scholarly journals The Double Six of Lines and a Theorem in Euclidean Plane Geometry

1952 ◽  
Vol 1 (1) ◽  
pp. 1-7 ◽  
Author(s):  
John Dougall
Keyword(s):  
Projective Space ◽  
Euclidean Plane ◽  
Natural Extension ◽  
Plane Geometry ◽  
Three Dimensions ◽  
Classical Theorem ◽  
Invariant Relation ◽  
Geometrical Form

The object of the present paper is to establish the equivalence of the well-known theorem of the double-six of lines in projective space of three dimensions and a certain theorem in Euclidean plane geometry. The latter theorem is of considerable interest in itself for two reasons. In the first place, it is a natural extension of Euler's classical theorem connecting the radii of the circumscribed and the inscribed (or the escribed) circles of a triangle with the distance between their centres. Secondly, it gives in a geometrical form the invariant relation between the circle circumscribed to a triangle and a conic inscribed in the triangle. For a statement of the theorem, see § 13 (4).

scholarly journals Discourse on service modularity: investigating service delivery process

2019 ◽  
Author(s):  
Ilona Skačkauskienė ◽  
Jurga Vestertė
Keyword(s):  
Service Delivery ◽  
Service Provider ◽  
Service Management ◽  
Natural Extension ◽  
Abductive Reasoning ◽  
Three Dimensions ◽  
Systemic Analysis ◽  
Provider Perspectives ◽  
Service Offering ◽  
E Mail

*E-mail: [email protected] Abstract. Purpose – the purpose of this research is to determine aspects of the service delivery process what must be considered for modularisation. The aim is reached through (1) investigating the process construct; (2) describing and schematizing service delivery process through integration of customer and provider perspectives; (3) ascertaining the as-pects of the service delivery process modularisation. Research methodology – the article is built on an overview of the scientific literature dealing with the topic, using meth-ods of comparative analysis, systemic analysis, abstraction, synthesis, abductive reasoning. Findings – for achieving service modularisation, the service provider may apply standardisation and automation methods on three dimensions of service delivery process: (1) service offering; (2) parts of the service process that are managed by the provider; (3) organisational structure of the provider. Research limitations – the study examines the aspects of modularity only on the conceptual level. A natural extension of this research is an empirical investigation of the introduced approaches. Practical implications – the proposed approaches help practitioners in the decision-making process for a service delivery process modularisation. Originality/Value – the study approaches the modularisation of the service delivery process considering the customer and service provider perspectives and fills the gap in the literature on service modularisation management. Keywords: services; service modularity; service delivery process; service management.

scholarly journals Formalization of the arithmetization of Euclidean plane geometry and applications

2019 ◽  
Vol 90 ◽  
pp. 149-168 ◽  
Author(s):  
Pierre Boutry ◽  
Gabriel Braun ◽  
Julien Narboux
Keyword(s):  
Euclidean Plane ◽  
Plane Geometry

scholarly journals One Parametric Motion in Klein Spaces

1950 ◽  
Vol 1 ◽  
pp. 19-23
Author(s):  
Minoru Kurita
Keyword(s):  
Euclidean Space ◽  
Projective Space ◽  
Euclidean Plane ◽  
Parametric Motion ◽  
Fixed Curve ◽  
Fixed Center

On the euclidean plane one-parametric motion is in general a roulett motion, exceptions being a translation at each instant and a rotation with a fixed center; here we mean by a roulett motion a motion in which a certain curve rolls on another fixed curve without slipping. In this paper we extend this fact to the case of Klein spaces and investigate in detail especially the cases of the euclidean space and the projective space.

Theoretical Relationships for the Lift and Drag of an Aerofoil Structure

1923 ◽  
Vol 27 (154) ◽  
pp. 512-518
Author(s):  
H. Glauert
Keyword(s):  
Physical Properties ◽  
Relative Motion ◽  
Three Dimensions ◽  
General Nature ◽  
Essential Characteristic ◽  
Dominant Component ◽  
Induced Drag ◽  
Vortex Theory ◽  
Geometrical Form ◽  
Lift And Drag

It is a fact of common experience that a body in motion relative to a gas or liquid is subject to a resultant force, and it is customary to resolve this force into two components, the drag opposing the relative motion and the lift at right-angles to the direction of this motion. In general the drag is the pre-dominant component, but the class of bodies known as aerofoils and used for the construction of aeroplane wings, is such that the lift is considerably in excess of the drag. The present discussion relates solely to this class of bodies whose essential characteristic is the production of a large lift correlated with a relatively small drag. It is a matter of very considerable importance to develop a theory which will explain the origin of the forces experienced by an aerofoil, and will provide a method of calculating the characteristics of any aerofoil structure from a knowledge of its geometrical form and of the physical properties of the fluid through which it moves. It is proposed to discuss the behaviour of aerofoils in two and in three dimensions, and in particular to discuss the vortex theory of lift and induced drag. The general nature of the flow pattern on which the theory is based has been described by Lanchester, but the mathematical development of the thcorv is due to Prandtl and his colleagues.

Euclidean constructions

1954 ◽  
Vol 47 (4) ◽  
pp. 231-233
Author(s):  
Robert C. Yates
Keyword(s):  
Euclidean Plane ◽  
Plane Geometry

In the spirit of the old-time revival and the spring tonic, I feel it periodically necessary to “reaffirm the faith” and refresh myself in the fundamental constructions of Euclidean plane geometry. It seems always such a satisfying experience that I wish to share it. My refreshment takes the following form.

On a new analytical representation of curves in space

1947 ◽  
Vol 43 (4) ◽  
pp. 455-458 ◽  
Author(s):  
D. Pedoe
Keyword(s):  
Projective Space ◽  
Analytical Representation ◽  
Three Dimensions ◽  
Image Position

It was in a paper bearing this title that Cayley(1) first considered the problem of representing a curve in projective space of three dimensions by means of the complex of lines which meet the curve. He took the conic given by the equationsand found that the linewith dual Grassmann coordinates (…,pij,…), whereintersects the conic if, and only if,where F(u0, u1) is homogeneous and of degree 2 in both sets of indeterminates u0 and u1 and G(…,pij,…) is a form of degree 2 in the pij. Both F(u0, u1) and G(…,pij,…) are easily determined in this case.

The Elements of Non-Euclidean Plane Geometry and Trigonometry

1919 ◽  
Vol 9 (141) ◽  
pp. 374
Author(s):  
Philip E. B. Jourdain ◽  
H. S. Carslaw
Keyword(s):  
Euclidean Plane ◽  
Plane Geometry

Is a linear space contained in a submanifold? – On the number of derivatives needed to tell

1999 ◽  
Vol 1999 (508) ◽  
pp. 53-60
Author(s):  
J. M Landsberg
Keyword(s):  
Projective Space ◽  
Linear Space ◽  
General Problem ◽  
General Point ◽  
Classical Theorem ◽  
Linear Spaces ◽  
Dual Varieties

Abstract Let Xn ⊂ or Xn ⊂ ℙn + a be a patch of a C∞ submanifold of an affine or projective space such that through each point x ∈ X there exists a line osculating to order n + 1 at x. We show that X is uniruled by lines, generalizing a classical theorem for surfaces. We describe two circumstances that imply linear spaces of dimension k osculating to order two must be contained in X, shedding light on some of Ein's results on dual varieties. We present some partial results on the general problem of finding the integer m0 = m0(k, n, a) such that there exist examples of patches Xn ⊂ ℙn + a, having a linear space L of dimension k osculating to order m0 — 1 at each point such that L is not locally contained in X, but if there are k-planes osculating to order m0 at each point, they are locally contained in X. The same conclusions hold in the analytic category and complex analytic category if there is a linear space osculating to order m at one general point x ∈ X.

Hitting probabilities for random ellipses and ellipsoids

1993 ◽  
Vol 30 (04) ◽  
pp. 971-974 ◽  
Author(s):  
Andrei Duma ◽  
Marius Stoka
Keyword(s):  
Distribution Function ◽  
Euclidean Plane ◽  
Three Dimensions ◽  
Dependence Structure ◽  
Rectangular Lattice ◽  
Hitting Probabilities ◽  
Mean And Variance ◽  
Function Density

Let denote a rectangular lattice in the Euclidean plane E 2, generated by (a × b) rectangles. In this paper we consider the probability that a random ellipse having main axes of length 2α and 2ß, with intersects . We regard the lattice as the union of two orthogonal sets and of equidistant lines and evaluate the probability that the random ellipse intersects or . Moreover, we consider the dependence structure of the events that the ellipse intersects or . We study further the case when the main axes of the ellipse are parallel to the lines of the lattice and satisfy 2ß = min (a, b) < 2α = max (a, b). In this case, the probability of intersection is 1, and there exist almost surely two perpendicular segments in within the ellipse. We evaluate the distribution function, density, mean and variance of the length of these segments. We conclude with a generalization of this problem in three dimensions.

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