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)

Francisco José Ayala, especialista en evolució i en filosofia de la biologia

Dins del XI Congrés Internacional d'Ontologia, Francisco José Ayala (Universitat de Califòrnia, Irvine) va oferir una xerrada sobre la biologia i la cultura humanes a la Facultat de Filosofia i Lletres el 6 d'octubre. Ayala ha desenvolupat tota la seva carrera acadèmica als Estats Units, tot esdevenint un dels científics més prestigiosos de l'actualitat.Dentro del XI Congreso Internacional de Ontología, Francisco José Ayala (Universidad de California, Irvine) ofreció una charla sobre la biología y la cultura humanas en la Facultad de Filosofía y Letras el 6 de octubre. Ayala ha desarrollado su carrera académica en Estados Unidos, convirtiéndose en uno de los científicos más prestigiosos de hoy en día.As part of the 11th International Conference on Ontology, Francisco José Ayala (University of California, Irvine) gave a talk on human biology and culture at the Faculty of Philosophy and Arts on 6 October. Ayala has carried out his entire academic career in the United States and is now one of the most prestigious scientists in the world

A Singular web service for geometric computations

Outsourcing algebraic computations in dynamic geometry is a possible strategy used when software distribution constraints apply. Either if the target user machine has hardware limitations, or if the computer algebra system cannot be easily (or legally) packaged inside the geometric software, this approach can solve current shortcomings in dynamic environments. We report the design and implementation of a web service using Singular, a program specialized in ideal theory and commutative algebra. Besides its canonical address, a virtual appliance and a port to a low-cost ARM based computer are also provided. Any interactive geometric environment can then outsource computations where Singular is used, and incorporate their results into the system. In particular, we illustrate the capabilities of the web service by extending current abilities of GeoGebra to deal with algebraic loci and envelopes by means of a recent algorithm for studying parametric polynomial systems

Using Maple's RegularChains library to automatically classify plane geometric loci

We report a preliminary discussion on the usability of the RegularChains library of Maple for the automatic computation of plane geometric loci and envelopes in graphical interactive environments. We describe a simple implementation of a recently proposed taxonomy of algebraic loci, and its extension to envelopes of families of curves is also discussed. Furthermore, we sketch how currently unsolvable problems in interactive environments can be approached by using the RegularChains library

On the Unavoidable Uncertainty of Truth in Dynamic Geometry Proving

The aim of this note is to discuss some issues posed by the emergency of universal interfaces able to decide on the truth of geometric statements. More specifically, we consider a recent GeoGebra module allowing general users to verify standard geometric theorems. Working with this module in the context of Varignon’s theorem, we were driven – by the characteristics of the GeoGebra interface– to perform a quite detailed study of the very diverse fate of attempting to automatically prove this statement, when using two different construction procedures.We highlight the relevance –for the theorem proving output– of expression power of the dynamic geometry interface, and we show that the algorithm deciding about the truth of some –even quite simple– statements can fall into a not true and not false situation, providing a source of confusion for a standard user and an interesting benchmark for geometers interested in discovering new geometric facts

Computing envelopes in dynamic geometry environments

We review the behavior of standard dynamic geometry software when computing envelopes, relating these approaches with the various definitions of envelope. Special attention is given to the recently released version of GeoGebra 5.0, that implements a recent parametric polynomial solving algorithm, allowing sound computations of envelopes of families of plane curves. Specific details on this novel approach are provided in this paper