A Geometric Approach to the Conic Sections

1966 ◽  
Vol 59 (4) ◽  
pp. 348-350
Author(s):  
Maurice Marie Byrne
Keyword(s):  
Geometric Approach ◽  
Geometric Construction ◽  
Conic Sections ◽  
Algebra Ii ◽  
Definition Of

Connie Byron, a student in an Algebra II class,* attempted a demonstration of how the locus of points fulfills the definition of a hyperbola by a geometric construction, based on the use of a circle as representing the constant difference. Although not conclusive in determining the locus at the time, this method was the stimulus for further study and investigation of a unique geometric approach to each of the conic sections.

On the symplectically invariant variational principle and generating functions

1990 ◽  
Vol 431 (1883) ◽  
pp. 403-417 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Generating Function ◽  
Generating Functions ◽  
Geometric Approach ◽  
Canonical Transformations ◽  
Weyl Transform ◽  
Analogous Relation ◽  
Definition Of ◽  
Symplectic Invariance

The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.

Lines as “Foci” for Conic Sections

2019 ◽  
Vol 112 (4) ◽  
pp. 312-316
Author(s):  
Wayne Nirode
Keyword(s):  
Spherical Geometry ◽  
Geometric Object ◽  
Plane Geometry ◽  
Conic Sections ◽  
Geometric Objects ◽  
Definition Of ◽  
Taxicab Geometry

One of my goals, as a geometry teacher, is for my students to develop a deep and flexible understanding of the written definition of a geometric object and the corresponding prototypical diagram. Providing students with opportunities to explore analogous problems is an ideal way to help foster this understanding. Two ways to do this is either to change the surface from a plane to a sphere or change the metric from Pythagorean distance to taxicab distance (where distance is defined as the sum of the horizontal and vertical components between two points). Using a different surface or metric can have dramatic effects on the appearance of geometric objects. For example, in spherical geometry, triangles that are impossible in plane geometry (such as triangles with three right or three obtuse angles) are now possible. In taxicab geometry, a circle now looks like a Euclidean square that has been rotated 45 degrees.

Synthesis of Planar Compliances With Mechanisms Having Six Compliant Components: Geometric Approach

2020 ◽  
Vol 12 (3) ◽  
Author(s):  
Shuguang Huang ◽  
Joseph M. Schimmels
Keyword(s):  
Compliant Mechanism ◽  
Geometric Approach ◽  
Geometric Construction ◽  
Elastic Component ◽  
Parallel Mechanisms ◽  
Elastic Joints

Abstract In this paper, the synthesis of any planar compliance with a six-component compliant mechanism is addressed. The mechanisms studied are either serial mechanisms with six elastic joints or parallel mechanisms with six springs. For each type of mechanism, conditions on the mechanism configurations that must be satisfied to realize a given compliance are developed. The geometric significance of each condition is identified and graphically represented. Geometric construction-based synthesis procedures for both types of mechanism are developed. These procedures allow one to select each elastic component from a restricted space based on its geometry.

scholarly journals New algebraic and geometric constructs arising from Fibonacci numbers

2020 ◽  
Vol 24 (23) ◽  
pp. 17497-17508 ◽  
Author(s):  
Fabio Caldarola ◽  
Gianfranco d’Atri ◽  
Mario Maiolo ◽  
Giuseppe Pirillo
Keyword(s):  
Fibonacci Numbers ◽  
Geometric Construction ◽  
Binary Alphabet ◽  
The Family ◽  
Definition Of ◽  
Sturmian Words ◽  
Regular Octagon

AbstractFibonacci numbers are the basis of a new geometric construction that leads to the definition of a family $$\{C_n:n\in \mathbb {N}\}$$ { C n : n ∈ N } of octagons that come very close to the regular octagon. Such octagons, in some previous articles, have been given the name of Carboncettus octagons for historical reasons. Going further, in this paper we want to introduce and investigate some algebraic constructs that arise from the family $$\{C_n:n\in \mathbb {N}\}$$ { C n : n ∈ N } and therefore from Fibonacci numbers: From each Carboncettus octagon $$C_n$$ C n , it is possible to obtain an infinite (right) word $$W_n$$ W n on the binary alphabet $$\{0,1\}$$ { 0 , 1 } , which we will call the nth Carboncettus word. The main theorem shows that all the Carboncettus words thus defined are Sturmian words except in the case $$n=5$$ n = 5 . The fifth Carboncettus word $$W_5$$ W 5 is in fact the only word of the family to be purely periodic: It has period 17 and periodic factor 000 100 100 010 010 01. Finally, we also define a further word $$W_{\infty }$$ W ∞ named the Carboncettus limit word and, as second main result, we prove that the limit of the sequence of Carboncettus words is $$W_{\infty }$$ W ∞ itself.

scholarly journals On the Geometric Approach to the Boundary Problem in Supergravity

Universe ◽  
2021 ◽  
Vol 7 (12) ◽  
pp. 463
Author(s):  
Laura Andrianopoli ◽  
Lucrezia Ravera
Keyword(s):  
Cosmological Constant ◽  
Boundary Problem ◽  
Geometric Approach ◽  
Geometric Construction ◽  
Boundary Values ◽  
Negative Cosmological Constant ◽  
Boundary Terms ◽  
Bonnet Term ◽  
Topological Boundary ◽  
Field Strengths

We review the geometric superspace approach to the boundary problem in supergravity, retracing the geometric construction of four-dimensional supergravity Lagrangians in the presence of a non-trivial boundary of spacetime. We first focus on pure N=1 and N=2 theories with negative cosmological constant. Here, the supersymmetry invariance of the action requires the addition of topological (boundary) contributions which generalize at the supersymmetric level the Euler-Gauss-Bonnet term. Moreover, one finds that the boundary values of the super field-strengths are dynamically fixed to constant values, corresponding to the vanishing of the OSp(N|4)-covariant supercurvatures at the boundary. We then consider the case of vanishing cosmological constant where, in the presence of a non-trivial boundary, the inclusion of boundary terms involving additional fields, which behave as auxiliary fields for the bulk theory, allows to restore supersymmetry. In all the cases listed above, the full, supersymmetric Lagrangian can be recast in a MacDowell-Mansouri(-like) form. We then report on the application of the results to specific problems regarding cases where the boundary is located asymptotically, relevant for a holographic analysis.

Drawing, Geometry and Construction

2016 ◽  
pp. 608-641 ◽  
Author(s):  
Marco Canciani
Keyword(s):  
Survey Data ◽  
Geometric Construction ◽  
Original Design ◽  
Design Data ◽  
Design Idea ◽  
Definition Of ◽  
Architecture And Design ◽  
The Church ◽  
Construction Practice ◽  
Design Drawing

The link between the design drawing to an architectural work, sometimes goes through the definition of geometric paths which establish alignments, proportions, correspondences. The comparison of the geometric construction of survey data of an architecture and design data is very important for understanding the original design idea, highlighting not only the artist's modus progettandi, but also matches, modifications or changes respect of precisely geometric paths and its building architecture. In these studies, the church of San Carlo alle Quattro Fontane in Rome, by Francesco Borromini, is an exemplar case. The project of the church, built between 1638 and 1675 and characterized by a coffered vault with an oval planimetric shape, is documented by a consistent corpus of Borromini drawings. This research, based on survey data, can allow to make new contributions to Borromini work and formulate new hypotheses regarding his construction practice.

scholarly journals The geometry of the space of Cauchy data of nonlinear PDEs

2013 ◽  
Vol 11 (11) ◽  
Author(s):  
Giovanni Moreno
Keyword(s):  
Infinite Order ◽  
Geometric Approach ◽  
Cauchy Data ◽  
Nonlinear Pdes ◽  
General Notion ◽  
Geometric Framework ◽  
Transversality Conditions ◽  
Jet Bundles ◽  
First Order ◽  
Definition Of

AbstractFirst-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways — for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework for the space of Cauchy data for nonlinear PDEs. As an example, we derive a general notion of transversality conditions in the Calculus of Variations.

scholarly journals EXTERNAL FIELD INFLUENCE ON SEMIFLEXIBLE MACROMOLECULES: GEOMETRIC COUPLING

2011 ◽  
Vol 25 (22) ◽  
pp. 1809-1819
Author(s):  
STEFANO BELLUCCI ◽  
YEVGENY MAMASAKHLISOV ◽  
ARMEN NERSESSIAN
Keyword(s):  
External Field ◽  
Preliminary Analysis ◽  
Hamiltonian Formulation ◽  
Geometric Approach ◽  
Initial Energy ◽  
Arc Length ◽  
Dna Molecules ◽  
Initial Energy Density ◽  
Definition Of ◽  
Field Influence

We suggested a geometric approach to address the external field influence on the DNA molecules, described by the WLC model via geometric coupling. It consists in the introduction of the effective metrics depending on the potential of the external field, with further re-definition of the arc-length parameter and of the extrinsic curvatures of the DNA molecules. It yields the nontrivial impact of the external field in the internal energy of macromolecules. We give the Hamiltonian formulation of this model and perform its preliminary analysis in the redefinition of the initial energy density.

scholarly journals Definition of mass and production of equations of general relativity

2020 ◽  
Vol 5 (3) ◽  
Author(s):  
Constantinos Krikos
Keyword(s):  
Background Radiation ◽  
Cosmic Background Radiation ◽  
Flat Space ◽  
Density Parameter ◽  
Conic Sections ◽  
3D Space ◽  
A Value ◽  
Definition Of ◽  
Simply Connected ◽  
The Universe

From the measurements of the anisotropies of the cosmic background radiation at the present time, we get a value for the density parameter (Ω(t)) near to unit, i.e. Ω(t) ∼ 1. The value of the density parameter determines if the Universe is open (Ω(t) < 1), flat (Ω(t) = 1) or closed (Ω(t) > 1)). This result forces us to assume that the boundary of the Universe is a 2D flat space, i.e. the R2 , since its interior is a 3D space as we conceive it. The R2 space is characterized by isotropy and homogeneity. It is a simply connected space and that it does not exhibit any particular characteristic anywhere. These attributes are expressed by a circle of an infinite radius in the center of which is an observer, at every point in the Universe. Since circle is the geometric object from which all other conic sections produced, then we shall examine the equations that characterize them and the consequences of their mappings in the interior of the Universe through one to one correspondences.

Sign in / Sign up

Export Citation Format

Share Document