Special Issue "Discrete and Computational Geometry"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 31 July 2020.

Special Issue Information

Dear Colleagues,

Discrete and Computational Geometry is a well-stablished research field, lying between Mathematics and Computer Science and intertwining two disciplines.

On one hand, Discrete Geometry is a branch of Discrete Mathematics dealing with the study of geometric objects and properties. More concretely, it considers problems involving finite sets of simple geometric objects, like points, lines, circles, triangles, cubes, or hyperplanes, for which questions such as number, incidences, or relative position are typical objects of study.

On the other hand, Computational Geometry is a branch of Computer Science aiming to design efficient algorithms for solving geometric problems. An important part of this process is the study of the time and space complexity, the data structures to be used, or the complexity of the output.

This Special Issue aims to provide readers with original research articles and review articles presenting novel results in discrete and/or computational geometry. Potential topics include, but are not limited to, the following:

  • Discrete combinatorial geometry;
  • Design, analysis, and implementation of geometric algorithms and data structures;
  • Discrete geometric optimization;
  • Bounds for the computational complexity of geometric problems;
  • Experimental evaluation of geometric algorithms;
  • Geometric modeling and visualization;
  • Computational geometry multidisciplinary applications (robotics, geographic information systems, CAGD, etc.).

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 papers will be 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 monthly 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 1200 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.

Published Papers (5 papers)

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Research

Open AccessArticle
Quasi-Isometric Mesh Parameterization Using Heat-Based Geodesics and Poisson Surface Fills
Mathematics 2019, 7(8), 753; https://doi.org/10.3390/math7080753 - 17 Aug 2019
Abstract
In the context of CAD, CAM, CAE, and reverse engineering, the problem of mesh parameterization is a central process. Mesh parameterization implies the computation of a bijective map ϕ from the original mesh M R 3 to the planar domain ϕ ( [...] Read more.
In the context of CAD, CAM, CAE, and reverse engineering, the problem of mesh parameterization is a central process. Mesh parameterization implies the computation of a bijective map ϕ from the original mesh M R 3 to the planar domain ϕ ( M ) R 2 . The mapping may preserve angles, areas, or distances. Distance-preserving parameterizations (i.e., isometries) are obviously attractive. However, geodesic-based isometries present limitations when the mesh has concave or disconnected boundary (i.e., holes). Recent advances in computing geodesic maps using the heat equation in 2-manifolds motivate us to revisit mesh parameterization with geodesic maps. We devise a Poisson surface underlying, extending, and filling the holes of the mesh M. We compute a near-isometric mapping for quasi-developable meshes by using geodesic maps based on heat propagation. Our method: (1) Precomputes a set of temperature maps (heat kernels) on the mesh; (2) estimates the geodesic distances along the piecewise linear surface by using the temperature maps; and (3) uses multidimensional scaling (MDS) to acquire the 2D coordinates that minimize the difference between geodesic distances on M and Euclidean distances on R 2 . This novel heat-geodesic parameterization is successfully tested with several concave and/or punctured surfaces, obtaining bijective low-distortion parameterizations. Failures are registered in nonsegmented, highly nondevelopable meshes (such as seam meshes). These cases are the goal of future endeavors. Full article
(This article belongs to the Special Issue Discrete and Computational Geometry)
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Open AccessArticle
Construction and Application of Nine-Tic B-Spline Tensor Product SS
Mathematics 2019, 7(8), 675; https://doi.org/10.3390/math7080675 - 29 Jul 2019
Abstract
In this paper, we propose and analyze a tensor product of nine-tic B-spline subdivision scheme (SS) to reduce the execution time needed to compute the subdivision process of quad meshes. We discuss some essential features of the proposed SS such as continuity, polynomial [...] Read more.
In this paper, we propose and analyze a tensor product of nine-tic B-spline subdivision scheme (SS) to reduce the execution time needed to compute the subdivision process of quad meshes. We discuss some essential features of the proposed SS such as continuity, polynomial generation, joint spectral radius, holder regularity and limit stencil. Some results of the SS using surface modeling with the help of computer programming are shown. Full article
(This article belongs to the Special Issue Discrete and Computational Geometry)
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Open AccessArticle
A New Class of 2q-Point Nonstationary Subdivision Schemes and Their Applications
Mathematics 2019, 7(7), 639; https://doi.org/10.3390/math7070639 - 18 Jul 2019
Abstract
The main objective of this study is to introduce a new class of 2 q -point approximating nonstationary subdivision schemes (ANSSs) by applying Lagrange-like interpolant. The theory of asymptotic equivalence is applied to find the continuity of the ANSSs. These schemes can be [...] Read more.
The main objective of this study is to introduce a new class of 2 q -point approximating nonstationary subdivision schemes (ANSSs) by applying Lagrange-like interpolant. The theory of asymptotic equivalence is applied to find the continuity of the ANSSs. These schemes can be nicely generalized to contain local shape parameters that allow the user to locally adjust the shape of the limit curve/surface. Moreover, many existing approximating stationary subdivision schemes (ASSSs) can be obtained as nonstationary counterparts of the proposed ANSSs. Full article
(This article belongs to the Special Issue Discrete and Computational Geometry)
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Open AccessArticle
B-Spline Solutions of General Euler-Lagrange Equations
Mathematics 2019, 7(4), 365; https://doi.org/10.3390/math7040365 - 22 Apr 2019
Abstract
The Euler-Lagrange equations are useful for solving optimization problems in mechanics. In this paper, we study the B-spline solutions of the Euler-Lagrange equations associated with the general functionals. The existing conditions of B-spline solutions to general Euler-Lagrange equations are given. As part of [...] Read more.
The Euler-Lagrange equations are useful for solving optimization problems in mechanics. In this paper, we study the B-spline solutions of the Euler-Lagrange equations associated with the general functionals. The existing conditions of B-spline solutions to general Euler-Lagrange equations are given. As part of this work, we present a general method for generating B-spline solutions of the second- and fourth-order Euler-Lagrange equations. Furthermore, we show that some existing techniques for surface design, such as Coons patches, are exactly the special cases of the generalized Partial differential equations (PDE) surfaces with appropriate choices of the constants. Full article
(This article belongs to the Special Issue Discrete and Computational Geometry)
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Open AccessArticle
NLP Formulation for Polygon Optimization Problems
Mathematics 2019, 7(1), 24; https://doi.org/10.3390/math7010024 - 27 Dec 2018
Abstract
In this paper, we generalize the problems of finding simple polygons with minimum area, maximum perimeter, and maximum number of vertices, so that they contain a given set of points and their angles are bounded by α + π where α ( 0 [...] Read more.
In this paper, we generalize the problems of finding simple polygons with minimum area, maximum perimeter, and maximum number of vertices, so that they contain a given set of points and their angles are bounded by α + π where α ( 0 α π ) is a parameter. We also consider the maximum angle of each possible simple polygon crossing a given set of points, and derive an upper bound for the minimum of these angles. The correspondence between the problems of finding simple polygons with minimum area and maximum number of vertices is investigated from a theoretical perspective. We formulate these three generalized problems as nonlinear programming models, and then present a genetic algorithm to solve them. Finally, the computed solutions are evaluated on several datasets and the results are compared with those from the optimal approach. Full article
(This article belongs to the Special Issue Discrete and Computational Geometry)
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