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Geometry
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Feature Papers in Geometry
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Feature Papers in Geometry

A special issue of Geometry (ISSN 3042-402X).

Deadline for manuscript submissions: 31 December 2025 | Viewed by 1783

Special Issue Editor

Prof. Dr. Yang-Hui He

E-Mail Website
Guest Editor
1. London Institute for Mathematical Sciences, Royal Institution, London W1S 4BS, UK
2. Merton College, University of Oxford, Oxford OX1 4JD, UK
Interests: AI-assisted mathematics; mathematical physics; string theory; algebraic geometry; number theory
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

As the Editor-in-Chief of Geometry, I am pleased to announce the Special Issue "Feature Papers in Geometry", which will be a collection of high-quality papers (original research articles or comprehensive reviews) from top academics addressing the nature of geometry. I welcome the submission of manuscripts from Editorial Board Members and from outstanding scholars invited by the Editorial Board and the Editorial Office related to any of the topics covered in the scope of the journal (https://www.mdpi.com/journal/geometry).

You are invited to send short proposals for submissions to our Editorial Office (geometry@mdpi.com) for evaluation. Please note that selected full papers will still be subject to a thorough and rigorous peer-review.

Prof. Dr. Yang-Hui He
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. Geometry is an international peer-reviewed open access quarterly 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 1000 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

  • Euclidean geometry
  • differential geometry
  • algebraic geometry
  • complex geometry
  • discrete geometry
  • computational geometry
  • geometric group theory
  • convex geometry
  • geometric theory/algorithm
  • mathematical physics

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (3 papers)

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Research

25 pages, 368 KiB  
Article
LU Factorizations for ℕ × ℕ-Matrices and Solutions of the k[S]-Hierarchy and Its Strict Version
by G. F. Helminck and J. A. Weenink
Geometry 2025, 2(2), 4; https://doi.org/10.3390/geometry2020004 - 15 Apr 2025
Viewed by 83
Abstract
Let S be the N×N-matrix of the shift operator and let k denote the field of real or complex numbers. We consider two different deformations of the commutative algebra k[S], together with the evolution equations of [...] Read more.
Let S be the N×N-matrix of the shift operator and let k denote the field of real or complex numbers. We consider two different deformations of the commutative algebra k[S], together with the evolution equations of the deformations of the powers {Si,i⩾1}. They are called the k[S]-hierarchy and the strict k[S]-hierarchy. For suitable Banach spaces B, we explain how LU factorizations in GL(B) can be used to produce dressing matrices of both hierarchies. These dressing matrices correspond to bounded operators on B, a class far more general than the one used at a prior construction. This wider class of solutions of both hierarchies makes it possible to treat reductions of both systems. The matrix coefficients of these matrices are shown to be quotients of analytic functions. Moreover, we present a subgroup Gcpt(B) of GL(B) such that the procedure with LU factorizations works for each g∈Gcpt(B). Full article
(This article belongs to the Special Issue Feature Papers in Geometry)
17 pages, 7219 KiB  
Article
A Laguerre-Type Action for the Solution of Geometric Constraint Problems
by Nefton Pali
Geometry 2025, 2(1), 2; https://doi.org/10.3390/geometry2010002 - 18 Feb 2025
Viewed by 208
Abstract
A well-known idea is to identify spheres, points, and hyperplanes in Euclidean space Rn with points in real projective space. To address geometric constraints such as intersections, tangencies, and angle requirements, it is important to also encode the orientations of hyperplanes and [...] Read more.
A well-known idea is to identify spheres, points, and hyperplanes in Euclidean space Rn with points in real projective space. To address geometric constraints such as intersections, tangencies, and angle requirements, it is important to also encode the orientations of hyperplanes and spheres. The natural space for encoding such geometric objects is the real projective quadric with signature (n+1,2). In this article, we first provide a general formula for calculating the angles formed by the geometric objects encoded by the points of the quadric. The main result is the determination of a very simple parametrization of a Laguerre-type subgroup that acts transitively on the quadric while preserving the geometric nature of its points. That is, points of the quadric representing oriented spheres, points, and oriented hyperplanes in Rn are mapped by the action to points of the same geometric type. We also provide simple parametrizations of the isotropies of the action. The action described in this article provides the foundation for an effective solution to geometric constraint problems. Full article
(This article belongs to the Special Issue Feature Papers in Geometry)
16 pages, 304 KiB  
Article
Trigonometric Polynomial Points in the Plane of a Triangle
by Clark Kimberling and Peter J. C. Moses
Geometry 2024, 1(1), 27-42; https://doi.org/10.3390/geometry1010005 - 23 Dec 2024
Viewed by 677
Abstract
It is well known that the four ancient Greek triangle centers and others have homogeneous barycentric coordinates that are polynomials in the sidelengths a,b, and c of a triangle ABC. For example, the circumcenter is represented by [...] Read more.
It is well known that the four ancient Greek triangle centers and others have homogeneous barycentric coordinates that are polynomials in the sidelengths a,b, and c of a triangle ABC. For example, the circumcenter is represented by the polynomial a(b2+c2−a2). It is not so well known that triangle centers have barycentric coordinates, such as tanA : tan B : tan C, that are also representable by polynomials, in this case, by p(a, b, c) : p(b, c, a) : p(c, a, b), where p(a, b, c)=a(a2+b2−c2)(a2+c2−b2). This paper presents and discusses the polynomial representations of triangle centers that have barycentric coordinates of the form f(a, b, c) : f(b, c, a) : f(c, a, b), where f depends on one or more of the functions in the set {cos, sin, tan, sec, csc, cot}. The topics discussed include infinite trigonometric orthopoints, the n-Euler line, and symbolic substitution. Full article
(This article belongs to the Special Issue Feature Papers in Geometry)
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