Graphic Algorithms for Constructing a Quadric, Given Nine Points

2019 ◽  
Vol 7 (2) ◽  
pp. 3-12 ◽  
Author(s):  
Виктор Короткий ◽  
Viktor Korotkiy
Keyword(s):  
Computer Graphics ◽  
Spatial Configuration ◽  
Second Order ◽  
Algebraic Solution ◽  
Conic Section ◽  
General Point ◽  
Principal Axes ◽  
Constructive Version ◽  
The 19Th Century ◽  
Set Of Points

The fundamental issue of constructing a nine-point quadric was frequently discussed by mathematicians in the 19th century. They failed to find a simple linear geometric dependence that would join ten points of a quadric (similar to Pascal's theorem, which joins six points of a conic section). Nevertheless, they found different algorithms for a geometrically accurate construction (using straightedge and compass or even using straightedge alone) of any number of points of a quadric that passes through nine given points. While the algorithms are quite complex, they can be implemented only with the help of computer graphics. The paper proposes a simplified computer-based realization of J.H. Engel’s well-known algorithm, which makes it possible to define the ninepoint quadric metric. The proposed graphics algorithm can be considered an alternative to the algebraic solution of the stated problem. The article discusses two well-known graphical algorithms for constructing a quadric (the Rohn — Papperitz algorithm and the J.H. Engel algorithm) and proposes a simplified version of the J.H. algorithm. For its constructive implementation using computer graphics. All algorithms allow you to determine the set of points and the set of flat sections of the surface of the second order, given by nine points. The Rohn — Papperitz algorithm, based on the spatial configuration of Desargues, is best suited for its implementation on an axonometric drawing using 3D computer graphics. Algorithm J.H. Engel allows you to solve a problem on the plane. The proposed simplified constructive version of the algorithm J.H. Engel is supplemented with an algorithm for constructing the principal axes and symmetry planes of a quadric, given by nine points. The construction cannot be performed with a compass and a ruler, since this task reduces to finding the intersection points of two second-order curves with one known general point (third degree task). For its constructive solution, a computer program is used that performs the drawing of a second order curve defined by an arbitrarily specified set of five points and tangents (both real and imaginary). The proposed graphic algorithm can be considered as an alternative to the algebraic solution of the problem.

scholarly journals Microwave Spectrum and Quadrupole Coupling Constants of 2-Bromothiophene

1975 ◽  
Vol 30 (4) ◽  
pp. 541-548 ◽  
Author(s):  
P. J. Mjöberg ◽  
W. M. Ralowski ◽  
S. O. Ljunggren
Keyword(s):  
Ground State ◽  
Second Order ◽  
Coupling Constants ◽  
Order Perturbation ◽  
Vibrationally Excited ◽  
Rotational Constants ◽  
Isotopic Species ◽  
Principal Axes ◽  
Quadrupole Coupling ◽  
Hyperfine Splittings

Abstract The microwave spectra of the two 79Br and 81Br isotopic species of 2-bromothiophene have been measured in the region 18000-40000 MHz.For both isotopic species, the rotational constants of the ground state and one vibrationally excited state were determined, as well as the centrifugal distortion coefficients of the ground state. The ground state rotational constants in MHz are as follows:C4H332S79Br C4H332S81BrA = 5403.432 ±0.111 5403.563 ±0.095,B = 1139.0689±0.0010 1126.5173±0.0011 C = 940.5142±0.0018 931.9315±0.0009.In order to perform a second-order perturbation treatment of the quadrupole interaction, the matrix elements of products of direction cosines in terms of the symmetric top wave functions have been derived. By the first-and second-order perturbation analysis of the hyperfine splittings of the rotational lines, the nuclear quadrupole coupling constants have been determined. The values in MHz areXaa = 592.7 ±1.5 493.7 ±1.5,Xbb = -295.3 ±0.6 -245.6 ±0.7, Xcc = -297.4 ±1.6 -248.1 ±1.6,Xab = 80 ±9 64±8 ,in the principal axes system of the molecule.

On the Interconnection of a Range and the Second-Order Cluster

2016 ◽  
Vol 4 (2) ◽  
pp. 8-18 ◽  
Author(s):  
Графский ◽  
O. Grafskiy
Keyword(s):  
Computer Graphics ◽  
Computational Geometry ◽  
Second Order ◽  
Multimedia Systems ◽  
First Order ◽  
Dual Form ◽  
Eastern State ◽  
The Subject ◽  
General Abstract ◽  
Far Eastern

In accordance with “Specialized sections of affine, projective and computational geometry” syllabus for Master’s degree program in “Multimedia systems and computer graphics” developed at the Far Eastern State Transport University, the subject “Projective theory of the second-order curves” is considered [4; 14; 18]. Both at the sources mentioned and the textbook [11] projective method of the second-order curves formation as a range of the second order and its dual form – a second-order cluster (with regard to well-known theorems and relations, including Pascal and Brianchon theorems) is discernible. However, the graphical interpretations represented at the sources mentioned have general abstract character: to form the secondorder range two projective clusters of the first-order with the corresponding right lines are defined, and to design the second-order range – two projective series with the corresponding points. Techniques of high value can be observed when constructing outlines with the second-order curves; in this case, depending on engineering discriminant values, these curves can be constructed both using Pascal lines and qualities of the engineering discriminant itself, that is paying attention to the fact that tangents to the second-order curves makes the second-order cluster. Naturally, intent arises not to set the corresponding points on projective ranges, but to get them by elaboration, disclosing upon that regularities when constructing different second-order curves (the first aspect of research). The second aspect is in the consider - ation of the particular cases which would have definite secondorder clusters. In this case the task would be to model the secondorder range as a dual form of cluster. Thus it would be possible to get the interconnection of the definite cluster and the second-order cluster.

SECOND ORDER NUCLEAR QUADRUPOLE EFFECTS IN SINGLE CRYSTALS: PART I. THEORETICAL

1953 ◽  
Vol 31 (5) ◽  
pp. 820-836 ◽  
Author(s):  
G. M. Volkoff
Keyword(s):  
Magnetic Field ◽  
Electric Field ◽  
Single Crystals ◽  
Electric Quadrupole ◽  
Second Order ◽  
Order Effects ◽  
Axially Symmetric ◽  
Crystalline Electric Field ◽  
Resonance Frequencies ◽  
Principal Axes

The dependence of electric quadrupole splitting of nuclear magnetic resonance absorption lines in single crystals on crystal orientation in an external magnetic field is investigated theoretically following earlier work of Pound, of Volkoff, Petch, and Smellie, and of Bersohn. Explicit formulae are given, applicable to non axially symmetric crystalline electric field gradients (η ≠ 0), and valid up to terms of the second order in the quadrupole coupling constant [Formula: see text], for the dependence of the absorption frequencies on the angle of rotation of the crystal about any arbitrary axis perpendicular to the magnetic field. Some formulae including third order effects in Cz are also given. It is shown that an experimental study of the dependence of this splitting on the angles of rotation about any two arbitrary mutually perpendicular axes is sufficient, when second order effects are measurable, to yield the values of | Cz |, η, and the orientation of the principal axes of the electric field gradient tensor at the nuclear sites. In the case that the direction of one of the principal axes is known from crystal symmetry, a single rotation about this axis gives the complete information.A new method of determining nuclear spin I is proposed which depends on comparing first and second order shifts of the resonance frequencies of the strong inner line components. The method will be of interest in those cases where the total number 2I of line components can not be unambiguously ascertained owing to the outer line components being excessively broadened and weakened by crystal imperfections.

History of the Historiography of Spanish Textbooks and Treatises on International Law of the 19th Century

2013 ◽  
Vol 17 (1) ◽  
pp. 1-22
Author(s):  
Yolanda Gamarra Chopo
Keyword(s):  
Nineteenth Century ◽  
Sixteenth Century ◽  
International Law ◽  
Late Development ◽  
Spanish History ◽  
Principal Axes ◽  
History Of ◽  
The Subject ◽  
The 19Th Century

The bibliography of Spanish international law textbooks is a good indicator of the evolution of the historiography of international law. Spanish historiography, with its own special features, was a recipient of the great debates concerning naturalism v. positivism and universalism v. particularism that flourished in European and American historiography in the nineteenth century. This study is articulated on four principal axes. The first states how the writings of the philosophes continued to dominate the way in which the subject was conceived in mid-nineteenth century Spain. Secondly, it explores the popularization and democratization of international law through the work of Concepcion Arenal and the heterodox thought of Rafael Maria de Labra. Thirdly, it examines the first textbooks of international law with their distinct natural law bias, but imbued with certain positivist elements. These textbooks trawled sixteenth century Spanish history, searching for the origins of international law and thus demonstrating the historical civilizing role of Spain, particularly in America. Fourthly, it considers the vision of institutionist, heterodox reformers and bourgeois liberals who proclaimed the universality of international law, not without some degree of ambivalence, and their defence of Spain as the object of civilization and also a civilizing subject. In conclusion, the article argues that the late development of textbooks was a consequence of the late institutionalization of the study of international law during the last decade of the nineteenth century. Nevertheless, the legacy of the nineteenth century survives in the most progressive of contemporary polemics for a new international law.

First-order definability in modal logic

1975 ◽  
Vol 40 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. I. Goldblatt
Keyword(s):  
Modal Logic ◽  
Binary Relation ◽  
Second Order ◽  
Order Conditions ◽  
Kripke Models ◽  
Order Condition ◽  
First Order ◽  
Axiom Systems ◽  
Set Of Points ◽  
First Order Definability

In the early days of the development of Kripke-style semantics for modal logic a great deal of effort was devoted to showing that particular axiom systems were characterised by a class of models describable by a first-order condition on a binary relation. For a time the approach seemed all encompassing, but recent work by Thomason [6] and Fine [2] has shown it to be somewhat limited—there are logics not determined by any class of Kripke models at all. In fact it now seems that modal logic is basically second-order in nature, in that any system may be analysed in terms of structures having a nominated class of second-order individuals (subsets) that serve as interpretations of propositional variables (cf. [7]). The question has thus arisen as to how much of modal logic can be handled in a first-order way, and precisely which modal sentences are determined by first-order conditions on their models. In this paper we present a model-theoretic characterisation of this class of sentences, and show that it does not include the much discussed LMp → MLp.Definition 1. A modal frame ℱ = 〈W, R〉 consists of a set W on which a binary relation R is defined. A valuation V on ℱ is a function that associates with each propositional variable p a subset V(p) of W (the set of points at which p is “true”).

Second Order Curves on Computer Screen

2018 ◽  
Vol 6 (2) ◽  
pp. 100-112 ◽  
Author(s):  
Виктор Короткий ◽  
Viktor Korotkiy ◽  
Е. Усманова ◽  
E. Usmanova
Keyword(s):  
Computer Graphics ◽  
Computer Program ◽  
Geometric Modeling ◽  
Main Tool ◽  
Second Order ◽  
Modern Computer ◽  
Order Curve ◽  
Junction Points ◽  
Tools And Techniques ◽  
Straight Lines

Modern computer graphics is based on methods of computational geometry. The curves and surfaces’ description is based on apparatus of spline functions, which became the main tool for geometric modeling. Methods of projective geometry are almost not applying. One of the reasons for this is impossibility to exactly construct a second-order curve passing through given points and tangent to given straight lines. To eliminate this defect a computer program for second order curves construction has been developed. The program performs the construction of second-order curve’s metric (center, vertices, asymptotes, foci) for following combinations: • The second-order curve is given by five points; • The second-order curve is given by five tangent lines; • The second-order curve is given by a point and two tangent lines with points of contact indicated on them; • The parabola is given by four tangent lines; • The parabola is given by four points. In this paper are presented algorithms for construction a metric for each combination. After construction the metric the computer program written in AutoLISP language and using geometrically exact projective algorithms which don’t require algebraic computations draws a second-order curve. For example, to construct vertices and foci of two parabolas passing through four given points, it is only necessary to draw an arbitrary circle and several straight lines. To construct a conic metric passing through five given points, it is necessary to perform only three geometrically exact operations: to construct an involution of conjugate diameters, to find the main axes and asymptotes; to note the vertices of desired second-order curve. Has been considered the architectural appearance of a new airport in Simferopol. It has been demonstrated that a terminal facade’s wavelike form can be obtained with a curve line consisting of conic sections’ areas with common tangent lines at junction points. The developed computer program allows draw second-order curves. The program application will promote the development of computer graphics’ tools and techniques.

scholarly journals POINTS ON HYPERBOLAS AT RATIONAL DISTANCE

2012 ◽  
Vol 08 (04) ◽  
pp. 911-922
Author(s):  
EDRAY HERBER GOINS ◽  
KEVIN MUGO
Keyword(s):  
Elliptic Curve ◽  
Rational Numbers ◽  
Rational Points ◽  
Conic Section ◽  
Distance Set ◽  
Set Of Points ◽  
Distance Sets

Richard Guy asked for the largest set of points which can be placed in the plane so that their pairwise distances are rational numbers. In this article, we consider such a set of rational points restricted to a given hyperbola. To be precise, for rational numbers a, b, c, and d such that the quantity D = (ad - bc)/(2a2) is defined and non-zero, we consider rational distance sets on the conic section axy + bx + cy + d = 0. We show that, if the elliptic curve Y2 = X3 - D2X has infinitely many rational points, then there are infinitely many sets consisting of four rational points on the hyperbola such that their pairwise distances are rational numbers. We also show that any rational distance set of three such points can always be extended to a rational distance set of four such points.

On the application of the theory of elliptic functions to the rotation of a rigid body round a fixed point

1851 ◽  
Vol 5 ◽  
pp. 797-800
Keyword(s):  
Fixed Point ◽  
Complete Solution ◽  
Elliptic Functions ◽  
The Body ◽  
Conic Section ◽  
First Order ◽  
Concentric Spheres ◽  
Principal Axes ◽  
Revolving Body ◽  
Second Degree

In the introduction to his investigation, the author, after noticing the investigations of D’Alembert and Euler, and the solution of this problem by Lagrange, refers more particularly to the memoir of Poinsot, in which the motion of a body round a fixed point, and free from the action of accelerating forces, is reduced to the motion of a certain ellipsoid whose centre is fixed, and which rolls without sliding on a plane fixed in space; and likewise to the researches of Maccullagh, in which, by adopting an ellipsoid the reciprocal of that chosen by Poinsot, he deduced those results which long before had been arrived at by the more operose methods of Euler and Lagrange; observing, however, that it is to Legendre that we are indebted for the happy conception of substituting, as a means of investigation, an ideal ellipsoid having certain relations with the actually revolving body. He then states, that several years ago he was led to somewhat similar views, from remarking the identity which exists between the formulæ for finding the position of the principal axes of a body and those for determining the symmetrical diameters of an ellipsoid; and further observing that the expression for the perpendicular from the centre on a tangent plane to an ellipsoid, in terms of the cosines of the angles which it makes with the axes, is precisely the same in form as that which gives the value of the moment of inertia round a line passing through the origin. Guided by this analogy, he was led to assume an ellipsoid the squares of whose axes should be directly proportional to the moments of inertia round the coinciding principal axes of the body. This is also the ellipsoid chosen by Maccullagh. Although it may at first sight appear of little importance which of the ellipsoids—the inverse of Poinsot, or the direct of Maccullagh and the author—is chosen as the geometrical substitute for the revolving body, it is by no means a matter of indifference when we come to treat of the properties of the integrals which determine the motion. Generally those integrals depend on the properties of those curves of double flexure in which cones of the second degree are generally intersected by concentric spheres; and it so happens that the direct ellipsoid of moments is intersected by a concentric sphere in one of these curves. By means of the properties of these curves a complete solution may be obtained even in the most general cases, to which only an approximation has hitherto been made. In the first section of the paper, the author establishes such properties as he has subsequently occasion to refer to, of cones of the second degree, and of the curves of double curvature in which these surfaces may be intersected by concentric spheres, some, of which he believes will not be found in any published treatise on the subject. He considers that he has been so fortunate as to be the first to obtain the true representative curve of elliptic functions of the first order. It is shown that any spherical conic section, the tangents of whose principal semiarcs are the ordinates of an equilateral hyperbola whose transverse semiaxis is 1, may be rectified by an elliptic function of the first order, and the quadrature of such a curve may be effected by a function of the same order, when the cotangents of the halves of the principal arcs are the ordinates of the same equilateral hyperbola.

SECOND ORDER NUCLEAR QUADRUPOLE EFFECTS IN SINGLE CRYSTALS: PART II. EXPERIMENTAL RESULTS FOR SPODUMENE

1953 ◽  
Vol 31 (5) ◽  
pp. 837-858 ◽  
Author(s):  
H. E. Petch ◽  
N. G. Cranna ◽  
G. M. Volkoff
Keyword(s):  
Principal Axis ◽  
Order Theory ◽  
Resonance Absorption ◽  
Second Order ◽  
The Other ◽  
Oxygen Ion ◽  
Principal Axes ◽  
Nuclear Quadrupole ◽  
Spin Determination ◽  
Resonance Absorption Line

Experiments on the splitting of the Al27 resonance absorption line in a single crystal of LiAl(SiO3)2 (spodumene) are described, and are used to illustrate the second order theory of Part I. [Formula: see text] for Al27 nuclei in spodumene is found to be 2950 ± 20 kc./sec. [Formula: see text] at Al sites is found to be 0.94 ± 0.01. The x principal axis (corresponding to the smallest eigenvalue [Formula: see text]) of [Formula: see text] at the Al sites is found to coincide with the b axis of the monoclinic spodumene crystal. The other two principal axes lie in the ac plane with the y axis (corresponding to the intermediate eigenvalue [Formula: see text]) making an angle of [Formula: see text] with the crystal c axis towards the a axis. The y principal axis at the Al sites and the z principal axis at the Li sites appear to point at the projection of the nearest oxygen ion in each case. The new method of spin determination proposed in Part I is checked by confirming the known value I = 5/2 for Al27.

scholarly journals POINT SET LABELING WITH SPECIFIED POSITIONS

2002 ◽  
Vol 12 (01n02) ◽  
pp. 29-66 ◽  
Author(s):  
SRINIVAS R. DODDI ◽  
MADHAV V. MARATHE ◽  
BERNARD M. MORET
Keyword(s):  
Fixed Point ◽  
Computer Graphics ◽  
Best Approximation ◽  
Maximum Size ◽  
Map Labeling ◽  
Combinatorial Properties ◽  
The Best Approximation ◽  
Point Set ◽  
Set Of Points ◽  
Aesthetic Criterion

Motivated by applications in cartography and computer graphics, we study a version of the map-labeling problem that we call the k-Position Map-Labeling Problem: given a set of points in the plane and, for each point, a set of up to k allowable positions, place uniform and non-intersecting labels of maximum size at each point in one of the allowable positions. This version combines an aesthetic criterion and a legibility criterion and comes close to actual practice while generalizing the fixed-point and slider models found in the literature. We present a general heuristic that given an ∊ > 0, runs in time O(n log n + n log (R*/ ∊) log (k)), where R* is the size of the optimal label, and guarantees a constant approximation for any regular labels. For circular labels, our technique yields a (3.6 + ∊)-approximation, improving in the case of arbitrary placement over the previous bound of approximately 19.5 obtained by Strijk and Wolff.28 We then extend our approach to arbitrary positions, obtaining an algorithm that is easy to implement and also substantially improves the best approximation bounds. Our technique combines several geometric and combinatorial properties, which may be of independent interest.

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