scholarly journals On the relation between GDOP and the volume described by the user-to-satellite unit vectors for GNSS positioning

GPS Solutions ◽  
2017 ◽  
Vol 21 (3) ◽  
pp. 1139-1147 ◽  
Author(s):  
Nuria Blanco-Delgado ◽  
Fernando Duarte Nunes ◽  
Gonzalo Seco-Granados
Keyword(s):  
Gnss Positioning ◽  
Satellite Unit ◽  
Unit Vectors

scholarly journals Some Improvements on Relativistic Positioning Systems

2018 ◽  
Vol 3 (1) ◽  
pp. 161-166 ◽  
Author(s):  
Màrius Josep Fullana i Alfonso ◽  
Diego Pascual Sáez Milán ◽  
Josep Vicent Arnau i Córdoba ◽  
Neus Puchades Colmenero
Keyword(s):  
Transformation Matrix ◽  
Positioning Systems ◽  
Satellite Unit ◽  
Unit Vectors

AbstractWe make some considerations about Relativistic Positioning Systems (RPS). Four satellites are needed to position a user. First of all we define the main concepts. Errors should be taken into account. Errors depend on the Jacobian transformation matrix. Its Jacobian is proportional to the tetrahedron volume whose vertexes are the four tips of the receiver-satellite unit vectors. If the four satellites are seen by the user on a circumference in the sky, then, the Jacobian and the tetrahedron volume vanish. The users we consider are spacecraft. Spacecraft to be positioned cannot be close to a null Jacobian satellites-user configuration. These regions have to be avoided choosing an appropriate set of four satellites which are not seen too close to the same circumference in the sky. Errors also increase as the user spacecraft separates from the emission satellite region, since the tetrahedron volume decreases.We propose a method to autonomously potion a user-spacecraft which can test our method. This positioning should be compared with those obtained by current methods. Finally, a proposal to position a user-spacecraft moving far from Earth, with suitable devices (autonomous), is presented.

Crystallochemical classifications of phyllosilicates based on the unified system of projection of chemical composition: I. The mica group

Clay Minerals ◽  
1990 ◽  
Vol 25 (1) ◽  
pp. 73-81 ◽  
Author(s):  
A. Wiewióra
Keyword(s):  
Chemical Composition ◽  
Divalent Cations ◽  
Layer Charge ◽  
Vector Representation ◽  
Similar Formula ◽  
Crystallochemical Formula ◽  
Trivalent Cations ◽  
Image Position ◽  
Composition I ◽  
Unit Vectors

AbstractA unified system of vector representation of chemical composition is proposed for the phyllosilicates based on projection of the composition, as given by crystallochemical formula, onto a field with orthogonal axes chosen for octahedral divalent cations, R2+, and Si (X, Y, respectively), and oblique axes for octahedral trivalent cations, R3+, and vacancies, □, (V, Z, respectively). Point coordinates for each set of axes were used to define the direction and length of the unit vectors for phyllosilicates belonging to different groups. Parallel to these fundamental directions the composition isolines were drawn in the projection fields. Applied to micas, this system enables control of the chemical composition by the general crystallochemical formula covering all varieties of Li-free dioctahedral and trioctahedral micas:where z (number of vacancies) = (y-x+ m)/2; m (layer charge) =1; u+y+z = 3. There is a similar formula for vacancy-free lithian micas:where w = m — x+y;m=1; u+y+w = 3, and for Li-free brittle micas:where z = (y — x+m)/2; m = 2; u+y+z = 3. Projection fields were used to classify micas.

Consistency conditions for a finite set of projections of a function

1979 ◽  
Vol 85 (1) ◽  
pp. 61-68 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Lebesgue Measure ◽  
Convex Subset ◽  
Compact Convex ◽  
Unit Vector ◽  
Compact Convex Subset ◽  
Finite Collection ◽  
Finite Set ◽  
Almost All ◽  
Dimensional Lebesgue Measure ◽  
Unit Vectors

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.

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