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The Concept of Modeling Packing and Covering Problems Using Modern Computational Geometry Software
A class of geometric packing and covering problems is considered. A new concept of their mathematical modeling using a special class of functions is...
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Optimized ellipse packings in regular polygons
We present model development and numerical solution approaches to the problem of packing a general set of ellipses without overlaps into an optimized...
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Efficiency of Packings and Coverings with a Sequence of Convex Disks
We search for those convex disks which are: 1) the least efficient for packing the plane, 2) the least efficient for covering it. Thus, in a certain...
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Packing ovals in optimized regular polygons
We present a model development framework and numerical solution approach to the general problem-class of packing convex objects into optimized convex...
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Coverings
The concept of covering is dual to that of packing. In this chapter, we shall define lattice covering, general covering, covering density, thinnest...
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Φ-Functions of 2D Objects with Boundaries Being Second-Order Curves
An approach to constructing analytical conditions of non-intersection and inclusion of non-oriented convex 2D objects is considered, the boundaries...
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Extremal Properties of Regular Polyhedra
On the surface of a ball the problems of the densest circle packing and the thinnest circle covering with 4, 6, or 12 congruent circles lead to...
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Packing ellipses in an optimized convex polygon
Packing ellipses with arbitrary orientation into a convex polygonal container which has a given shape is considered. The objective is to find a...
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Challenges to the Development of Effective Creativity
One of the key problems for students, especially from develo** countries during pandemic is a lack of the electronic materials. Obviously, there...
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A Diffuse Interface Framework for Modeling the Evolution of Multi-cell Aggregates as a Soft Packing Problem Driven by the Growth and Division of Cells
We present a model for cell growth, division and packing under soft constraints that arise from the deformability of the cells as well as of a...
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Sparse Balanced Layout of Ellipsoids*
The authors consider the problem of generating spheroidal voids in a three- dimensional domain of complex geometry, with regard for the constraints...
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Creating Constellation Patterns I: Composition
Constellation patterns are a family of Islamic geometric patterns that combine different stars in a tightly interconnected matrix with few subsidiary...
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Computational Conformal Geometric Methods for Vision
Conformal geometry studies the geometric properties of objects invariant under conformal transformation group. It is a powerful theoretic tool to...
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Penguin Huddling: A Continuum Model
Penguins huddling in a cold wind are represented by a two-dimensional, continuum model. The huddle boundary evolves due to heat loss to the huddle...
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Notes
The subject of polarity, in particular the specific problem of the product of the volume of polar convex bodies, which comes up in the inequality...
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Recent Developments of Surface Parameterization Methods Using Quasi-conformal Geometry
Surface parameterization is of fundamental importance for many tasks in computer vision and imaging. In recent years, computational quasi-conformal...
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Some Theorems from Elementary Geometry
In this chapter we compile the necessary preliminaries from elementary geometry. They are mainly about some well-known concepts and theorems,...
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Combination Theorems in Groups, Geometry and Dynamics
The aim of this chapter is to give a survey of combination theorems occurring in hyperbolic geometry, geometric group theory and complex dynamics,...
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Recent Developments of Surface Parameterization Methods Using Quasi-conformal Geometry
Surface parameterization is of fundamental importance for many tasks in computer vision and imaging. In recent years, computational quasi-conformal...
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Random heterogeneous microstructure construction of composites via fractal geometry
The microstructures of a composite determine its macroscopic properties. In this study, microstructures with particles of arbitrary shapes and sizes...