A completion of hypotheses method for 3D-geometry. 3D-extensions of Ceva and Menelaus theorems

A method that automates hypotheses completion in 3D-Geometry is presented. It consists of three processes: defi ning the geometric objects in the confi guration; determining the hypothesis conditions of the confi guration (through a point-on-object declaration method); and applying an algebraic automatic theorem proving method to obtain and prove the sufficiency of complementary hypothesis conditions. To avoid as much as possible the appearance of rational expressions, projective coordinates are used (although affine and Euclidean problems can also be treated). A Maple implementation of the method has been used to extend to 3D classic 2D geometric theorems like Ceva's and Menelaus'

Connecting the 3D DGS Calques3D with the CAS Maple

Many (2D) Dynamic Geometry Systems (DGSs) are able to export numeric coordinates and equations with numeric coefficients to Computer Algebra Systems (CASs). Moreover, different approaches and systems that link (2D) DGSs with CASs, so that symbolic coordinates and equations with symbolic coefficients can be exported from the DGS to the CAS, already exist. Although the 3D DGS Calques3D can export numeric coordinates and equations with numeric coefficients to Maple and Mathematica, it cannot export symbolic coordinates and equations with symbolic coefficients. A connection between the 3D DGS Calques3D and the CAS Maple, that can handle symbolic coordinates and equations with symbolic coefficients, is presented here. Its main interest is to provide a convenient time-saving way to explore problems and directly obtain both algebraic and numeric data when dealing with a 3D extension of "ruler and compass geometry". This link has not only educational purposes but mathematical ones, like mechanical theorem proving in geometry, geometric discovery (hypotheses completion), geometric loci finding... As far as we know, there is no comparable "symbolic" link in the 3D case, except the prototype 3D-LD (restricted to determining algebraic surfaces as geometric loci)

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)

Some Reflections About the Success and Impact of the Computer Algebra System DERIVE with a 10-Year Time Perspective

The computer algebra system DERIVE had a very important impact in teaching mathematics with technology, mainly in the 1990’s. The authors analyze the possible reasons for its success and impact and give personal conclusions based on the facts collected. More than 10 years after it was discontinued it is still used for teaching and several scientific papers (most devoted to educational issues) still refer to it. A summary of the history, journals and conferences together with a brief bibliographic analysis are included.Depto. de Estadística e Investigación OperativaFac. de Ciencias MatemáticasTRUEMinisterio de Ciencia e Innovación (MICINN)Comunidad de Madridpu

An algebraic taxonomy for locus computation in dynamic geometry

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 GröbnerCover 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

Utilización de metodologías activas de enseñanza para el aprendizaje de las matemáticas, centradas en el estudiante y desarrolladas en el espacio innovador de una hiperaula

La inauguración de la hiperaula de la Facultad de Educación-CFP (UCM) en abril de 2019 y su prioridad de uso para el Máster en Formación del Profesorado de Educación Secundaria, Bachillerato, Formación Profesional y Enseñanza de Idiomas durante el curso académico 2019-2020, supone un marco experimental absolutamente novedoso. Estas circunstancias permiten a la responsable del proyecto y a varios profesores participantes en el mismo, algunos de ellos docentes en el mencionado Máster y pertenecientes a diferentes Departamentos y Facultades de la UCM, así como a varios alumni y personal de administración y servicios implicados en el proyecto, utilizar este innovador espacio para la enseñanza de la asignatura de "Innovación docente e iniciación a la investigación educativa" (especialidad de matemáticas del Máster en Formación del Profesorado), empleando para ello metodologías activas de enseñanza centradas en el estudiante

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 Gröbner 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

Some reflections on dynamic geometry systems (the GUI for ADG)

Dynamic Geometry Systems (DGS) are the Graphic User Interfaces for Automatic Deduction in Geometry (ADG). We will develop below some reflections regarding how geometric data are introduced to the most frequently adopted DGS and how the constructive geometric processes (that are similar to the programs of "usual" computer languages) are stored. The later can be used as input for ADG processes