Strand 5:

Co-operation between DGS and CAS

ICTMT 5, Klagenfurt, 6-9 August 2001

(Schedule, tentative as of 8.06.2001)

Chair: Martín Garbayo

Monday 14:15 - 15:00 Chair: M. Garbayo

Plenary:

Co-operation Between Dynamic Geometry Systems and Computer Algebra Systems - Investigating, Guessing, Checking and Proving with the computer

Eugenio Roanes Lozano (Spain, eroanes@eucmos.sim.ucm.es)

Monday 15:15 - 16:00 Chair: M. Garbayo

Bridging the Gap between Dynamic Geometry and Computer Algebra: The Case of Loci Discovery.

José Luis Valcarce Gómez (Spain, fbotana@uvigo.es)

Monday 16:15 - 17:00 Chair: M. Garbayo

A Computer Learning Environment in Linear Algebra using CAS MuPAD

Wolfgang Fraunholz*, Frank Postel, Germany (Germany, w.fraunholz@uni-koblenz.de)

Tuesday 8:30 - 9:15 Chair: M. Garbayo

A study of conics with Maple V and Cabri-Géomètre II

Yuriko Baldin (Brazil, dyyb@power.ufscar.br)

Tuesday 9:30 - 10:15 Chair: M. Garbayo

The Three and Four Bar Linkages Revisited: Graphs and Equations

Francisco Botana Ferreiro (Spain, fbotana@uvigo.es)

Abstracts:

Plenary: Eugenio Roanes-Lozano, Spain:

Co-operation Between Dynamic Geometry Systems and Computer Algebra Systems - Investigating, Guessing, Checking and Proving with the computer

Computer Algebra Systems (CASs), like Maple, Derive, Mathematica, Axiom, Macsyma, Reduce, MuPad..., are specialised in exact and algebraic calculations. They use Exact Arithmetic and can handle non-assigned variables (i.e. variables in the "mathematical" sense, not in the usual sense in Computer Science). Many extensions like symbolic differentiation and integration, linear and non-linear equation and polynomial systems solving, 2D and 3D plotting... are usually included too. II, Cinderella, Euklid, Dr. Geo, WinGeom..., are specialised in rule and compass Geometry. The adjective dynamic comes from the fact that, once a construction is finished, the first objects drawn (points) can be dragged and dropped with the mouse, subsequently changing the whole construction. They usually incorporate animation and tracing too.

Unfortunately CASs and DGSs have evolved independently. Some CASs like Maple include specific and powerful packages devoted to Euclidean Geometry, but no CAS has incorporated Dynamic Geometry capabilities. On the other hand, Dynamic Geometry Systems can't handle (at least from the point of view of the user) non-assigned variables. Therefore, what can be saved from a DGS is only live graphic (to be read by the DGS), a geometric algorithm (script or macro, to be interpreted by the DGS) or a dead (fixed) graphic in one of the standard graphic formats. More precisely, what is missing in the DGSs is the possibility to handle and export parametric data about the plot: co-ordinates of points (allowing parameters as co-ordinates), equations of objects (allowing parameters as coefficients), length of objects (depending on parameters)...

Some DGSs (like Cabri Geometry II or Cinderella) include theorem-checking capabilities. This theorem-checking is based in altering the initial data: they find counterexamples if the result is false and suppose that the result is true if they find no counterexample (i.e., they are not "proofs" from the mathematical point of view). This lack of co-operation is more surprising in cases like the TI-92, where both technologies are simultaneously available. A straightforward application of this co-operation would be to treat with the computer the whole mathematical process of

discovery (or re-discovery):

Investigating - Guessing - Checking - Proving.

The talk will begin presenting an overview of the main capabilities of CASs and DGSs. A basic introduction to Automatic Theorem Proving in Geometry (Gröbner bases method and Wu's pseudoremainder method) will follow. The missing co-operation between CASs and DGSs will be detailed afterwards. Finally, the (ideal) whole mathematical process of discovery mentioned above will be presented. All steps will be illustrated with adequate examples.

Yuriko Baldin, Brazil:

A study of conics with Maple V and Cabri-Géomètre II

The usual presentation of conics in elementary instruction is based on the plane geometry, starting from focal properties and then connecting geometry to algebra by means of quadratic expressions.With 3-dimensional approach, conics are presented as plane sections of a symmetric cone and the fundamental focal properties are usually hard to be understood by students. Nevertheless, the most beautiful and motivating applications of conics to real world problems demand the conics to be worked out in 3-dimensional settings. In this paper, we present a study with combined use of CAS(Maple V) and DGS(Cabri-Géomètre) which integrates both approaches in the classroom, stressing the capabilities of each program suited to specific situations. We include useful exercises on Dandelin constructions with Maple V and Cabri-Géomètre, which would help teachers to construct concrete teaching material on the subject.

Francisco Botana Ferreiro, Spain:

The Three and Four Bar Linkages Revisited: Graphs and Equations

This paper reviews the behavior of current dynamic geometry systems (The Geometer's Sketchpad, Cabri Géomètre, Cinderella, Geometry Expert and Locus) when dealing with two simple linkages: the three and four bar linkages. The different approaches to numerical generation of loci are discussed, highlighting their success and limitations. Dynamic linkage generation can be used in engineering education and real design, overcoming the needs of books for designers.

Wolfgang Fraunholz*, Frank Postel, Germany:

A Computer Learning Environment in Linear Algebra using CAS MuPAD

The Computer Learning Environment in Linear Algebra offers an introduction to Linear Algebra (vector space, basis, dimension, matrices, determinants, systems of linear equations, linear operators, dot product, vector product). Representing the development and the examples, the solution of exercises step by step and the controlling of solutions is done by the Computer Algebra System MuPAD. Important is also a three-dimensional graphic tool, which visualises vectors, vector algebra, linear equations, mappings in three dimensions. The talk will give aspects of math education (Wolfgang Fraunholz) as well as those of programming and software (Frank Postel).

José Luis Valcarce Gómez, Spain:

Bridging the Gap between Dynamic Geometry and Computer Algebra: The Case of Loci Discovery

A basic problem in elementary geometry consists of finding the equation of a locus, given some conditions defining it. This problem remains unsolved in the field of mathematics education from a technological point of view: no friendly tool exists that allows a student to specify the conditions of a locus in a diagram and it returns the equation of the locus. Numerical approaches to this problem have been tackled in cuurent dynamic geometry environments but they share an essential incompleteness: an object must be constrained to move along a predefined path in order to get the trace of some other object. This paper describes a symbolic-dynamic approach to this problem: a computer algebra system solves it within a dynamic geometry environment.