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Mathematics
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Symbolic Computation for Mathematical Visualization
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Symbolic Computation for Mathematical Visualization

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (20 January 2023) | Viewed by 10565

Special Issue Editor

Prof. Dr. Juana Sendra

E-Mail Website
Guest Editor
Dpto. Matemática Aplicada a las TIC. ETSI Sistemas de Telecomunicación. Universidad Politécnica de Madrid, Madrid, Spain
Interests: visualization of curve and surface properties: algorithms and applications; symbolic computation for computer-aided geometric design; symbolic computation for algebraic geometry; visualization of matrix properties: algorithms and applications; symbolic computation for linear algebra and multilinear algebra; hybrid (approximate and symbolic) methods for the visualization of mathematical properties and proofs

Special Issue Information

Dear Colleagues,

The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves and surfaces. Richard S. Palais, in his work “The Visualization of Mathematics: Towards a Mathematical Exploratorium. Notices of the American Mathematical Society 46, 647-658, 1999”, indicated:

“Creating visualizations of implicitly defined curves and surfaces leads to many interesting

problems, both for the mathematician and for the programmer. The need to solve the equations involved numerically is one major difficulty. Constructing reliable and efficient algorithms for finding all the solutions when there are no restrictions on permitted singularities is not a completely solved problem. This is so even for the important special case of interest to algebraic geometers, namely, when the objects in question are defined as the solutions of polynomial equations. Another difficulty is that implicitly defined surfaces do not come equipped with a natural grid, and so special techniques (such as so-called “ray-tracing” methods) must be used to render them. Because of these challenges (and the importance of algebraic geometry), …….”

 Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers. It consists mainly of algorithm design and software development for the study of properties, as well as their applications, of explicitly given algebraic varieties.

The visualization of mathematics, consisting of the visualization of objects, and mathematical structures and methods, focuses on the development of intuition and mental models that
provide answers to questions that may arise within and outside of the mathematics.

The central goal of this Special Issue is to collect original contributions in the common intersection area of symbolic computation, algebraic geometry and/or visualization of mathematics.

Prof. Juana Sendra
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • constructive algebraic geometry
  • mathematical visualization
  • symbolic computation
  • hybrid computation
  • geometric computation
  • non-linear algebra
  • linear and multilinear algebra

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Published Papers (4 papers)

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Research

29 pages, 1172 KiB  
Article
Visualizing a Cubic Linkage through the Use of CAS and DGS
by Tomás Recio, Rafael Losada-Liste, Luis Felipe Tabera and Carlos Ueno
Mathematics 2022, 10(15), 2550; https://doi.org/10.3390/math10152550 - 22 Jul 2022
Cited by 1 | Viewed by 2209
Abstract
Our goal is to discuss the different issues that arise when attempting to visualize a joints-and-bars cube through GeoGebra, a widespread program that combines dynamic geometry (DGS) and computer algebra systems (CAS). As is standard in the DGS framework, the performance of the [...] Read more.
Our goal is to discuss the different issues that arise when attempting to visualize a joints-and-bars cube through GeoGebra, a widespread program that combines dynamic geometry (DGS) and computer algebra systems (CAS). As is standard in the DGS framework, the performance of the graphic model (i.e., the positions of the other vertices when dragging a given one) must correspond to a mathematically rigorous, symbolic computation-driven output. This requirement poses both computational algebraic geometry and dynamic geometry programming challenges that will be described, together with the corresponding proposed solutions. Among these, we include a complete determination of the dimension of the cubic linkage from an algebraic perspective, and introduce advanced 3D visualizations of this structure by using the GeoGebra software. Full article
(This article belongs to the Special Issue Symbolic Computation for Mathematical Visualization)
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19 pages, 1718 KiB  
Article
Asymptotic Behavior of a Surface Implicitly Defined
by Elena Campo-Montalvo, Marián Fernández de Sevilla and Sonia Pérez-Díaz
Mathematics 2022, 10(9), 1445; https://doi.org/10.3390/math10091445 - 25 Apr 2022
Cited by 3 | Viewed by 1380
Abstract
In this paper, we introduce the notion of infinity branches and approaching surfaces. We obtain an algorithm that compares the behavior at the infinity of two given algebraic surfaces that are defined by an irreducible polynomial. Furthermore, we show that if two surfaces [...] Read more.
In this paper, we introduce the notion of infinity branches and approaching surfaces. We obtain an algorithm that compares the behavior at the infinity of two given algebraic surfaces that are defined by an irreducible polynomial. Furthermore, we show that if two surfaces have the same asymptotic behavior, the Hausdorff distance between them is finite. All these concepts are new and represent a great advance for the study of surfaces and their applications. Full article
(This article belongs to the Special Issue Symbolic Computation for Mathematical Visualization)
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29 pages, 5608 KiB  
Article
Discovering Geometric Inequalities: The Concourse of GeoGebra Discovery, Dynamic Coloring and Maple Tools
by Tomás Recio, Rafael Losada, Zoltán Kovács and Carlos Ueno
Mathematics 2021, 9(20), 2548; https://doi.org/10.3390/math9202548 - 11 Oct 2021
Cited by 3 | Viewed by 3325
Abstract
Recently developed GeoGebra tools for the automated deduction and discovery of geometric statements combine in a unique way computational (real and complex) algebraic geometry algorithms and graphic features for the introduction and visualization of geometric statements. In our paper we will explore the [...] Read more.
Recently developed GeoGebra tools for the automated deduction and discovery of geometric statements combine in a unique way computational (real and complex) algebraic geometry algorithms and graphic features for the introduction and visualization of geometric statements. In our paper we will explore the capabilities and limitations of these new tools, through the case study of a classic geometric inequality, showing how to overcome, by means of a double approach, the difficulties that might arise attempting to ‘discover’ it automatically. On the one hand, through the introduction of the dynamic color scanning method, which allows to visualize on GeoGebra the set of real solutions of a given equation and to shed light on its geometry. On the other hand, via a symbolic computation approach which currently requires the (tricky) use of a variety of real geometry concepts (determining the real roots of a bivariate polynomial p(x,y) by reducing it to a univariate case through discriminants and Sturm sequences, etc.), which leads to a complete resolution of the initial problem. As the algorithmic basis for both instruments (scanning, real solving) are already internally available in GeoGebra (e.g., via the Tarski package), we conclude proposing the development and merging of such features in the future progress of GeoGebra automated reasoning tools. Full article
(This article belongs to the Special Issue Symbolic Computation for Mathematical Visualization)
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19 pages, 1954 KiB  
Article
On the Implicit Equation of Conics and Quadrics Offsets
by Jorge Caravantes, Gema M. Diaz-Toca, Mario Fioravanti and Laureano Gonzalez-Vega
Mathematics 2021, 9(15), 1784; https://doi.org/10.3390/math9151784 - 28 Jul 2021
Viewed by 2338
Abstract
A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such [...] Read more.
A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such as point positioning or solving intersection purposes. It is based on several classical results in “A Treatise on the Analytic Geometry of Three Dimensions” by G. Salmon for offsets to non-degenerate conics and central quadrics. Full article
(This article belongs to the Special Issue Symbolic Computation for Mathematical Visualization)
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