Miguel A. Abánades

An algebraic taxonomy for locus computation in dynamic geometry

Abstract The automatic determination of geometric loci is an important issue in Dynamic Geometry. In Dynamic Geometry systems, it is often the case that locus determination is purely graphical, producing an output that is not robust enough and not reusable by the given software. Parts of the true locus may be missing, and extraneous objects can be appended to it as side products of the locus determination process. In this paper, we propose a new method for the computation, in dynamic geometry, of a locus defined by algebraic conditions. It provides an analytic, exact description of the sought locus, making possible a subsequent precise manipulation of this object by the system. Moreover, a complete taxonomy, cataloging the potentially different kinds of geometric objects arising from the locus computation procedure, is introduced, allowing to easily discriminate these objects as either extraneous or as pertaining to the sought locus. Our technique takes profit of the recently developed GrobnerCover algorithm. The taxonomy introduced can be generalized to higher dimensions, but we focus on 2-dimensional loci for classical reasons. The proposed method is illustrated through a web-based application prototype, showing that it has reached enough maturity as to be considered a practical option to be included in the next generation of dynamic geometry environments.

Exact internet accessible computation of paths of points in planar linkages and diagrams

Dynamic Geometry, also known as Interactive Geometry, refers to computer programs where accurate construction of (generally) planar drawings can be made. The key characteristic of this software is that, when dragging certain elements of the configuration, the geometric properties of the construction are preserved. In this paper, we describe an educational web‐based application that complements standard dynamic geometry programs in a mathematically sound manner. We put the focus on computing the geometric locus of distinguished points in linkages and other geometric constrained configurations, since knowing the equations of such loci is a typical engineering task. The tool is located at http://nash.sip.ucm.es/LAD/LADucation.html.

Adding remote computational capabilities to Dynamic Geometry Systems

A Dynamic Geometry System (DGS) is a computer application that allows the exact drawing and dynamic manipulation of geometric constructions. DGS have been the paradigm of new technologies applied to Math education, but some authors have claimed that some symbolic capabilities should be added to this systems. We present an example of communication between the commercial DGS Cabri, The Geometers Sketchpad and Cinderella and two Computer Algebra Systems (CAS), Mathematica and CoCoA. The tool is a web application designed to symbolically process locus, proof and discovery tasks on geometric diagrams. Named LAD (Locus-Assertion-Discovery), it is a remote add-on for the three DGS. LAD is a prototype oriented to research users. We also describe LADucation, a one-click educational version of LAD. By just uploading the file generated by the considered DGS, graphs and equations of geometric loci are computed.

Using a free open source software to teach mathematics

We present the experience of the authors teaching mathematics to freshmen engineering students with the help of the open source computer algebra system Sage. We describe some teaching resources and present an ad hoc distribution of Sage used by the authors.© 2012 Wiley Periodicals, Inc. Comput Appl Eng Educ 22:728–735, 2014; View this article online at wileyonlinelibrary.com/journal/cae; DOI 10.1002/cae.21565

Computing Locus Equations for Standard Dynamic Geometry Environments

GLI (Geometric Locus Identifier), an open web-based tool to determine equations of geometric loci specified using Cabri Geometryand The Geometers Sketchpad, is described. A geometric construction of a locus is uploaded to a Java Servlet server, where two computer algebra systems, CoCoA and Mathematica, following the Groebner basis method, compute the locus equation and its graph. Moreover, an OpenMath description of the geometric construction is given. GLI can be efficiently used in mathematics education, as a supplement of the locus functions of the standard dynamic geometry systems. The system is located at http://nash.sip.ucm.es/GLI/GLI.html .

Automatic deduction in (dynamic) geometry: Loci computation

A symbolic tool based on open source software that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction is presented. The 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, namely with the open source dynamic geometry system GeoGebra or using the common file format for dynamic geometry developed by the Intergeo project. Locus computation algorithms based on Automatic Deduction techniques are recalled and presented as basic for an efficient treatment of advanced methods in dynamic geometry. Moreover, an algorithm to eliminate extraneous parts in symbolically computed loci is discussed. The algorithm, based on a recent work on the Grobner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Several examples are shown in detail.

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.

On a generalized name entity recognizer based on Hidden Markov Models

This paper presents a Named Entity Recognition (NER) system based on Hidden Markov Models. The system design is language independent, and the target language and scope of the NER is determined by the training corpus. The NER is formed by two subsystems that detect and label the entities independently. Each subsystem implements a different approach of that statistical theory, showing that each component may complement the results of the other one. Unlike most of the previous works, two labels are returned when the components provide different results. This redundancy is an advantage when human supervision is mandatory at the end of the process such as in intelligence environments.

First Steps on Using OpenMath to Add Proving Capabilities to Standard Dynamic Geometry Systems

A prototype for a web application designed to symbolically process locus, proof and discovery tasks on geometric diagrams created with the commercial dynamic geometry systems Cabri, The Geometers Sketchpadand Cinderellais presented. The application, named LAD (acronym for Locus-Assertion-Discovery) and thought of as a remote add-onfor the considered DGS, follows the Groebner basis method relying on CoCoA and a Mathematica kernel for the involved symbolic computations. From the DGS internal textual representation of a geometric diagram, an OpenMath (i.e. semantic based) description of the requested task is created using the elements in the plangeoOpenMath content dictionaries. A review of the elements included in these CDs is given and two new elements proposed, namely locusand discovery. Everything is finally thoroughly illustrated with examples. LAD is freely accessible at http://nash.sip.ucm.es/LAD/LAD.html .

A dynamic symbolic geometry environment based on the GröbnerCover algorithm for the computation of geometric loci and envelopes

An enhancement of the dynamic geometry system GeoGebra for the automatic symbolic computation of algebraic loci and envelopes is presented. Given a GeoGebra construction, the prototype, after rewriting the construction as a polynomial system in terms of variables and parameters, uses an implementation of the recent GrobnerCover algorithm to obtain the algebraic description of the sought locus/envelope as a locally closed set. The prototype shows the applicability of these techniques in general purpose dynamic geometry systems.