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

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