The Elliptic Polarity of Screws

1985 ◽  
Vol 107 (3) ◽  
pp. 377-386 ◽  
Author(s):  
H. Lipkin ◽  
J. Duffy
Keyword(s):  
Euclidean Space ◽  
Conformal Mapping ◽  
Complex Plane ◽  
Robot Manipulators ◽  
Constrained Motion ◽  
Planar Representation ◽  
Invariant Properties ◽  
Quaternion Representation

The nature and invariant properties of the elliptic polarity of screws in Euclidean space is established using a novel series of mappings, central to which is a quaternion representation. The role of the elliptic polarity in modeling constrained motion is detailed and illustrated by way of an example which has application in the control of robot manipulators. Ball’s planar representation of the two system of screws is generalized and is shown to be a representation on a complex plane where the elliptic polarity is a conformal mapping.

Circles on the Complex Plane

2021 ◽  
pp. 3-12
Author(s):  
A. Girsh
Keyword(s):  
Euclidean Space ◽  
Complex Plane ◽  
Euclidean Plane ◽  
Complex Space ◽  
Radical Center ◽  
Special Cases ◽  
Different Types ◽  
Different Dimensions ◽  
The Given ◽  
Made In

The Euclidean plane and Euclidean space themselves do not contain imaginary elements by definition, but are inextricably linked with them through special cases, and this leads to the need to propagate geometry into the area of imaginary values. Such propagation, that is adding a plane or space, a field of imaginary coordinates to the field of real coordinates leads to various variants of spaces of different dimensions, depending on the given axiomatics. Earlier, in a number of papers, were shown examples for solving some urgent problems of geometry using imaginary geometric images [2, 9, 11, 13, 15]. In this paper are considered constructions of orthogonal and diametrical positions of circles on a complex plane. A generalization has been made of the proposition about a circle on the complex plane orthogonally intersecting three given spheres on the proposition about a sphere in the complex space orthogonally intersecting four given spheres. Studies have shown that the diametrical position of circles on the Euclidean E-plane is an attribute of the orthogonal position of the circles’ imaginary components on the pseudo-Euclidean M-plane. Real, imaginary and degenerated to a point circles have been involved in structures and considered, have been demonstrated these circles’ forms, properties and attributes of their orthogonal position. Has been presented the construction of radical axes and a radical center for circles of the same and different types. A propagation of 2D mutual orthogonal position of circles on 3D spheres has been made. In figures, dashed lines indicate imaginary elements.

Variable Structure Hybrid Control of Manipulators in Unconstrained and Constrained Motion

1996 ◽  
Vol 118 (2) ◽  
pp. 327-332 ◽  
Author(s):  
Robert R. Y. Zhen ◽  
Andrew A. Goldenberg
Keyword(s):  
Control Method ◽  
Hybrid Control ◽  
Sliding Mode ◽  
Robot Manipulators ◽  
Hierarchical Control ◽  
Variable Structure Control ◽  
Variable Structure ◽  
Control Algorithms ◽  
Constrained Motion ◽  
Structure Control

This paper addresses the problem of robust hybrid position and force control of robot manipulators. Variable structure control with sliding mode is used to implement the hybrid control strategy. Two variable structure control algorithms are developed in task space. One of the algorithms is based on hierarchical control method, and the other is developed for control of robot manipulators used to carried out both unconstrained and constrained tasks.

scholarly journals The group of isometries on Hardy spaces of the n-ball and the polydisc

1980 ◽  
Vol 21 (2) ◽  
pp. 199-204 ◽  
Author(s):  
Earl Berkson ◽  
Horacio Porta
Keyword(s):  
Euclidean Space ◽  
Unit Ball ◽  
Complex Plane ◽  
Hardy Spaces ◽  
Inner Product ◽  
Euclidean Norm ◽  
Dimensional Euclidean Space ◽  
Open Unit ◽  
Open Unit Ball ◽  
Group Of Isometries

Let C be the complex plane, and U the disc |Z| < 1 in C. Cn denotes complex n-dimensional Euclidean space, <, > the inner product, and | · | the Euclidean norm in Cn;. Bn will be the open unit ball {z ∈ Cn:|z| < 1}, and Un will be the unit polydisc in Cn. For l ≤ p < ∞, p ≠ 2, Gp(Bn) (resp., Gp(Un)) will denote the group of all isometries of Hp(Bn) (resp., Hp(Un)) onto itself, where Hp(Bn) and HP(Un) are the usual Hardy spaces.

Airfoils with separation and the resulting wakes

1986 ◽  
Vol 163 ◽  
pp. 323-347 ◽  
Author(s):  
Tuncer Cebeci ◽  
R. W. Clark ◽  
K. C. Chang ◽  
N. D. Halsey ◽  
K. Lee
Keyword(s):  
Boundary Layer ◽  
Conformal Mapping ◽  
Flow Separation ◽  
Close Agreement ◽  
Eddy Viscosity ◽  
Lift Coefficient ◽  
Inviscid Flow ◽  
Interaction Method ◽  
Large Flow

A viscous/inviscid interaction method is described and has been used to calculate flows around four distinctly different airfoils as a function of angle of attack. It comprises an inviscid-flow method based on conformal mapping, a boundary-layer procedure based on the numerical solution of differential equations and an algebraic eddy viscosity. The results are in close agreement with experiment up to angles close to stall. In one case, where the airfoil thickness is large, small difficulties were experienced and are described. The method is shown to be capable of obtaining results with large flow separation and quantifies the role of transition on the lift coefficient.

scholarly journals The Scepticism of Descartes’s Meditations

2011 ◽  
Vol 67 (2) ◽  
pp. 271-279
Author(s):  
James Thomas
Keyword(s):  
Euclidean Space ◽  
Thirteenth Century ◽  
Sensory Experience ◽  
Controlling Factor ◽  
New Science ◽  
The Will ◽  
Aristotelian Science ◽  
The Subject

What I’m suggesting is that the model for Descartes’s defence of Renaissance science would be Aquinas’s own defence of thirteenth-century Aristotelian science, except that the coherence of the will took on the role of the consistency of concepts, as the controlling factor in the analyses of all types of science. As a result, the new science would incorporate the awareness of Platonic ideas and the divisibility of Euclidean space as equally valid input into a dialectical knowledge of sensory experience. You can read the early arguments to doubt the reality of sensory experience and reason as a way of dividing out the experience of the will in affirming or denying an object’s nature, as the subject for subsequent inquiry.

scholarly journals On Certain Mappings of Riemannian Manifolds

1965 ◽  
Vol 25 ◽  
pp. 121-142
Author(s):  
Minoru Kurita
Keyword(s):  
Differential Geometry ◽  
Euclidean Space ◽  
Conformal Mapping ◽  
Riemannian Manifolds ◽  
Stereographic Projection ◽  
Point Of View ◽  
The Other ◽  
Central Projection ◽  
Special Cases ◽  
Special Case

In this paper we consider certain tensors associated with differentiable mappings of Riemannian manifolds and apply the results to a p-mapping, which is a special case of a subprojective one in affinely connected manifolds (cf. [1], [7]). The p-mapping in Riemannian manifolds is a generalization of a conformal mapping and a projective one. From a point of view of differential geometry an analogy between these mappings is well known. On the other hand it is interesting that a stereographic projection of a sphere onto a plane is conformal, while a central projection is projectve, namely geodesic-preserving. This situation was clarified partly in [6]. A p-mapping defined in this paper gives a precise explanation of this and also affords a certain mapping in the euclidean space which includes a similar mapping and an inversion as special cases.

scholarly journals On the derivatives at the origin of entire harmonic functions

1979 ◽  
Vol 20 (2) ◽  
pp. 147-154 ◽  
Author(s):  
D. H. Armitage
Keyword(s):  
Entire Function ◽  
Euclidean Space ◽  
Complex Plane ◽  
Harmonic Functions ◽  
Image Position ◽  
Derivatives Of

If f is an entire function in the complex plane such thatwhere 0 ≤ α < 1, and all the derivatives of f at 0 are integers, then it is easy to show that f is a polynomial (see e.g. Straus [10]). The best possible result of this type was proved by Pólya [9]. The main aim of this paper is to prove two analogous results for harmonic functions defined in the whole of the Euclidean space Rn, where n ≥ 2 (i.e. entire harmonic functions).

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