scholarly journals Proving and Generalizing Desargues’ Two-Triangle Theorem in 3-Dimensional Projective Space

Geometry ◽  
2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Dimitrios Kodokostas
Keyword(s):  
Projective Space ◽  
Arbitrary Number ◽  
Three Dimensional ◽  
3 Dimensional ◽  
Dimensional Projective Space ◽  
Set Of Points

With the use of only the incidence axioms we prove and generalize Desargues’ two-triangle Theorem in three-dimensional projective space considering an arbitrary number of points on each one of the two distinct planes allowing corresponding points on the two planes to coincide and three points on any of the planes to be collinear. We provide three generalizations and we define the notions of a generalized line and a triangle-connected plane set of points.

Transformations depending on sets of associated points

1952 ◽  
Vol 48 (3) ◽  
pp. 383-391
Author(s):  
T. G. Room
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Three Dimensional ◽  
Cubic Curves ◽  
Birational Transformations ◽  
Quadric Surfaces ◽  
Base Points ◽  
Dimensional Projective Space ◽  
Quartic Curves

This paper falls into three sections: (1) a system of birational transformations of the projective plane determined by plane cubic curves of a pencil (with nine associated base points), (2) some one-many transformations determined by the pencil, and (3) a system of birational transformations of three-dimensional projective space determined by the elliptic quartic curves through eight associated points (base of a net of quadric surfaces).

scholarly journals Classification of 3T1R Parallel Manipulators Based on Their Wrench Graph

2016 ◽  
Vol 9 (1) ◽  
Author(s):  
Semaan Amine ◽  
Ossama Mokhiamar ◽  
Stéphane Caro
Keyword(s):  
Projective Space ◽  
Parallel Manipulator ◽  
Three Dimensional ◽  
Parallel Manipulators ◽  
Geometrical Properties ◽  
Cayley Algebra ◽  
Dimensional Projective Space ◽  
Grassmann Cayley Algebra

This paper presents a classification of 3T1R parallel manipulators (PMs) based on the wrench graph. By using the theory of reciprocal screws, the properties of the three-dimensional projective space, the wrench graph, and the superbracket decomposition of Grassmann–Cayley algebra, six typical wrench graphs for 3T1R parallel manipulators are obtained along with their singularity conditions. Furthermore, this paper shows a way in which each of the obtained typical wrench graphs can be used in order to synthesize new 3T1R parallel manipulator architectures with known singularity conditions and with an understanding of their geometrical properties and assembly conditions.

scholarly journals On principal surfaces of quasi-special complexes with twofold fibration in $P_3$

1987 ◽  
pp. 120
Author(s):  
Ye.N. Ishchenko
Keyword(s):  
Projective Space ◽  
Three Dimensional ◽  
Dimensional Projective Space ◽  
Principal Surfaces

We prove that, if one of principal surfaces of quasi-special complexes with twofold fibration in three-dimensional projective space $P_3$ is a quadric, then this complex is complemented by second complex, which allows twofold fibration as well.

scholarly journals A quadratic transformation of the three-dimensional space of all conics through two points of a projective plane

2009 ◽  
Vol 42 (3) ◽  
Author(s):  
Giorgio Donati
Keyword(s):  
Projective Space ◽  
Projective Plane ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The Other ◽  
Quadratic Transformation ◽  
Pencil Of Conics ◽  
Dimensional Projective Space ◽  
Three Dimensional Space

AbstractUsing the Steiner’s method of projective generation of conics and its dual we define two projective mappings of a double contact pencil of conics into itself and we prove that one is the inverse of the other. We show that these projective mappings are induced by quadratic transformations of the three-dimensional projective space of all conics through two distinct points of a projective plane.

Maximal Strictly Partial Spreads

1978 ◽  
Vol 30 (03) ◽  
pp. 483-489 ◽  
Author(s):  
Gary L. Ebert
Keyword(s):  
Projective Space ◽  
Partial Spread ◽  
3 Dimensional ◽  
Partial Spreads ◽  
Maximal Partial Spread ◽  
Dimensional Projective Space

Let ∑ = PG(3, q) denote 3-dimensional projective space over GF(q). A partial spread of ∑ is a collection W of pairwise skew lines in ∑. W is said to be maximal if it is not properly contained in any other partial spread. If every point of ∑ is contained in some line of W, then W is called a spread. Since every spread of PG(3, q) consists of q2 + 1 lines, the deficiency of a partial spread W is defined to be the number d = q2 + 1 — |W|. A maximal partial spread of ∑ which is not a spread is called a maximal strictly partial spread (msp spread) of ∑.

THE INVARIANT MEASURES AT WEAK DISORDER FOR THE TWO-LINE ANDERSON MODEL

2004 ◽  
Vol 16 (05) ◽  
pp. 639-673
Author(s):  
T. C. DORLAS ◽  
J. V. PULÉ
Keyword(s):  
Projective Space ◽  
Anderson Model ◽  
Invariant Measures ◽  
Three Dimensional ◽  
Weak Disorder ◽  
Single Chain ◽  
Zero Energy ◽  
Dimensional Projective Space

We study the invariant measures in the weak disorder limit, for the Anderson model on two coupled chains. These measures live on a three-dimensional projective space, and we use a total set of functions on this space to characterize the measures. We find that at several points of the spectrum, there are anomalies similar to that first found by Kappus and Wegner for the single chain at zero energy.

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