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3D Cartesian Ovals and other Algebraic Surfaces with webMathematica

Francisco Botana
Organization: University of Vigo, Spain
Department: Dept. of Applied Mathematics

2002 Applications of Computer Algebra Conference (ACA '2002)
Conference location

Volos, Greece

Loci generation is a key characteristic of dynamic geometry environments. The approach taken by currents systems such as CABRI, The Geometer's Sketchpad, Geometry Expert or Cinderella, is based in tracing an element while other is being dragged. Lugares has shown that a symbolic, computer-algebra based approach can efficiently deal with this task, extending its applicability to a wider class of problems and returning the loci equations.

In this talk I will describe how the symbolic-dynamic approach to loci generation can be extended to deal with 3D geometric constructions. A new (as far as I know) family of algebraic surfaces, the 3D Cartesian ovals, will be used to illustrate the proposed techniques. Roughly speaking, the process of 3D loci generation follows these steps. A pseudo linguistic description of a 3D geometric construction is given to Mathematica. The construction consists of some basic points and some properties bounding an indeterminate point X. Mathematica rewrites the specified properties as polynomials, and, using the Groebner basis method, deduces all the polynomial consequences of the conditions. These consequences include polynomials containing the locus of X and, perhaps, degenerated conditions. Finally, the locus polynomials are plotted using the standard Mathematica facilities.

A prototype for 3D loci discovery, 3D-LD, is at http://rosalia.uvigo.es/sdge/web/3D. It uses webMathematica, thus allowing its remote testing and the interaction with the obtained surfaces via the applet LiveGraphics3D. Current limitations and further developments of 3D-LD will be discussed. Furthermore, first results on interfacing 3D-LD with an interactive 3D dynamic geometry environment, Calques3D, will also be presented.

*Wolfram Technology > Special Versions > webMathematica

computer algebra, dynamic geometry environments, 3D Cartesian ovals