scholarly journals Combining Methods of Euclidean, Affine, and Projective Geometries in Solving Geometric Problems

2021 ◽  
Vol 24 (3) ◽  
pp. 563-580
Author(s):  
Vadim Vasilievich Shurygin ◽  
Vadim Vadimovich Shyrygin
Keyword(s):  
Projective Geometry ◽  
Scalar Product ◽  
The Other ◽  
Geometric Problems ◽  
Projective Geometries ◽  
Combining Methods

The aim of the paper is to demonstrate how the techniques of one of the geometries indicated in the title can be used for solving problems formulated in the framework of one of the other geometries. In particular, it is shown how problems formulated in the framework of affine or projective geometry can be solved with an appropriate choice of Euclidean scalar product.

Projective Geometries that are Disjoint Unions of Caps

1980 ◽  
Vol 32 (6) ◽  
pp. 1299-1305 ◽  
Author(s):  
Barbu C. Kestenband
Keyword(s):  
Finite Field ◽  
Projective Geometry ◽  
Prime Power ◽  
Disjoint Union ◽  
Hermitian Matrices ◽  
Incidence Relation ◽  
Projective Geometries ◽  
Image Position ◽  
Natural Way ◽  
Square Matrix

We show that any PG(2n, q2) is a disjoint union of (q2n+1 − 1)/ (q − 1) caps, each cap consisting of (q2n+1 + 1)/(q + 1) points. Furthermore, these caps constitute the “large points” of a PG(2n, q), with the incidence relation defined in a natural way.A square matrix H = (hij) over the finite field GF(q2), q a prime power, is said to be Hermitian if hijq = hij for all i, j [1, p. 1161]. In particular, hii ∈ GF(q). If if is Hermitian, so is p(H), where p(x) is any polynomial with coefficients in GF(q).Given a Desarguesian Projective Geometry PG(2n, q2), n > 0, we denote its points by column vectors:All Hermitian matrices in this paper will be 2n + 1 by 2n + 1, n > 0.

An Easy Proof for Some Classical Theorems in Plane Geometry

1992 ◽  
Vol 35 (4) ◽  
pp. 560-568 ◽  
Author(s):  
C. Thas
Keyword(s):  
Projective Plane ◽  
Projective Geometry ◽  
The Other ◽  
Plane Geometry ◽  
The Real ◽  
Easy Proof ◽  
Special Cases ◽  
Euclidean Planes ◽  
Straight Lines ◽  
Short Method

AbstractThe main result of this paper is a theorem about three conies in the complex or the real complexified projective plane. Is this theorem new? We have never seen it anywhere before. But since the golden age of projective geometry so much has been published about conies that it is unlikely that no one noticed this result. On the other hand, why does it not appear in the literature? Anyway, it seems interesting to "repeat" this property, because several theorems in connection with straight lines and (or) conies in projective, affine or euclidean planes are in fact special cases of this theorem. We give a few classical examples: the theorems of Pappus-Pascal, Desargues, Pascal (or its converse), the Brocard points, the point of Miquel. Finally, we have never seen in the literature a proof of these theorems using the same short method see the proof of the main theorem).

scholarly journals Undecidable theories of Lyndon algebras

2001 ◽  
Vol 66 (1) ◽  
pp. 207-224 ◽  
Author(s):  
Vera Stebletsova ◽  
Yde Venema
Keyword(s):  
Projective Geometry ◽  
Algebraic Logic ◽  
Equational Theory ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
Projective Geometries ◽  
Interesting Connection

AbstractWith each projective geometry we can associate a Lyndon algebra. Such an algebra always satisfies Tarski's axioms for relation algebras and Lyndon algebras thus form an interesting connection between the fields of projective geometry and algebraic logic. In this paper we prove that if G is a class of projective geometries which contains an infinite projective geometry of dimension at least three, then the class L(G) of Lyndon algebras associated with projective geometries in G has an undecidable equational theory. In our proof we develop and use a connection between projective geometries and diagonal-free cylindric algebras.

Periodic Linear Transformations of Affine and Projective Geometries

1950 ◽  
Vol 2 ◽  
pp. 149-151 ◽  
Author(s):  
Ernst Snapper
Keyword(s):  
Number Theory ◽  
Projective Geometry ◽  
Irreducible Polynomial ◽  
Galois Field ◽  
Linear Transformations ◽  
Finite Projective Geometry ◽  
Projective Geometries ◽  
Periodic Linear

Introduction. In a paper called “A Theorem in Finite Projective Geometry and some Applications to Number Theory” [Trans. Amer. Math. Soc, vol. 43 (1938), 377-385], J. Singer proved that the finite projective geometry PG(s — 1,pn), that is the projective geometry of dimension s — 1 whose coordinate field is the Galois field GF(pn), admits a collineation L of period q = (psn — 1)/ (pn — 1). Since this q is the number of points of PG(s — 1, pn), Singer's result states that the points of PG(s — 1, pn) are cyclically arranged. Singer's construction of L uses the notion of a “primitive irreducible polynomial of degree 5 belonging to a field GF(pn) which defines a PG(s — 1, pn).”

A Lemma on Projective Geometries as Modular and/or Arguesian Lattices

1983 ◽  
Vol 26 (3) ◽  
pp. 283-290 ◽  
Author(s):  
Alan Day
Keyword(s):  
Projective Geometry ◽  
Modular Lattice ◽  
General Position ◽  
Projective Geometries

AbstractA projective geometry of dimension (n - 1) can be defined as modular lattice with a spanning n-diamond of atoms (i.e.: n + 1 atoms in general position whose join is the unit of the lattice). The lemma we show is that one could equivalently define a projective geometry as a modular lattice with a spanning n-diamond that is (a) is generated (qua lattice) by this n-diamond and a coordinating diagonal and (b) every non-zero member of this coordinatizing diagonal is invertible. The lemma is applied to describe certain freely generated modular and Arguesian lattices.

Pointwise bounds for the solutions of certain boundary-value problems

1951 ◽  
Vol 208 (1093) ◽  
pp. 170-175 ◽  
Keyword(s):  
Boundary Value Problems ◽  
Function Space ◽  
Scalar Product ◽  
Boundary Value ◽  
The Other ◽  
Green Functions ◽  
Regular Functions ◽  
Cutting Out ◽  
Neumann Type ◽  
The One

The boundary-value problems considered are of the Dirichlet-Neumann type. The method now given for obtaining pointwise bounds of the solution and its derivatives is a compromise between the methods of Diaz and Greenberg on the one hand and Maple and Synge on the other; it appears to be simpler than either. The solution having been located on a hypercircle in function space, the pointwise bounds are obtained by taking the scalar product of the solution by certain vectors (Green’s vectors). Divergence of integrals due to the poles of the Green functions is avoided by the use of regular functions matching the Green functions on the boundary (Diaz-Greenberg device) instead of by cutting out spheres from the domain (Maple-Synge device).

scholarly journals Constructive Geometric Models with Imaginary Objects

2020 ◽  
pp. short27-1-short27-9
Author(s):  
Denis Voloshinov ◽  
Alexandra Solovjeva
Keyword(s):  
Projective Geometry ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Higher Order ◽  
Second Order ◽  
The Other ◽  
Significant Obstacle ◽  
The Relationship ◽  
Planar Drawing ◽  
Three Dimensional Space

The article is devoted to the consideration of a number of theoretical questions of projective geometry related to specifying and displaying imaginary objects, especially, conics. The lack of development of appropriate constructive schemes is a significant obstacle to the study of quadratic images in three-dimensional space and spaces of higher order. The relationship between the two circles, established by the inversion operation with respect to the other two circles, in particular, one of which is imaginary, allows obtain a simple and effective method for indirect setting of imaginary circles in a planar drawing. The application of the collinear transformation to circles with an imaginary radius also makes it possible to obtain unified algorithms for specifying and controlling imaginary conics along with usual real second-order curves. As a result, it allows eliminate exceptional situations that arise while solving problems with quadratic images in spaces of second and higher order.

scholarly journals On von Staudt’s Geometrie der Lage

1900 ◽  
Vol 1 (20) ◽  
pp. 323-331
Author(s):  
C. A. Scott
Keyword(s):  
Projective Geometry ◽  
The Other ◽  
The Ideal

Having adjoined the ideal elements to the visible universe, von Staudt proves the usual theorems of projective geometry, noting the different cases that arise from different arrangements of the elements. As the only means of comparing figures is by projection, all the proofs depend on this; in many cases on the theorem that a given cast (Wurf) is unaltered when any two elements are interchanged, provided the other two are interchanged also (G. 59), which can be written symbolically(ABCD) = (BADC) = (CDAB) = (DCBA),while if the cast is harmonic, viz. AG harmonic with respect to BD, (ABCD) = (CBAD) = (ADCB).

Notes on Screw Product Operations in the Formulations of Successive Finite Displacements

1997 ◽  
Vol 119 (4) ◽  
pp. 434-439 ◽  
Author(s):  
C. Huang
Keyword(s):  
Scalar Product ◽  
Direction Cosine ◽  
Linear Formulation ◽  
The Other ◽  
Other Hand ◽  
Finite Displacements ◽  
Definition Of

Geometrical interpretations of two line-based formulations of successive finite displacements, Dimentberg’s formulation and a linear formulation, are discussed in this paper. The interpretations are based on the fact that the pitch of the screw product of two unit line vectors is consistent with Parkin’s definition of pitch. Finite twists in Dimentberg’s formulation are shown to be the screw product of unit line vectors divided by the scalar product of unit line vectors. On the other hand, Finite twists in the linear formulation are interpreted as the screw product of unit line vectors divided by the scalar product of direction-cosine vectors.

scholarly journals SVD on Intersection Spaces

2018 ◽  
Author(s):  
Mazen Ali ◽  
Anthony Nouy
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Tensor Product ◽  
Scalar Product ◽  
Truncation Error ◽  
Tensor Products ◽  
The Other ◽  
Low Rank ◽  
Differentiable Functions ◽  
Low Rank Approximations

We are interested in applying SVD to more general spaces, the motivating example being the Sobolev space $H^1(\Omega)$ of weakly differentiable functions over a domain $\Omega\subset\R^d$. Controlling the truncation error in the energy norm is particularly interesting for PDE applications. To this end, one can apply SVD to tensor products in $H^1(\Omega_1)\otimes H^1(\Omega_2)$ with the induced tensor scalar product. However, the resulting space is not $H^1(\Omega_1\times\Omega_2)$ but is instead the space $H^1_{\text{mix}}(\Omega_1\times\Omega_2)$ of functions with mixed regularity. For large $d>2$ this poses a restrictive regularity requirement on $u\in H^1(\Omega)$. On the other hand, the space $H^1(\Omega)$ is not a tensor product Hilbert space, in particular $\|\cdot\|_{H^1}$ is not a reasonable crossnorm. Thus, we can not identify $H^1(\Omega)$ with the space of Hilbert Schmidt operators and apply SVD. However, it is known that $H^1(\Omega)$ is isomorph (here written for $d=2$) to the Banach intersection space $$H^1(\Omega_1\times\Omega_2)=H^1(\Omega_1)\otimes L_2(\Omega_2)\cap L_2(\Omega_1)\otimes H^1(\Omega_2)$$ with equivalent norms. Each of the spaces in the intersection is a tensor product Hilbert space where SVD applies. We investigate several approaches to construct low-rank approximations for a function $u\in H^1(\Omega_1\times\Omega_2)$.

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