Dual Problems with Conics

2020 ◽  
Vol 8 (1) ◽  
pp. 15-24
Author(s):  
A. Girsh
Keyword(s):  
Euclidean Plane ◽  
Complex Geometry ◽  
Intersection Point ◽  
Dual Problems ◽  
Geometric Problems ◽  
Straight Line ◽  
Solution Methods ◽  
Method Of Solution ◽  
Straight Lines ◽  
Geometric Picture

The problem for construction of straight lines, which are tangent to conics, is among the dual problems for constructing the common elements of two conics. For example, the problem for construction of a chordal straight line (a common chord for two conics) ~ the problem for construction of an intersection point for two conics’ common tangents. In this paper a new property of polar lines has been presented, constructive connection between polar lines and chordal straight lines has been indicated, and a new way for construction of two conics’ common chords has been given, taking into account the computer graphics possibilities. The construction of imaginary tangent lines to conic, traced from conic’s interior point, as well as the construction of common imaginary tangent lines to two conics, of which one lies inside another partially or thoroughly is considered. As you know, dual problems with two conics can be solved by converting them into two circles, followed by a reverse transition from the circles to the original conics. This method of solution provided some clarity in understanding the solution result. The procedure for transition from two conics to two circles then became itself the subject of research. As and when the methods for solving geometric problems is improved, the problems themselves are become more complex. When assuming the participation of imaginary images in complex geometry, it is necessary to abstract more and more. In this case, the perception of the obtained result’s geometric picture is exposed to difficulties. In this regard, the solution methods’ correctness and imaginary images’ visualization are becoming relevant. The paper’s main results have been illustrated by the example of the same pair of conics: a parabola and a circle. Other pairs of affine different conics (ellipse and hyperbola) have been considered in the paper as well in order to demonstrate the general properties of conics, appearing in investigated operations. Has been used a model of complex figures, incorporating two superimposed planes: the Euclidean plane for real figures, and the pseudo-Euclidean plane for imaginary algebraic figures and their imaginary complements.

scholarly journals Involutes of Pseudo-Null Curves in Lorentz–Minkowski 3-Space

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1256
Author(s):  
Rafael López ◽  
Željka Milin Šipuš ◽  
Ljiljana Primorac Gajčić ◽  
Ivana Protrka
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Reconstruction Process ◽  
Null Curve ◽  
Straight Line ◽  
Null Curves ◽  
Straight Lines

In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. The first result of the paper is that the involutes of pseudo-null curves are null curves, more precisely, null straight lines. Furthermore, a method of reconstruction of a pseudo-null curve from a given null straight line as its involute is provided. Such a reconstruction process in Euclidean plane generates an evolute of a curve, however it cannot be applied to a straight line. In the case presented, the process is additionally affected by a choice of different null frames that every null curve allows (in this case, a null straight line). Nevertheless, we proved that for different null frames, the obtained pseudo-null curves are congruent. Examples that verify presented results are also given.

scholarly journals Detour Extra Straight Lines in the Euclidean Plane

2020 ◽  
pp. 237-249
Author(s):  
I. Szalay ◽  
B. Szalay
Keyword(s):  
Euclidean Plane ◽  
Euclidean Geometry ◽  
The Other ◽  
Real Numbers ◽  
Other Hand ◽  
Straight Line ◽  
Axiom Systems ◽  
Straight Lines

Using the theory of exploded numbers by the axiom-systems of real numbers and Euclidean geometry, we explode the Euclidean plane. Exploding the Euclidean straight lines we get super straight lines. The extra straight line is the window phenomenon of super straight line. In general, the extra straight lines are curves in Euclidean sense, but they have more similar properties to Euclidean straight lines. On the other hand, with respect of parallelism we find a surprising property: there are detour straight lines.

Automatic Tread Gaging Machine

1979 ◽  
Vol 7 (1) ◽  
pp. 31-39
Author(s):  
G. S. Ludwig ◽  
F. C. Brenner
Keyword(s):  
Experimental Tests ◽  
Automatic Machine ◽  
Wear Rates ◽  
Handling Device ◽  
Straight Line ◽  
Component Systems ◽  
Before And After ◽  
Good Repeatability ◽  
The Difference ◽  
Straight Lines

Abstract An automatic tread gaging machine has been developed. It consists of three component systems: (1) a laser gaging head, (2) a tire handling device, and (3) a computer that controls the movement of the tire handling machine, processes the data, and computes the least-squares straight line from which a wear rate may be estimated. Experimental tests show that the machine has good repeatability. In comparisons with measurements obtained by a hand gage, the automatic machine gives smaller average groove depths. The difference before and after a period of wear for both methods of measurement are the same. Wear rates estimated from the slopes of straight lines fitted to both sets of data are not significantly different.

Discussion of Criteria of Shear Strength Reduction FEM

2011 ◽  
Vol 243-249 ◽  
pp. 1279-1282
Author(s):  
Li Hong Chen ◽  
Shu Yu ◽  
Hong Tao Zhang
Keyword(s):  
Shear Strength ◽  
Complex Geometry ◽  
Strength Reduction ◽  
Nonlinear Material ◽  
Elastoplastic Model ◽  
Material Models ◽  
Main Technique ◽  
Failure Point ◽  
Straight Lines ◽  
Shear Strength Reduction

Shear strength reduction finite element method (SSRFEM) has been a main technique for stability analysis of slope. Although SSRFEM has advantages to deal with complex geometry and nonlinear material, the criteria for failure is still argued. Ideal elastoplastic model and rheological model were both adopted, and the results of computation showed that using the intersection of two straight lines as failure point was more appropriate. The usage and advantage of two different material models was compared.

scholarly journals An Improved Path-Generating Regulator for Two-Wheeled Robots to Track the Circle/Arc Passage

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Jun Dai ◽  
Naohiko Hanajima ◽  
Toshiharu Kazama ◽  
Akihiko Takashima
Keyword(s):  
Control Method ◽  
Convergence Property ◽  
Tangential Direction ◽  
Navigation Problem ◽  
Wheeled Robots ◽  
Robot Trajectory ◽  
Look Ahead ◽  
Straight Line ◽  
The Family ◽  
Straight Lines

The improved path-generating regulator (PGR) is proposed to path track the circle/arc passage for two-wheeled robots. The PGR, which is a control method for robots so as to orient its heading toward the tangential direction of one of the curves belonging to the family of path functions, is applied to navigation problem originally. Driving environments for robots are usually roads, streets, paths, passages, and ridges. These tracks can be seen as they consist of straight lines and arcs. In the case of small interval, arc can be regarded as straight line approximately; therefore we extended the PGR to drive the robot move along circle/arc passage based on the theory that PGR to track the straight passage. In addition, the adjustable look-ahead method is proposed to improve the robot trajectory convergence property to the target circle/arc. The effectiveness is proved through MATLAB simulations on both the comparisons with the PGR and the improved PGR with adjustable look-ahead method. The results of numerical simulations show that the adjustable look-ahead method has better convergence property and stronger capacity of resisting disturbance.

A New Acoustic Energy-Based Method to Estimate Pre-Loads on Cored Rocks

2018 ◽  
pp. 281-325
Author(s):  
Murat Karakus ◽  
Ashton Ingerson ◽  
William Thurlow ◽  
Michael Genockey ◽  
Jesse Jones
Keyword(s):  
Intersection Point ◽  
Acoustic Energy ◽  
Stress Axis ◽  
Irreversible Damage ◽  
Micro Cracks ◽  
Stress Curve ◽  
Sudden Release ◽  
The University ◽  
Straight Lines ◽  
Ae Signals

The Acoustic Emission (AE) due to the sudden release of energy from the micro-fracturing within the rock under loading has been used to estimate pre-load. Once the pre-load is exceeded an irreversible damage occurs at which AE signals significantly increase. This phenomenon known as Kaiser Effect (KE) can be recognised as an inflexion point in the cumulative AE hits versus stress curve. In order to determine the value of pre-load (sm) accurately, the curve may be approximated by two straight lines. The intersection point projected onto the stress axis indicates the pre-load. However, in some cases locating the point of inflexion is not easy. To overcome this problem we have developed a new method, The University of Adelaide Method (UoA), which use cumulative acoustic energy. Unlike existing methods, the UoA method emphasises the energy of each AE, the square term of the amplitude of each AE. As the axial pre-load is exceeded, the micro cracks become larger than the existing fractures and therefore energy released with new and larger cracks retain higher acoustic energy.

An n-Vertex Theorem for Convex Space Curves

1985 ◽  
Vol 37 (2) ◽  
pp. 217-237
Author(s):  
Tibor Bisztriczky
Keyword(s):  
Topological Space ◽  
Euclidean Plane ◽  
Convex Surface ◽  
Space Curves ◽  
Straight Line ◽  
History Of ◽  
General Convex ◽  
Vertex Theorem ◽  
The Subject ◽  
Three Space

The classical four-vertex theorem states that a simple closed convex C2 curve in the Euclidean plane has at least four vertices (points of extreme curvature). This theorem has many generalizations with regard to both the curve and the topological space and for a history of the subject, we refer to [4] and [1]. The particular generalization of concern, credited to H. Mohrmann, is the following n-vertex theorem.Let a simple closed C3 curve on a closed convex surface be intersected by a suitable plane in n points. Then the curve has at least n inflections (vertices).The closed convex surface in the preceding is defined as having at most two points in common with any straight line. Presently, we extend this result to curves on more general convex surfaces in a real projective three-space P3.

Appendix, containing the Investigation of a Formula for the Rectification of any Arch of an Ellipse

1805 ◽  
Vol 5 (2) ◽  
pp. 271-293
Keyword(s):  
Finite Number ◽  
The Other ◽  
Algebraic Equations ◽  
Conic Sections ◽  
Straight Line ◽  
Curve Line ◽  
Straight Lines

It is now generally understood, that by the rectification of a curve line, is meant, not only the method of finding a straight line exactly equal to it, but also the method of expressing it by certain functions of the other lines, whether straight lines or circles, by which the nature of the curve is defined. It is evidently in the latter sense that we must understand the term rectification, when applied to the arches of conic sections, seeing that it has hitherto been found impossible, either to exhibit straight lines equal to them, or to express their relation to their co-ordinates, by algebraic equations, consisting of a finite number of terms.

A theorem of M. L. Urquhart's and some consequences

2007 ◽  
Vol 91 (520) ◽  
pp. 39-50
Author(s):  
R. T. Leslie
Keyword(s):  
Euclidean Geometry ◽  
Straight Line ◽  
Elementary Theorem ◽  
Straight Lines

In an obituary of M. L. Urquhart in [1], David Elliott quotes him as claiming that Urquhart's theorem (below) is the most elementary theorem of Euclidean Geometry ‘since it involves only the concepts of straight line and distance’.Urquhart's theoremLet AC and AE be two straight lines.Let B be a point on AC, D a point on AE, and suppose that BE and CD intersect at F.If AB + BF = AD + DF then AC + CF = AE + EF. (1)

scholarly journals II. Second memoir on plane stigmatics

1867 ◽  
Vol 15 ◽  
pp. 192-203
Keyword(s):  
General Expression ◽  
Fundamental Lemma ◽  
Double Points ◽  
Straight Line ◽  
Straight Lines ◽  
Stigmatic Point ◽  
Coordinate Geometry

Let there be two groups of points upon a plane, termed, for distinction, indices and stigmata respectively, bearing such relations to each other that any one index determines the position of n stigmata, and any one stigma determines the position of m indices. The theory of these relations between indices and stigmata constitutes plane stigmatics . Each related pair of index X and stigma Y constitutes a stigmatic point , henceforth written “the s. point ( xy )." The straight lines joining any index with each of its corresponding stigmata are termed ordinates . If, when the index moves upon a straight line, the ordinate remains parallel to some other straight line, the relation between index and stigma is that expressed by the relation between abscissa and ordinate in the coordinate geometry of Descartes. When only one index corresponds to one stigma and conversely, and both indices and stigmata lie always on one and the same straight line, or the indices upon one and the stigmata upon another, the relations between indices and stigmata are those between homologous points in the homographic geometry of Chasles. The general expression of the stigmatic relation is obtained by a generalization of Chasles’s fundamental lemma in his theory of characteristics ( Comptes Rendus , June 27, 1864, vol. lviii. p. 1175), clinants being substituted for scalars. It results that in certain forms of the law of coordination , which “ coordinates ” the stigmata with the indices, there may be solitary indices which have no corresponding stigmata, and solitary stigmata which have no corresponding indices, and also double points in which the index coincides with its stigma (76). The particular case in which one index corresponds to one stigma and conversely, and no solitary index or stigma occurs, is termed a stigmatic line (henceforth written “s. line”), because the Cartesian case is that of a Cartesian straight line in ordinary coordinate geometry, but in the general s. line the figures described by index and stigma may be any directly similar plane figures (77). The investigation of this particular case occupies almost the whole of the Introductory Memoir . When one index corresponds to one stigma and conversely, but there is one solitary index and one solitary stigma, we have s. homography , provided the solitary index is distinct from the solitary stigma (79), and s. involution when the solitary index coincides with the solitary stigma (78), so called because they generalize the relations treated of under these names by Chasles.

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