A Line Geometric Foundation for Finite Screw Systems Associated With Spatial Linkages

2011 ◽  
Author(s):  
Chintien Huang ◽  
Tzu-Cheng Hsing
Keyword(s):  
Linear Complex ◽  
Linear Complexes ◽  
Intersection Operation ◽  
Linear Congruences ◽  
Spatial Linkages ◽  
Overconstrained Linkages

This paper investigates the line varieties corresponding to finite screw systems associated with spatial linkages. This research is based on the correspondence between a screw and a linear complex, and a screw system corresponds to the intersection of the linear complexes. In finite kinematics, two screw systems associated with the finite motions of the revolute-revolute (R-R) and prismatic-revolute (P-R) open chains have been discovered. These two screw systems also led to the discovery of the screw systems associated with the finite motions of the spatial 4R, spatial RPRP, and other overconstrained linkages. By using the intersection operation of linear complexes, this paper finds the linear reguli corresponding to the finite motions of R-R and P-R chains. Then we utilize the sum operation of the linear reguli corresponding to the R-R and P-R chains to obtain the hyperbolic linear congruences corresponding to the finite motions of the spatial 4R and RPRP linkages. The result presented here serves as a line geometric foundation for finite screw systems associated with spatial linkages. In addition, with CAD drawings, this paper enables the visualization of the obtained line varieties and their operations.

scholarly journals Pencils of Null Polarities

1959 ◽  
Vol 11 ◽  
pp. 614-620
Author(s):  
Seymour Schuster
Keyword(s):  
Complex Projective Space ◽  
Three Dimensions ◽  
Central Question ◽  
Linear Complex ◽  
Different Types ◽  
Linear Complexes ◽  
Linear Congruences ◽  
Null Polarity ◽  
Complex Projective

A theorem due to von Staudt states that a null polarity in complex projective space of three dimensions is determined by a selfpolar skew pentagon. By allowing an element of the self-polar pentagon to vary in a suitable manner we can arrive at a family of ∞1 null polarities, which we term a pencil of null polarities. Each polarity of the pencil distinguishes a linear complex as the class of self-polar lines. Thus, associated with the pencil is a family of ∞1 linear complexes, which we term a pencil of linear complexes.It is the purpose of this paper to continue an earlier investigation of pencils of polarities (2), by applying analogous techniques to the study of pencils of null polarities and pencils of linear complexes.Since it develops that the lines common to all linear complexes of a pencil are the lines of a linear congruence, the central question has been: How many of the different types of linear congruences can be achieved in this manner? Happily, it can be reported that the classification of pencils of null polarities yields all of the three types of linear congruences (4, pp. 140-141).

The Linear Complexes belonging to the Invariant System of Three Quadrics

1923 ◽  
Vol 42 ◽  
pp. 29-40 ◽  
Author(s):  
J. Williamson
Keyword(s):  
Invariant Theory ◽  
London Mathematical Society ◽  
Point Of View ◽  
Mathematical Society ◽  
Invariant System ◽  
Linear Complex ◽  
Projective Invariant ◽  
Linear Complexes

In the Proceedings of the London Mathematical Society, Ser. 2, Vol. 20 (1921), pp. 465–489, Professor H. W. Turnbull has studied the projective invariant theory of three quadrics. The following paper is based on this work and develops one definite section of the theory. From the geometrical point of view the linear complex is now seen to be fundamental in the study of three arbitrary quadrics; particularly when their (2, 2, 2) invariant φ123 vanishes.

scholarly journals The Complete System of Linear Complexes and one Quaternary Quadric

1930 ◽  
Vol 2 (2) ◽  
pp. 61-70
Author(s):  
P. N. Das Gupta ◽  
H. W. Turnbull
Keyword(s):  
Complete System ◽  
Linear Complex ◽  
Linear Complexes

In the following pages we give a complete system of projective concomitants for any number of linear complexes and one quadric, in a quaternary field. The investigation follows the earlier1 paper on mixed quaternary forms, in which the corresponding system, omitting the quadric, was given. Such a system, for one quadric and one linear complex was given by Weitzenböck who used complex symbols. A detailed investigation of the case of two linear complexes and a quadric, together with their geometry, has been given elsewhere.

scholarly journals Influence of Copper(I) Halides on the Reactivity of Aliphatic Carbodiimides

2020 ◽  
Vol 3 (1) ◽  
pp. 20
Author(s):  
Valentina Ferraro ◽  
Marco Bortoluzzi
Keyword(s):  
Density Functional Theory ◽  
Electronic Structure ◽  
General Formula ◽  
Electron Density ◽  
Density Functional ◽  
Functional Theory ◽  
Fukui Functions ◽  
Bond Lengths ◽  
Linear Complexes ◽  
Nitrogen Bond

The influence of copper(I) halides CuX (X = Cl, Br, I) on the electronic structure of N,N′-diisopropylcarbodiimide (DICDI) and N,N′-dicyclohexylcarbodiimide (DCC) was investigated by means of computational DFT (density functional theory) methods. The coordination of the considered carbodiimides occurs by one of the nitrogen atoms, with the formation of linear complexes having a general formula of [CuX(carbodiimide)]. Besides varying the carbon–nitrogen bond lengths, the thermodynamically favourable interaction with Cu(I) reduces the electron density on the carbodiimides and alters the energies of the (NCN)-centred, unoccupied orbitals. A small dependence of these effects on the choice of the halide was observable. The computed Fukui functions suggested negligible interaction of Cu(I) with incoming nucleophiles, and the reactivity of carbodiimides was altered by coordination mainly because of the increased electrophilicity of the {NCN} fragments.

The elliptic products of Jacobi and the theory of linear congruences

1926 ◽  
Vol 23 (4) ◽  
pp. 337-355
Author(s):  
P. A. MacMahon
Keyword(s):  
Elliptic Functions ◽  
Linear Congruences ◽  
Image Position ◽  
Theory Of Numbers

In the application of Elliptic Functions to the Theory of Numbers the two formulae of Jacobiare of great importance.

Discrete positions method in kinematics and control of spatial linkages

1980 ◽  
Vol 15 (1) ◽  
pp. 47-60 ◽  
Author(s):  
M.S Konstantinov ◽  
M.D Markov
Keyword(s):  
Spatial Linkages ◽  
And Control
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