Automatic Deduction in Dynamic Geometry using Sage

We present a symbolic tool that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction. The main prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction. In one worksheet, diagrams constructed with the open source dynamic geometry system GeoGebra are accepted. In this worksheet, Groebner bases are used to either compute the equation of a geometric locus in the case of a locus construction or to determine the truth of a general geometric statement included in the GeoGebra construction as a boolean variable. In the second worksheet, locus constructions coded using the common file format for dynamic geometry developed by the Intergeo project are accepted for computation. The prototype and several examples are provided for testing. Moreover, a third Sage worksheet is presented in which a novel algorithm to eliminate extraneous parts in symbolically computed loci has been implemented. The algorithm, based on a recent work on the Groebner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Detailed examples are discussed.Comment: In Proceedings THedu'11, arXiv:1202.453

Some issues on the automatic computation of plane envelopes in interactive environments

This paper addresses some concerns, and describes some proposals, on the elusive concept of envelope of an algebraic family of varieties, and on its automatic computation. We describe how to use the recently developed Gröbner Cover algorithm to study envelopes of families of algebraic curves, and we give a protocol toward its implementation in dynamic geometry environments. The proposal is illustrated through some examples. A beta version of GeoGebra is used to highlight new envelope abilities in interactive environments, and limitations of our approach are discussed, since the computations are performed in an algebraically closed field

A Parametric Approach to 3D Dynamic Geometry

Dynamic geometry systems are computer applications allowing the exact on-screen drawing of geometric diagrams and their interactive manipulation by mouse dragging. Whereas there exists an extensive list of 2D dynamic geometry environments, the number of 3D systems is reduced. Most of them, both in 2D and 3D, share a common approach, using numerical data to manage geometric knowledge and elementary methods to compute derived objects. This paper deals with a parametric approach for automatic management of 3D Euclidean constructions. An open source library, implementing the core functions in a 3D dynamic geometry system, is described here. The library deals with constructions by using symbolic parameters, thus enabling a full algebraic knowledge about objects such as loci and envelopes. This parametric approach is also a prerequisite for performing automatic proof. Basic functions are defined for symbolically checking the truth of statements. Using recent results from the theory of parametric polynomial systems solving, the bottleneck in the automatic determination of geometric loci and envelopes is solved. As far as we know, there is no comparable library in the 3D case, except the paramGeo3D library (designed for computing equations of simple 3D geometric objects, which, however, lacks specific functions for finding loci and envelopes)

La evolución del concepto Influencer y su repercusión en los eventos de moda. Tendencias de la comunicación y la creatividad en el S.XXI

Actualmente el fenómeno de los influencers está en auge. Las marcas recurren cada vez más a ellos para dar visibilidad o ampliar el público de determinados eventos en la industria de la moda, en la que centraremos nuestro estudio. Analizaremos el nacimiento del concepto, así como su evolución y su utilidad a nivel empresarial y, más en concreto, en los eventos de moda.Grado en Publicidad y Relaciones Pública