Francisco Botana

A dynamic-symbolic interface for geometric theorem discovery

This paper describes Discover, a program for learning and teaching geometry with the help of a computer. The program is a dynamic geometry environment that can communicate with Mathematica, using its symbolic capabilities to perform geometric discovery or rediscovery. Discover is specially suited to be used as a learning tool for geometry from the ages of 12 up to University. It permits the replacement of the traditional ruler and compass by electronic substitutes, as in standard dynamic geometry environments. Through its link with the computer algebra software, it enhances the process of conjecturing and proving. The results can be expressed in natural language or through the use of equations. The mathematical methods that Discover uses are sound, although not complete. Despite this last fact, almost all parts of the school curricula in plane geometry can be adequately treated with the program.

A software tool for the investigation of plane loci

We describe the capabilities of Lugares for obtaining the equations and graphs of plane loci. Lugares is a Windows program written in Prolog that offers a standard dynamic geometry environment and uses the symbolic capabilities of CoCoA, a computer algebra system specialized in Groebner basis computations, or Mathematica. The main novelty in Lugares consists of the link between the dynamic geometry paradigm and a symbolic approach to automatic discovery in geometry. Through this link Lugares determines as a locus almost any algebraic curve specified by geometric conditions.

Automated Theorem Proving in GeoGebra: Current Achievements

GeoGebra is an open-source educational mathematics software tool, with millions of users worldwide. It has a number of features (integration of computer algebra, dynamic geometry, spreadsheet, etc.), primarily focused on facilitating student experiments, and not on formal reasoning. Since including automated deduction tools in GeoGebra could bring a whole new range of teaching and learning scenarios, and since automated theorem proving and discovery in geometry has reached a rather mature stage, we embarked on a project of incorporating and testing a number of different automated provers for geometry in GeoGebra. In this paper, we present the current achievements and status of this project, and discuss various relevant challenges that this project raises in the educational, mathematical and software contexts. We will describe, first, the recent and forthcoming changes demanded by our project, regarding the implementation and the user interface of GeoGebra. Then we present our vision of the educational scenarios that could be supported by automated reasoning features, and how teachers and students could benefit from the present work. In fact, current performance of GeoGebra, extended with automated deduction tools, is already very promising—many complex theorems can be proved in less than 1 second. Thus, we believe that many new and exciting ways of using GeoGebra in the classroom are on their way.

Interactive versus Symbolic Approaches to Plane Loci Generation in Dynamic Geometry Environments

This paper reviews current approaches to plane loci generation within dynamic geometry environments. Such approaches are classified as interactive, when just a plot of the locus is shown, and symbolic, if, in addition to plotting the locus, its equation is also given. It is shown how symbolic approaches outperform the interactive ones when dealing with loci which are algebraic curves. Additionally, two experimental improvements are reported: i) an efficient computer algebra system allows symbolically generated loci to behave as dynamic objects, and ii) a general purpose computer algebra system is used to remove spurious parts of some loci.

Automatic determination of envelopes and other derived curves within a graphic environment

Dynamic geometry programs provide environments where accurate construction of geometric configurations can be done. Nevertheless, intrinsic limitations in their standard development technology mostly produce objects that are equationally unknown and so can not be further used in constructions. In this paper, we pursue the development of a geometric system that uses in the background the symbolic capabilities of two computer algebra systems, CoCoA and Mathematica. The cooperation between the geometric and symbolic modules of the software is illustrated by the computation of plane envelopes and other derived curves. These curves are described both graphically and analytically. Since the equations of these curves are known, the system allows the construction of new elements depending on them.

A web-based intelligent system for geometric discovery

An open web-based tool for automatic discovery in elementary Euclidean geometry, webDiscovery, is described. It is based in recent findings in automatic discovery in geometry. A user-defined geometric construction is uploaded to a Java Servlet server, where two computer algebra systems, CoCoA and Mathematica, return the discovered facts about the construction. webDiscovery can be efficiently used in mathematics education, linkage design and testing and computer aided geometric design. The system can be tested at rosalia.uvigo.es/sdge/web/2D.

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.

Where the Truth Lies (in Automatic Theorem Proving in Elementary Geometry)

In this paper we use a new integrated theorem prover (GDI), codeveloped by the second author, to discuss a geometric result due to Maclane, the 83 theorem, which has been declared to be true, according some authors, while other claim it is false. Our approach is based in Grobner bases computations and illustrates the controversial concept of truth in the algebraic automatic theorem proving model, as well as some of the new features provided by GDI. The potential applications to computer graphics of the idea behind these rather unique features, are also briefly discussed.

Towards solving the dynamic geometry bottleneck via a symbolic approach

The goal of this paper is to report on a prototype of a new dynamic geometry software, GDI (Geometria Dinamica Inteligente). We will describe how, apart from being a standard dynamic environment for elementary geometry, GDI addresses some key problems of the dynamic geometry paradigm, by including enhanced tools for loci generation and automatic proving, plus another distinguished feature, namely, a discovery option, allowing the user to find complementary hypotheses for arbitrary statements to become true. The key technique for all these improvements is the development of an automatic “bridge” between the graphic and the algebraic counterparts of the program (calling on an external computer algebra system).