A Geometric Approach to the Heine-Borel Theorem

1971 ◽  
Vol 23 (5) ◽  
pp. 845-848
Author(s):  
R. B. Killgrove ◽  
Jason Frand ◽  
William Giles ◽  
Henry Bray
Keyword(s):  
Geometric Approach ◽  
Plane Geometry ◽  
Topological Properties ◽  
Absolute Plane ◽  
Topological Plane ◽  
The Absolute

In a topological plane with strong enough topological properties one can use [6] open triangular regions to define a base for the topology. Similarly, one can use these regions to define boundedness of a set. In this setting we show that in the absolute plane geometry, the holding of the Heine-Borel theorem is equivalent to every four points being contained in some such region and that this second condition is equivalent to the parallel postulate. Thus we give two new conditions equivalent to the parallel postulate.

scholarly journals Classification of Geometries with Projective Metric

1909 ◽  
Vol 28 ◽  
pp. 25-41 ◽  
Author(s):  
D. M. Y. Sommerville
Keyword(s):  
Dihedral Angle ◽  
Plane Geometry ◽  
Three Dimensions ◽  
Angular Measurement ◽  
Plane Angle ◽  
Degenerate Form ◽  
Projective Metric ◽  
The Absolute ◽  
Straight Lines

In the Cayley-Klein projective metric it is ordinarily assumed that the measure of angles, plane and dihedral, is always elliptic, i.e. in a sheaf of planes or lines there is no actual plane or line which makes an infinite angle with the others. With this restriction there are only three kinds of geometry—Parabolic, Hyperbolic and Elliptic, i.e. the geometries of Euclid, Lobachevskij and Riemann ; and the form of the absolute is also limited. Thus in plane geometry the only degenerate form of the absolute which is possible is two coincident straight lines and a pair of imaginary points ; in three dimensions the absolute cannot be a ruled quadric, other than two coincident planes. If, however, this restriction as to angular measurement is removed, there are 9 systems of plane geometry and 27 in three dimensions; for the measure of distance, plane angle and dihedral angle may be parabolic, hyperbolic, or elliptic.

LINEAR LAYOUT OF GENERALIZED HYPERCUBES

2003 ◽  
Vol 14 (01) ◽  
pp. 137-156 ◽  
Author(s):  
KOJI NAKANO
Keyword(s):  
Edge Length ◽  
Topological Properties ◽  
Induced Subgraph ◽  
Absolute Value ◽  
Bisection Width ◽  
The Absolute ◽  
The Difference

This paper deals with two kinds of generalized hypercubes: a d-dimensional c-ary clique [Formula: see text] and a d-dimensional c-ary array [Formula: see text]. A d-dimensional c-ary clique [Formula: see text] has nodes labeled by cd integers from 0 to cd - 1 and two nodes are connected by an edge if and only if the c-ary representations of their labels differ by one and only one digit. A d-dimensional c-ary array [Formula: see text] also has nodes labeled by cd integers from 0 to cd - 1, and two nodes are connected if and only if the c-ary representations of their labels differ by one and only one digit and the absolute value of the difference in that digit is 1. Further, an n-node c-ary clique [Formula: see text] is the induced subgraph of [Formula: see text] with nodes labeled by integers from 0 to n - 1. The main contribution of this paper is to clarify several topological properties of [Formula: see text] and [Formula: see text] in terms of their linear layouts. For this purpose, we first prove that [Formula: see text] is a maximum subgraph of [Formula: see text], that is, [Formula: see text]has the largest number of edges over all n-node subgraphs of [Formula: see text], whenever n ≤ m. Using this fact, we show the exact values of the bisection width, cut width, and total edge length of [Formula: see text]. We also show the exact value of the bisection width of [Formula: see text] and nearly tight values of the cut width and the total edge length of [Formula: see text].

scholarly journals An Efficient Closed Form Solution to the Absolute Orientation Problem for Camera with Unknown Focal Length

Sensors ◽  
2021 ◽  
Vol 21 (19) ◽  
pp. 6480
Author(s):  
Kai Guo ◽  
Hu Ye ◽  
Zinian Zhao ◽  
Junhao Gu
Keyword(s):  
Closed Form ◽  
Focal Length ◽  
Closed Form Solution ◽  
Geometric Approach ◽  
Polynomial Equation ◽  
Form Solution ◽  
Camera Model ◽  
Absolute Orientation ◽  
Camera Position ◽  
The Absolute

In this paper we propose an efficient closed form solution to the absolute orientation problem for cameras with an unknown focal length, from two 2D–3D point correspondences and the camera position. The problem can be decomposed into two simple sub-problems and can be solved with angle constraints. A polynomial equation of one variable is solved to determine the focal length, and then a geometric approach is used to determine the absolute orientation. The geometric derivations are easy to understand and significantly improve performance. Rewriting the camera model with the known camera position leads to a simpler and more efficient closed form solution, and this gives a single solution, without the multi-solution phenomena of perspective-three-point (P3P) solvers. Experimental results demonstrated that our proposed method has a better performance in terms of numerical stability, noise sensitivity, and computational speed, with synthetic data and real images.

scholarly journals Distance in the absolute plane and Cauchy functional equations

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3585-3592 ◽  
Author(s):  
Miodrag Mateljevic ◽  
Miljan Knezevic ◽  
Marek Svetlik
Keyword(s):  
Distance Function ◽  
Functional Equations ◽  
Half Line ◽  
Constructive Approach ◽  
Absolute Plane ◽  
The Absolute

Let A denotes the absolute plane and da the distance function on it. Using a constructive approach which leads to the functional equations, we study which conditions on a ?measure? of segments on a given half-line l in the absolute plane are essential to be the restriction of da on l.

scholarly journals A Pure Geometric Approach to Cosmology

1999 ◽  
Vol 183 ◽  
pp. 316-316 ◽  
Author(s):  
M.I. Wanas
Keyword(s):  
Geometric Approach ◽  
Big Bang ◽  
Space Time ◽  
Material Distribution ◽  
Absolute Parallelism ◽  
Matter Tensor ◽  
Cosmic Space ◽  
The Absolute ◽  
The Universe

It is well known that standard Big-Bang cosmology suffers from certain problems, e.g. singularity, horizon, flatness, … In the present work it is claimed that the appearence of some of these problems is due to two main assumptions. The first is the assumption that the 4-dimentional Riemannian (RIE)-geometry gives a complete description of the cosmic space-time. The second is the assumption that the material distribution in the universe is described by a phenomenological (PH)-matter tensor. It is shown that, by relaxing these two assumptions, some of the problems of the standard Big-Bang cosmology could be avoided. The following table summarises some results in favour of the above claim. The absolute parallelism (AP)-geometry is used to construct some of the theories mentioned in the table.

scholarly journals Some New Variants of Relative Regularity via Regularly Closed Sets

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Sehar Shakeel Raina ◽  
A. K. Das
Keyword(s):  
Topological Property ◽  
Topological Properties ◽  
Complete Regularity ◽  
Completely Regular ◽  
Closed Sets ◽  
Relative Property ◽  
Relative Version ◽  
The Absolute

Every topological property can be associated with its relative version in such a way that when smaller space coincides with larger space, then this relative property coincides with the absolute one. This notion of relative topological properties was introduced by Arhangel’skii and Ganedi in 1989. Singal and Arya introduced the concepts of almost regular spaces in 1969 and almost completely regular spaces in 1970. In this paper, we have studied various relative versions of almost regularity, complete regularity, and almost complete regularity. We investigated some of their properties and established relationships of these spaces with each other and with the existing relative properties.

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