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### The geometric thickness of low degree graphs

Christian A. Duncan, David Eppstein, Stephen G. Kobourov
2004 Proceedings of the twentieth annual symposium on Computational geometry - SCG '04
We prove that the geometric thickness of graphs whose maximum degree is no more than four is two.  ...  The results have implications in graph theory, graph drawing, and VLSI design.  ...  In a recent paper on geometric thickness  , one of the authors posed as an open problem bounding the geometric thickness of a graph as a function of its degree.  ...

### The Geometric Thickness of Low Degree Graphs [article]

Christian A. Duncan, David Eppstein (University of California, Irvine), Stephen G. Kobourov
2003 arXiv   pre-print
We prove that the geometric thickness of graphs whose maximum degree is no more than four is two. All of our algorithms run in O(n) time, where n is the number of vertices in the graph.  ...  We also show a variation using orthogonal edges for maximum degree four graphs that also uses an n x n grid. The results have implications in graph theory, graph drawing, and VLSI design.  ...  In a recent paper on geometric thickness  , one of the authors posed as an open problem bounding the geometric thickness of a graph as a function of its degree.  ...

### Separating Thickness from Geometric Thickness [article]

David Eppstein
2003 arXiv   pre-print
We show that graph-theoretic thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric thickness >= t.  ...  Do random four-regular graphs have bounded expected geometric thickness? More generally, do bounded degree graphs have bounded geometric thickness?  ...  Is there a 2inductive graph with geometric thickness greater than two? Can the geometric thickness of such graphs be unbounded?  ...

### Separating Thickness from Geometric Thickness [chapter]

David Eppstein
2002 Lecture Notes in Computer Science
We show that graph-theoretic thickness and geometric thickness are not asymptotically equivalent: for every t, there exists a graph with thickness three and geometric thickness ≥ t.  ...  Do random four-regular graphs have bounded expected geometric thickness? More generally, do bounded degree graphs have bounded geometric thickness?  ...  Can we find good upper bounds for geometric thickness as a combination of the graph-theoretic thickness and a slowly growing function of the graph size?  ...

### Degree correlations in random geometric graphs

A. Antonioni, M. Tomassini
2012 Physical Review E
The prototypical spatial net- work we assume is the Random Geometric Graph of which many properties are known.  ...  Here we present new results for the two-point degree correlation function in terms of the clustering coefficient of the graphs for two-dimensional space in particular, with extensions to arbitrary finite  ...  The Random Geometric Graph (RGG) is a standard spatial network model that plays a role for spatial networks similar to the one played by the Erdös-Rényi random graph for relational ones.  ...

### Page 143 of Journal of Spacecraft and Rockets Vol. 11, Issue 3 [page]

1974 Journal of Spacecraft and Rockets
As expected, the torque coefficient changes very rapidly with the degree of porosity and the geo- metry of the disk.  ...  Geometrical pattern of the surface has a very strong effect on C,,.  ...

### Packing plane spanning trees and paths in complete geometric graphs

Oswin Aichholzer, Thomas Hackl, Matias Korman, Marc van Kreveld, Maarten Löffler, Alexander Pilz, Bettina Speckmann, Emo Welzl
2017 Information Processing Letters
We consider the following question: How many edge-disjoint plane spanning trees are contained in a complete geometric graph GK_n on any set S of n points in general position in the plane?  ...  Further, we consider variants of this problem by bounding the diameter and the degree of the trees (in particular considering spanning paths).  ...  A concept between graph-theoretical thickness and book thickness was later developed by Dillencourt, Eppstein, and Hirschberg  : given an abstract graph G, the geometric thickness of G is the smallest  ...

### COMPARATIVE ANALYSIS OF AN IMPELLER GEOMETRY AT DIFFERENT HEADS OF A PUMP

2018 Theoretical & Applied Science
Calculations of the impellers geometry were carried out at volume fluid flow rate (water) of 300 m 3 /h. Profiles and geometric dimensions of the impeller blades are obtained in the spans: 0 and 1.  ...  It is determined that the radial impellers are recommended to be used at the pump head of 20 -25 m. The most stable characteristic of the pump operation is observed in this range of heads.  ...  The ratio of thickness to the tip diameter equal to 0.03 indicates about a low probability of blockage to fluid flow.  ...

### Geometric Biplane Graphs I: Maximal Graphs

Alfredo García, Ferran Hurtado, Matias Korman, Inês Matos, Maria Saumell, Rodrigo I. Silveira, Javier Tejel, Csaba D. Tóth
2015 Graphs and Combinatorics
We study biplane graphs drawn on a finite planar point set S in general position. This is the family of geometric graphs whose vertex set is S and can be decomposed into two plane graphs.  ...  We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over n-element point sets.  ...  Note that the above generalization of plane graphs is reminiscent to, although more restrictive than, the notion of thickness, geometric thickness, and book thickness, which are defined for abstract graphs  ...

### Anatomy of triply-periodic network assemblies: characterizing skeletal and inter-domain surface geometry of block copolymer gyroids

Ishan Prasad, Hiroshi Jinnai, Rong-Ming Ho, Edwin L. Thomas, Gregory M. Grason
2018 Soft Matter
This work was supported by the U.S.  ...  ACKNOWLEDGEMENTS The authors are grateful to R. Spontak for initial discussions regarding EMT reconstructions of triblock networks and R. Kusner for useful discussions regarding TPMS.  ...  Beyond the analysis of domain thickness, we demonstrate that this simple numerical approximation of the 1D skeletal graph allows for the quantitative characterization of other key geometrical properties  ...

### Numerical Research on the Effect of Impactor Shape and Core Characteristics on the Ballistic Impact Response of Marine Sandwich Composite Plates

2019 International journal of recent technology and engineering
The kinetic energy loss of the projectile during the impact is the criteria used to compare the different types of impactors.  ...  The dynamic response of composite plates to impactors of different shapes is investigated.  ...  The core material is normally low strength material, but its higher thickness provides the sandwich composite with high bending stiffness with overall low density.  ...

### A Raster-to-Vector Conversion Concept Based on Industrial Requirements

Rune Espelid
1988 IAPR International Workshop on Machine Vision Applications
During this process the geometric features are extracted from image contour lines.  ...  With IGES as the starting point, covering the CAD area, I discuss a number of such needs. A strategy for the design of a r-t-v system that matches the requirements listed is then proposed.  ...  They produces nodes of degree three, they have a free end or merge into another artifact (the Xeffect), and their length is proportional to the local line thickness.  ...

### Finding the Optimal Condition of Exterior Spray Coating for Cookware Products

Yaowarak Wongyai, Suksan Prombanpong, J. Zhou, H. Gong
2015 MATEC Web of Conferences
From the analysis of the experiment, it can be concluded that 23 cm of height, 95 degrees of gun angle, 5.4 bar of air pressure 1.95 seconds of spray time, 295 degrees of nozzle size is proposed.  ...  One is studied using an experimental design process, and the other is the geometric modeling for the process of exterior spray coating.  ...  The graph shown the co-responding influence of factors affected on the volume of dry film thickness This graph showed the co-responding influence of factors that affected on the dry film thickness.  ...

### Anatomy of triply-periodic network assemblies: Characterizing skeletal and inter-domain surface geometry of block copolymer gyroids [article]

Ishan Prasad and Hiroshi Jinnai and Rong-Ming Ho and Edwin L. Thomas and Gregory M. Grason
2018 arXiv   pre-print
heterogeneous packing of molecular constituents that form the networks.  ...  While TPNs are, in reality, volume-filling patterns of spatially-varying molecular composition, physical and structural models most often reduce their structure to lower-dimensional geometric objects:  ...  This work was supported by the U.S.  ...

### The large-scale geometry of right-angled Coxeter groups [article]

Pallavi Dani
2018 arXiv   pre-print
This is a survey of some aspects of the large-scale geometry of right-angled Coxeter groups.  ...  The emphasis is on recent results on their negative curvature properties, boundaries, and their quasi-isometry and commensurability classification.  ...  The notion of thickness was introduced in [BDM09] as a geometric obstruction to relative hyperbolicity. An idea of the definitions of thickness and strong algebraic thickness is given in Section 5.2.  ...
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