A Singular web service for geometric computations

Abstract

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.

On the Unavoidable Uncertainty of Truth in Dynamic Geometry Proving

“…In order to simplify the costly elimination process, we realize that IIdeal contains a polynomial d 1 > EliminationIdeal(IdealA,{t1,t2,t3,t4});EliminationIdeal(IdealB, {t1,t2,t3,t4}); <0> <t2> > EliminationIdeal (IIdeal, {t1,t2,t3,t4}); <0>…”

“…and the like) from a given list of relation questions, that are also automatically translated by the program into algebraic terms. Then, following different criteria, constraints and heuristics, a collection of algebraic methods in ADG, some of them using the own GeoGebra symbolic computation features, some others connecting with an external server (see, for instance, [1]) are activated and sequentially attempt dealing with the proposed question, until one of them eventually succeeds or the program yields a failure warning. In the successful case, the output is a grant/denial of the truth of the proposed statement, eventually including a list of non-degeneracy conditions for the validity of the proposition.…”

“…Roughly speaking, statements dealing with constructions that have multiple instances for a single value of the free parameters of the corresponding construction and which are neither true for all such instances nor false for all of them. Section 3 introduces a simple illustrative example 1 , where a statement, neither generally true nor generally false (if naively formulated), can be easily turned into a generally true one, by adding an "intuitive" and natural complementary hypothesis. Furthermore, it advances some proving related consequences when defining, in dynamic geometry, the midpoint of a segment.…”

Computing envelopes in dynamic geometry environments

“…GSP, although not having a specific command to deal with envelopes, gives some related advice about constructing envelopes. 5 There, the statement a geometric envelope can be thought of as the limit or edge of the locus of a line or a circle is succesfully An approximate envelope computed through definition E 3 in GeoGebra applied to find some envelopes. This protocol catches the concept behind definition E 3 , and it is a natural way to get the envelopes as curves.…”

“…Finally, the last part of Sections 3 and 4 address the second objective of this note, namely, to describe the basic issues behind a new command for envelope computation, featured in the new 5.0 version of GeoGebra (September 2014) and based on a series of recent contributions by the authors of this note and their collaborators [5][6][7][8][9][10][11]. The idea here is to present just a sketchy picture on how some key ingredients from effective algebraic geometry are put together to conform the algorithmic approach behind this GeoGebra command.…”