Project MTM2014-54141-P
Brief Description
- Summary
Under the term algebro-geometric constructions we include an extensive collection of
mathematical objects that are susceptible, under certain assumptions, of a computational or
algorithmic treatment. This treatment not only has practical implications because in many
cases it can be used to deep in the understanding of the purely mathematical object
considered.
In this project under the name of algebro-geometric constructions we will consider the study
of algebraic varieties, the use of geometric loci in the context of Dynamic Geometry, the
manipulation of curves and surfaces commonly used in Computer Aided Geometric Design
and the analysis of the systems of algebraic equations together with the underlying geometry
behind the calculation of the position and attitude in global navigation satellite systems.
In this project we will focus primarily on deepening the fundamentals behind the
computational treatment of these algebro-geometric constructions, constructions regarded as
algebraic varieties from different perspectives (symbolic, numerical, hybrid, geometric, etc.).
As a by-product several tools will arise that will be used for the study of specific geometric
algebro-constructions such as offsets, bisectors, etc. Finally, the tools developed along with
the study performed will be used in the development of new techniques in Dynamic
Geometry and in the modeling of global navigation satellite systems.
- Main Goals
The project is based on the following three general goals:
- General Goal I: Development of computational tools for the manipulation of algebraic varieties, including all those algorithmmic, algebraic,
geometric, numeric, etc, aspects that could be usefull in the study of them.
- General Goal II: Theoretical and algorithmic study of specific algebro-geometric constructions as geometric loci, offsets, movement of surfaces,
automatic reasoning, etc.
- General Goal III: Application of the algorithmic treatment of the algebro-geometric constructions to dynamic geometry and to model GPS systems.
- Specific goals
- Specific goal I.1: General algorithms for algebraic varieties. This goal includes the following aspects:
- Algorithmic identification of swung and tubular surfaces.
- Characterization and determination of optimality of parametrizations.
- Perturbed algebraic varieties.
- Analysis of radical varieties.
- Parametrizations and algebraic differential equations.
- Specific goal I.2: Computational tools for automated reasoning in Geometry.
This goal includes the following aspects:
- Study of aspects from real geometry in automatic reasoning in geometry,
currently non-existent since all the methodology related to Gröbner Bases has been
developed in the complex space.
- Development of exact automated reasoning methods based on the verification on a finite
number of particular cases. The idea is to explore a new method, which can be traced back to the
lemma of Schwartz-Zippel or Kortenkamp's thesis (ETH Zurich, 1999),
but whose fundamental and theoretical aspects have not been developed yet.
The development of this method, given the role of combinatorics, may require the use of
tropical geometry techniques.
- Specific goal I.3: Elimination and values. This goal includes the following aspects:
- Subresultants: formulas for values and interpolation.
- Numerical treatment and stability of the symbolic-numeric
algorithms for manipulating geometric entities by values.
- Tropical elimination and applications.
- Specific goal II.1: Algorithms for algebro-geometric constructions.
This goal includes the following aspects:
- Theoretical/algorithmic relationships among different algebro-geometric
constructions including offsets, conchoids, pedal varieties, convolutions, cissoids, etc.
- Theoretical and algorithmic analysis of bisectors of curves and algebraic surfaces.
- Direct computation of singularities of offsets of curves and surfaces.
- Algorithms for curves and surfaces in motion.
- Specific goal II.2: Similarity, morphology and symmetry for curves and surfaces.
The following questions are included here:
- Similarity of algebraic curves implicitly defined; similarity of space rational curves.
- Morphology of curves and surfaces: invariance in curves and surfaces.
- Symmetry computation: polynomially parametrized surfaces, quadrics and ruled surfaces.
- Specific goal II.3: Application of the methods developed in the general goal I to the development of
algorithms tailored to the objects involved in automated reasoning in the context of dynamic geometry.
The following aspects are included here:
- Automatic determination of geometric loci, envelopes, bisecting curves and surfaces, etc.
in Dynamic Geometry 2D and 3D, using the methods introduced in goal I.
- Automatic determination of the truth/falsehood of a conjecture, automatic deduction of properties,
automatic discovery of necessary and sufficient conditions for the truth of a statement initially false,
all in the context of Dynamic Geometry and using the methods introduced in goal I.
- Specific goal III.1: Software and algorithms for Dynamic Geometry.
This goal includes the following aspects:
- Study of the applicability and implementation of the results mentioned above to dynamic geometry software,
with special emphasis on GeoGebra and its 2D and 3D interfaces, and on all aspects related to its use in
tablets and touch screens.
- Design and development of servers for remote computations under the new development paradigm of GeoGebra,
HTML5.
- Specific goal III.2: GPS modeling.
This goal includes the following aspects:
- Efficient computation of position by global navigation satellite systems.
- Efficient computation of course by global satellite navigation systems by resolving ambiguities.
- Coordinate changes in Algebraic Geodesy.