Orientation-Dependent Chord Length Distribution Function for Right Prisms with Rectangular or Right Trapezoidal Bases

2020 ◽  
Vol 55 (6) ◽  
pp. 344-355
Author(s):  
V. K. Ohanyan ◽  
D. M. Martirosyan
Keyword(s):  
Distribution Function ◽  
Length Distribution ◽  
Chord Length ◽  
Chord Length Distribution

scholarly journals The chord-length distribution of a polyhedron

2020 ◽  
Vol 76 (4) ◽  
pp. 474-488
Author(s):  
Salvino Ciccariello
Keyword(s):  
Distribution Function ◽  
Length Distribution ◽  
Chord Length ◽  
Square Root ◽  
Trigonometric Functions ◽  
Chord Length Distribution ◽  
Analytic Expressions ◽  
Analytic Expression ◽  
Positive Integers ◽  
Inverse Trigonometric Functions

The chord-length distribution function [γ′′(r)] of any bounded polyhedron has a closed analytic expression which changes in the different subdomains of the r range. In each of these, the γ′′(r) expression only involves, as transcendental contributions, inverse trigonometric functions of argument equal to R[r, Δ1], Δ1 being the square root of a second-degree r polynomial and R[x, y] a rational function. As r approaches δ, one of the two end points of an r subdomain, the derivative of γ′′(r) can only show singularities of the forms |r − δ|−n and |r − δ|−m+1/2, with n and m appropriate positive integers. Finally, the explicit analytic expressions of the primitives are also reported.

scholarly journals Chord length distribution and the distance between two random points in a convex body in Rn

2020 ◽  
Vol 9 (2) ◽  
pp. 74-79
Author(s):  
Rafik Aramyan ◽  
Daniel Yeranyan
Keyword(s):  
Distribution Function ◽  
Convex Body ◽  
Length Distribution ◽  
Chord Length ◽  
Chord Length Distribution

In this article for n-dimensional convex body D the relation between the chord length distribution function and the distribution function of the distance between two random points in D was found. Also the relation between their moments was found.

scholarly journals The chord length distribution function for regular polygons

2009 ◽  
Vol 41 (2) ◽  
pp. 358-366 ◽  
Author(s):  
H. S. Harutyunyan ◽  
V. K. Ohanyan
Keyword(s):  
Distribution Function ◽  
Regular Polygon ◽  
Length Distribution ◽  
Chord Length ◽  
Regular Hexagon ◽  
Regular Polygons ◽  
Chord Length Distribution ◽  
Regular Pentagon ◽  
Regular Triangle ◽  
Elementary Expression

In this paper we obtain an elementary expression for the chord length distribution function of a regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a regular triangle, a square, a regular pentagon, and a regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively.

scholarly journals The chord length distribution function for regular polygons

2009 ◽  
Vol 41 (02) ◽  
pp. 358-366
Author(s):  
H. S. Harutyunyan ◽  
V. K. Ohanyan
Keyword(s):  
Distribution Function ◽  
Regular Polygon ◽  
Length Distribution ◽  
Chord Length ◽  
Regular Hexagon ◽  
Regular Polygons ◽  
Chord Length Distribution ◽  
Regular Pentagon ◽  
Regular Triangle ◽  
Elementary Expression

In this paper we obtain an elementary expression for the chord length distribution function of a regular polygon. The formula is derived using δ-formalism in Pleijel identity. In the particular cases of a regular triangle, a square, a regular pentagon, and a regular hexagon, our formula coincides with the results of Sulanke (1961), Gille (1988), Aharonyan and Ohanyan (2005), and Harutyunyan (2007), respectively.

scholarly journals The chord length distribution function of a non-convex hexagon

2018 ◽  
Vol 9 (1) ◽  
pp. 20-34
Author(s):  
Uwe Bäsel ◽  
Vittoria Bonanzinga ◽  
Andrei Duma
Keyword(s):  
Distribution Function ◽  
Density Function ◽  
Length Distribution ◽  
Expected Value ◽  
Chord Length ◽  
Chord Length Distribution

Abstract In this paper we obtain the chord length distribution function of a non-convex equilateral hexagon and then derive the associated density function. Finally, we calculate the expected value of the chord length.

scholarly journals Effect of Pre-Shear on Agglomeration and Rheological Parameters of Cement Paste

Materials ◽  
2020 ◽  
Vol 13 (9) ◽  
pp. 2173
Author(s):  
Mareike Thiedeitz ◽  
Inka Dressler ◽  
Thomas Kränkel ◽  
Christoph Gehlen ◽  
Dirk Lowke
Keyword(s):  
Yield Stress ◽  
Mixing Time ◽  
Length Distribution ◽  
Chord Length ◽  
Rheological Parameters ◽  
Reflectance Measurement ◽  
Time Dynamic ◽  
Chord Length Distribution ◽  
In Situ Investigation

Cementitious pastes are multiphase suspensions that are rheologically characterized by viscosity and yield stress. They tend to flocculate during rest due to attractive interparticle forces, and desagglomerate when shear is induced. The shear history, e.g., mixing energy and time, determines the apparent state of flocculation and accordingly the particle size distribution of the cement in the suspension, which itself affects suspension’s plastic viscosity and yield stress. Thus, it is crucial to understand the effect of the mixing procedure of cementitious suspensions before starting rheological measurements. However, the measurement of the in-situ particle agglomeration status is difficult, due to rapidly changing particle network structuration. The focused beam reflectance measurement (FBRM) technique offers an opportunity for the in-situ investigation of the chord length distribution. This enables to detect the state of flocculation of the particles during shear. Cementitious pastes differing in their solid fraction and superplasticizer content were analyzed after various pre-shear histories, i.e., mixing times. Yield stress and viscosity were measured in a parallel-plate-rheometer and related to in-situ measurements of the chord length distribution with the FBRM-probe to characterize the agglomeration status. With increasing mixing time agglomerates were increasingly broken up in dependence of pre-shear: After 300 s of pre-shear the agglomerate sizes decreased by 10 µm to 15 µm compared to a 30 s pre-shear. At the same time dynamic yield stress and viscosity decreased up to 30% until a state of equilibrium was almost reached. The investigations show a correlation between mean chord length and the corresponding rheological parameters affected by the duration of pre-shear.

scholarly journals Contact and Chord Length Distribution Functions of the Poisson-Voronoi Tessellation in High Dimensions

2010 ◽  
Vol 42 (1) ◽  
pp. 48-68 ◽  
Author(s):  
L. Muche
Keyword(s):  
Voronoi Tessellation ◽  
Length Distribution ◽  
Distribution Functions ◽  
Chord Length ◽  
Dimensional Euclidean Space ◽  
Spherical Contact ◽  
Chord Length Distribution ◽  
Special Cases ◽  
Contact Distribution ◽  
Contact Distribution Function

In this paper we present formulae for contact distributions of a Voronoi tessellation generated by a homogeneous Poisson point process in the d-dimensional Euclidean space. Expressions are given for the probability density functions and moments of the linear and spherical contact distributions. They are double and simple integral formulae, which are tractable for numerical evaluation and for large d. The special cases d = 2 and d = 3 are investigated in detail, while, for d = 3, the moments of the spherical contact distribution function are expressed by standard functions. Also, the closely related chord length distribution functions are considered.

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