N. Alamo

Optical properties of conics: a method for obtaining reflecting and focusing profiles

An original method is proposed for determining suitable profiles for two different problems: to produce a given family of wave fronts from a single point source after reflecting on it, and to focus the given wave front on a fixed point. The method uses the optical properties of conics to obtain both reflecting and focusing profiles as the envelope of a specific family of conic sections. It also offers a procedure for determining the caustic of the wave front.

Generalized antiorthotomics and their singularities

In this paper we construct, for a given wavefront and a source point, an uniparametric family of mirrors with the characteristic property of producing the given wavefront, after reflecting on each mirror, from the source point. This family is made up of the envelopes of certain families of hyperquadrics of revolution, and includes the antiorthotomic of the wavefront with respect to the source point. We prove that the singularities of these generalized antiorthotomics sweep out the caustic of the wavefront. We also give a method to determine the generalized antiorthotomics starting from the associated caustic.

A link between the bounds on relativistic velocities and areas of hyperbolic triangles

In this paper we show the mathematical equivalence between two well-known facts: the existence of an upper bound for the area in Lobachevskian geometry and the existence of a limit for relativistic velocities. The key point is that the space of relativistic velocities can be interpreted as a Lobachevskian space.

A method to construct refracting profiles

We propose an original method for determining suitable refracting profiles between two media to solve two related problems: to produce a given wavefront from a single point source after refraction at the refracting profile, and to focus a given wavefront at a fixed point. These profiles are obtained as envelopes of specific families of Cartesian ovals. We study the singularities of these profiles and give a method to construct them from the data of the associated caustic.

Reflecting profiles and generalized holodiagrams

Summary The concepts of reflecting profiles and that of holodiagrams are examined and links between them are found so that the classical Abramsons Holodiagram idea can be generalized. The broader concept of Generalized Holodiagrams encompasses then the possibility of using different wave front shapes, not only spherical. Many of the nice properties of the classical holodiagram are shared by its generalized versions.

Decision and classification problems using Mahalanobis statistical distance

We present a method that uses Mahalanobis distance to study some decision and classification problems. Four examples are used to show the diversity of possible applications of our method. Based on two simulated sets of data, we have calculated the corresponding Mahalanobis distances and their sum and differences Holodiagrams. These diagrams are valuable visualization tools for studying decision and classification problems in the proposed applications.

Refracting profiles and generalized holodiagrams

The recently developed concept of refracting profiles and that of refraction holodiagrams are combined so that the classical Abramson holodiagrams can be generalized taking into account a wider class of wave fronts and refraction at an interface, whenever regions of caustics are avoided. These holodiagrams are obtained as envelopes of specific families of Cartesian Ovals with an appropriate parametrization. Classical and reflecting holodiagrams are particular cases of this class. Several of the properties of the classical holodiagrams are shared by their richer generalized versions.

The cosmological principle and honeycombs

We present the possibility that the gravitational growth of primordial density fluctuations leads to what can be considered a weak version of the cosmological principle. The large-scale mass distribution associated with this principle must have the geometrical structure known as a regular honeycomb. We give the most important parameters that characterize the honeycombs associated with �

Reflecting profiles that produce a given phase distribution of light

We give a method to construct, for a given surface and a point source, a one-parametric family of reflecting profiles, each of them with the characteristic property of producing a predetermined phase distribution of light that from the point source is incident on that surface after reflection at the profile. The profiles are constructed as the envelopes of specific families of ellipsoids of revolution. We also study the singularities of these profiles.

Optical Properties of the Conics Derived from an Elementary Kinematic Consideration

In a recent article appearing in this journal,1 the optical property of the ellipse is derived from energy conservation. In this paper we derive this property as well as the analogous for the other conics, i.e., hyperbola and parabola, from a very elemental kinematic consideration. The same argument can be used to prove the converse — that only the conics have these optical properties.