Related papers: Where is the cone?

The conic sections, as well as the solids obtained by revolving these curves, and many of their surprising properties, were already studied by Greek mathematicians since at least the fourth century B.C. Some of these properties come to the…

We review, from a didactic point of view, the definition of a toric section and the different shapes it can take. We'll then discuss some properties of this curve, investigate its analogies and differences with the most renowned conic…

This is a paper about triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also…

The method of application of areas as presented in Euclid's Elements, is employed to generate the three conics as the loci of points with Cartesian coordinates satisfying quadratic equations with coefficients defined by the initial settings…

The main theorem of "S. J. Kov\'acs: The cone of curves of a K3 surface, Math. Ann. 300 (1994), no. 4, 681-691" is proved in arbitrary characteristic. The proof is essentially the same as in the original paper where it was stated only over…

An expository description of smooth cubic curves in the real or complex projective plane.

This paper first gives a brief overview over some interesting descriptions of conic sections, showing formulations in the three geometric algebras of Euclidean spaces, projective spaces, and the conformal model of Euclidean space. Second…

After having investigated the real conic sections and their isoptic curves in the hyperbolic plane $\bH^2$ we consider the problem of the isoptic curves of generalized conic sections in the extended hyperbolic plane. This topic is widely…

An efficient way to get implicit equations of conics on five points and quadrics on nine, using pencils of conics and quadrics, is revealed. Parallel axis right cones intersect on a conic. An example, to show how to place five coplanar…

A classical result of von Staudt states that if eight planes osculate a twisted cubic curve and we divide them into two groups of four, then the eight vertices of the corresponding tetrahedra lie on a twisted cubic curve. In the current…

There are two problems Analytical Geometry with facing anyone who studies this discipline: define the nature of the locus represented by the general equation 2do degree in two or three variables: That curve represents the plane? What…

Conics in the Euclidean space have been known for their geometrical beauty and also for their power to model several phenomena in real life. It usually happens that when thinking about the conics in a semi-Riemannian manifold, the equations…

Omar Khayyam's treatment of cubic equations by intersections of conic sections has often been read as an anticipation of analytic or coordinate geometry. This paper argues that such a reading obscures the conceptual structure of Khayyam's…

In this article, we study the geometry of plane curves obtained by three sections and another section given as their sum on certain rational elliptic surfaces. We make use of Mumford representations of semi-reduced divisors in order to…

Addressing a question of M. Stillman, it had been shown by Ein, Eisenbud, and the author that in a projective space of dimension at most 5, every arithmetically Cohen-Macaulay curve which is cut out by quadrics scheme- theoretically also…

In the paper we consider convex cones in infinite-dimensional real vector spaces which are endowed with no topology. The main purpose is to study an internal geometric structure of convex cones and to obtain an analytical description of…

In classical Euclidean geometry, there are several equivalent definitions of conic sections. We show that in the hyperbolic plane, the analogues of these same definitions still make sense, but are no longer equivalent, and we discuss the…

This paper gives a complete classification of conics in $PE_2(\mathbb{R})$. The classification has been made earlier (Reveruk [5]), but it showed to be incomplete and not possible to cite and use in further studies of properties of conics,…

The notion of rectifying curve in the Euclidean space is introduced by Chen as a curve whose position vector always lies in its rectifying plane spanned by the tangent and the binormal vector field t and n_2 of the curve. In this study, we…

We introduce the class of rational plane curves parameterizable by conics as an extension of the family of curves parameterizable by lines (also known as monoid curves). We show that they are the image of monoid curves via suitable…