algebraic invariant and the geometric invariant is established.

Conics-based stereo, motion estimation, and pose determination

Stereo vision, motion and structure parameter estimation, and pose determination are three important problems in 3-D computer vision. The first step in all of these problems is to choose and to extract primitives and their features in images. In most of the previous work, people usually use edge points or straight line segments as primitives and their local properties as features. Few methods have been presented in the literature using more compact primitives and their global features. This article presents an approach using conics as primitives. For stereo vision, a closed-form solution is provided for both establishing the correspondence of conics in images and the reconstruction of conics in space. With this method, the correspondence is uniquely determined and the reconstruction is global. It is shown that the method can be extended for higher degree (degree≥3) planar curves.For motion and structure parameter estimation, it is shown that, in general, two sequential images of at least three conics are needed in order to determine the camera motion. A complicated nonlinear system must be solved in this case. In particular, if we are given two images of a pair of coplanar conics, a closed-form solution of camera motion is presented. In a CAD-based vision system, the object models are available, and this makes it possible to recognize 3-D objects and to determine their poses from a single image.For pose determination, it is shown that if there exist two conics on the surface of an object, the object's pose can be determined by an efficient one-dimensional search. In particular, if two conics are coplanar, a closed-form solution of the object's pose is presented.

Analytic Systems of Central Conics in Space

Analytic Systems of Central Conics in Space

Analytic systems of central conics in space

The amount of literature dealing with conic sections, individual curves and systems of curves, in one plane, is vast. When however we are dealing with a number of conics, not in the same plane, the situation is quite different. Certain figures, as the focal conics of a set of confocal quadrics, are familiar enough, but very little has been done in the way of a systematic study of more general systems. There are some studies carried out with the aid of purely synthetic methods; the algebraic or analytic treatment lags behind. The first writer to suggest a reasonable set of coordinates for a conic in space was Spottiswoode(l). The totality of straight lines that intersect a conic in three-space generates a very special sort of quadratic complex. The coefficients determining the equation of this complex, when a straight line has the usual Plucker line coordinates, may be taken as the coordinates of the conic, a clumsy enough system. A much better technique, perhaps the best for algebraic purposes, was developed by Johnson(2). Here a conic is looked upon not as a locus, but as the envelop of its tangent planes. Thus its tangential equation aiiuiuuO =0 gives ten homogeneous coordinates, connected by a quartic identity aii = aii, aiil = 0 i, j = 1, 2, 3, 4.

Conics in Space, and their Representation by Points in Space of Nineteen Dimensions

Conics in Space, and their Representation by Points in Space of Nineteen Dimensions

Characteristics of complexes of conics in space of four dimensions

In this note we obtain a system of characteristics on which depend the main enumerative properties of complexes, or systems ∞3, of conics in space of four dimensions. The method used is applicable to a large number of problems of this nature, and we select this illustration as being analogous to congruences in ordinary space. In particular a finite number of conics pass through an arbitrary point, thereby defining the order, n, of the system. Linear complexes are those for which this order is unity. The conics of a complex satisfy eight simple conditions, in general conditions of contact with a fixed form or forms, but they may include conditions of incidence with a surface, each such counting once, with a curve, each such counting twice, or with fixed points each such counting thrice. In particular only incidence conditions can occur when the system is linear, for otherwise more than one conic would pass through certain ∞3 points of space. Points through which more than a finite number of conics pass are termed singular, as well as their loci. Directrix constructs are necessarily singular, but they do not necessarily exhaust the singular system, for the complex may possess ∞2 curves lying on a surface. In the linear case through a point of a singular curve pass ∞2 conics, and through a point of a singular surface ∞1 conics. The possibilities in the nonlinear cases are too numerous to be detailed. Similarly the system of planes of the conies may possess a singular curve, through which ∞2 of the planes pass.

Some Complexes of Conics in Space of Three Dimensions

Some Complexes of Conics in Space of Three Dimensions

Complexes of Conics and the Weddle Surface

A complex or system ∞3 of conics in space of four dimensions is such that a finite number of conics pass through an arbitrary point. Linear complexes are those for which this number is unity, and are such that their curves are defined by conditions of incidence with fixed surfaces, curves and points. In this paper are discussed briefly the linear complexes defined by the condition that their curves meet an irreducible curve in four points. Denoting by a curve of order m and genus p it is found that the curves in question are The complex associated with is considered in greater detail, since it is found to have an interesting connection with the well-known Weddle quartic surface of ordinary space. In fact the conics of the system touching a space (of three dimensions) do so in the points of such a surface. The main properties of this surface can be thence deduced. In addition we discuss certain results in connection with this curve . The paper closes with certain enumerative results which were obtained in the course of the researches giving the results recorded and which we believe are worth record.