Connecting the 3D DGS Calques3D with the CAS Maple

Authors: Eugenio Roanes-Lozano (1),
-------- Nicolas van Labeke (2),
         Eugenio Roanes-Macias (1)

         (1) Departamento de Algebra,
             Universidad Complutense de Madrid, Spain
             {eroanes,roanes}@mat.ucm.es

         (2) School of Computer Science and Information
             Systems, University of London
             n.vanlabeke@scre.ac.uk

Acknowledgments:
----------------

This work was partially supported by the research projects
MTM2004-03175 (Ministerio de Educacion y Ciencia, Spain) and
UCM2005-910563 (Comunidad de Madrid - Universidad Complutense de
Madrid, research group ACEIA).

Extended Abstract:
------------------

Both Computer Algebra Systems (CASs) and 2D Dynamic Geometry
Systems (DGSs) have reached a high level of development.
Moreover, some 3D DGS like Cabri3D and the free Calques3D, have
recently been developed.

Powerful packages devoted to Euclidean Geometry have been
developed in CASs like Maple or Derive, but CASs have
incorporated neither mouse drawing capabilities nor dynamic
capabilities.

Meanwhile, the well-known 2D DGSs (Cabri Geometry, The Geometer's
Sketchpad, Cinderella, GeoGebra...) do not provide algebraic
facilities.

There have been different attempts to connect 2D DGS with CAS (a
need underlined years ago by Tomas Recio). The possible
strategies for collaboration can be classified the following way
(Roanes-Lozano):

- To develop a new system that integrates a new DGS and a new CAS.
  Examples: Geometry Expert, Java Geometry Expert, The Algebraic
  Geometer.

- To develop a new DGS that can communicate with existing CASs.
  Examples: GEOTHER & Epsilon, The Algebraic Geometer, GDI.

- To develop an external translator that allows an existing DGS
  to communicate with an existing CAS, what has the advantage of
  reusing software. Example: paramGeo (by Roanes and Roanes, that
  connects The Geometer's Sketchpad v.3 and v.4 with Maple and
  Derive, and includes the development of the corresponding Derive
  and Maple geometric packages --denoted paramGeo).

Observe that The Algebraic Geometer appears in two classes, as it
incorporates a small internal CAS but can also communicate with
the external CASs Maple and Mathematica. Let us underline that in
the case of this DGS, the communication with the CAS is
bidirectional, that is, the results of the CAS can make the
construction in the DGS change! It is, as far as we know, the
only DGS with this surprising capability.

Let us also underline that the authors of GDI and webDiscovery
(Botana et al.) are working in the standardization of the
connection of 2D DGS and CAS using (and evolving) the standard
OpenMath. The goal is that, in the future, any 2D DGS with the
possibility to export in OpenMath format would be able to access
webDiscovery.

Roanes and Roanes have been working for some years in automatic
theorem proving and discovery in geometry, and have developed a
Maple package for 3D geometry (paramGeo3D), that they have
successfully used, for instance, in founding (and proving) some
new (!) 3D geometric theorems.

Regarding the 3D DGS - CAS connection, as far as we know, there
were no connections until van Labeke reformed Calques3D so that it
could provide a Mathematica-style output for the "History" of a
construction. This was used by Botana et al., that, adopting our
philosophy of software reuse, used this output in his system
3D-LD (that calls the CAS CoCoA and Mathematica) and is oriented
exclusively to the discovery of algebraic surfaces as loci of
points.

Now van Labeke has included in Calques3D the possibility to
export in Maple-style the "History" of a construction in
order to make it possible for Calques3D to collaborate with our
Maple's paramGeo3D package.

Our purpose in not only automatic-theorem proving and discovery
in geometry, but a wider one: to explore 3D geometric problems
(that can be drawn in a 3D extension of "rule-and-compass
geometry") and to automatically obtain their equations (for any
required purpose).

We have therefore developed a "translator" Maple package that
acts as a mediator between the orders exported by Calques 3D and
those understood by Maple (after loading paramGeo3D package). This
way many of Calques 3D's "commands" (those related with the "3D
rule-and-compass geometry" mentioned above) are now accessible
from Maple.

The talk will be illustrated with simple but interesting examples,
like:

 - prove that the diagonals of a parallelepiped are concurrent
 - obtain the coordinates of the common point to the diagonals of
   a parallelepiped
 - prove that the planes through two vertices and the midpoint of
   the opposite side of a tetrahedron do intersect
 - prove a 3D version of Desargues theorem (for triangles)
 - prove a 3D version of Ceva and Menelaos theorems
 - ...

that are firstly drawn with Calques3D, and which "History" files
are used by Maple to perform the corresponding computations.

This work has clearly a didactic application in geometric problems
exploration. Nevertheless, its main interest is to provide a
convenient time-saving way to introduce data when dealing with a
3D extension of "rule and compass Geometry", which has a wider
scope than only educational purposes.