Director Circle Research Articles

Dynamic construction of a family of octic curves as geometric loci

<abstract><p>We explore the construction of curves of degree 8 (octics) appearing as geometric loci of points defined by moving points on an ellipse and its director circle. To achieve this goal we develop different computer algebra methods, dealing with trigonometric or with rational parametric representations, as well as through implicit polynomial equations, of the given curves. Finally, we highlight the involved mathematical or computational issues arising when reflecting on the outputs obtained in each case.</p></abstract>

Ibn Sahl’s machines to draw conics continuously

The article describes the machines proposed by Ibn Sahl, who lived in the tenth century, to draw continuously the three irreducible conics. These machines are based on the use of a director circle which, just as the directrix of a parabola allows one to draw the conic as a locus, and to find the tangent line at each point. All this is demonstrated with a touchscreen: the visitor may choose a conic and start several animations which show the geometry of the conic and its envelope. The corresponding machine is operated by the visitor who, by moving an on-screen pencil with their finger, can draw the conic, simulating the actual working of the machine.

A new look at the Feynman ‘hodograph’ approach to the Kepler first law

Hodographs for the Kepler problem are circles. This fact, known for almost two centuries, still provides the simplest path to derive the Kepler first law. Through Feynman’s ‘lost lecture’, this derivation has now reached a wider audience. Here we look again at Feynman’s approach to this problem, as well as the recently suggested modification by van Haandel and Heckman (vHH), with two aims in mind, both of which extend the scope of the approach. First we review the geometric constructions of the Feynman and vHH approaches (that prove the existence of elliptic orbits without making use of integral calculus or differential equations) and then extend the geometric approach to also cover the hyperbolic orbits (corresponding to ). In the second part we analyse the properties of the director circles of the conics, which are used to simplify the approach, and we relate with the properties of the hodographs and Laplace–Runge–Lenz vector the constant of motion specific to the Kepler problem. Finally, we briefly discuss the generalisation of the geometric method to the Kepler problem in configuration spaces of constant curvature, i.e. in the sphere and the hyperbolic plane.

Director circles of conic sections

Given a conic section, the locus of a moving point in the plane of the conic section such that the two tangent lines drawn to the conic section from the moving point are all mutually perpendicular is a curve. In the case of an ellipse and hyperbola this curve is a circle referred to as the director circle. In the case of the parabola this curve coincides with the directrix of the parabola. The last section is devoted to the graphical illustrations of director circles for circles, parabolas, ellipses and hyperbolas using the built-in Maple V software of Scientific Work Place 3.0.

準円の性質を用いた楕円パターンの抽出法

The rapid and accurate detection of elliptical patterns in images is an important aspect of computer vision systems, and the present study proposes an efficient method for detecting them by using the director circle of the ellipse The method involves a Center Detecting algorithm that determines the coordinates of the center by using the feature that the edges of two pixels on the ellipse, which are symmetric in the center, have the opposite orientations, a Director Circle Detecting algorithm that detects the diameter of the director circle of the ellipse, and a Form Detecting algorithm that uses the director circle to determine the slope angle, semiminor diameter and semimajor diameter of the ellipse This method is applied to sample images and several experimental results are presented.

Effects of parallel compressions on ring-shaped polycrystalline ferrimagnetic samples

This work deals with the behavior of the macroscopic magnetic properties of ring-shaped polycrystalline yttrium-iron garnets (YIG) under the action of compressive stresses applied parallel to the measurement direction along the director circles. For this purpose, a high-pressure cell has been designed consisting of a hydrostatic pressure application on the outer side of the ring. The results are separated in two parts according to the sample used: The first one is concerned with YIG samples called ‘‘type D’’ and so defined because of the dispersed nature of their natural domain-wall configuration. The second one is concerned with YIG samples called ‘‘type T’’ and so defined because of the toroidal character of their natural domain-wall configuration. It is shown that the first category (type-D samples) gives results in good agreement with the classical theory of the magnetoelasticity, while the second (type-T samples) gives results altogether inconsistent. The difference of behavior between the two types of samples is explained by the difference between their domain-wall configuration in the spontaneous state. A well-tried theoretical model is used for the general interpretation.

Relaxation frequency as a function of domain wall topography in soft polycrystalline ferrites

The influence of domain wall configuration changes on the frequency spectrum of the complex permeability, μ ̌ = μ′−jμ″: ,in the domain wall relaxation range is investigated. The samples studied, polycrystalline yttrium iron garnet ferrites (YIG), having a toroidal shape are differentiated from one another by various intragranular defects. Mechanical compression of up to 120 bar, applied parallel to the axis of revolution of the toroid, is used to vary the domain wall configuration. The sample is magnetized along the director circles, so that the stress is perpendicular to the initial magnetization direction. We find that the perpendicular compression changes the frequency properties of μ̌. We observe a systematic decrease in the real part of μ̌ and a strong increase of the relaxation frequency of the domain wall motion only if the grains contain a high density of defects. Using a concept presented in previous work, the interpretation of the received results is proposed.

デカルトの卵形線の性質に関する考察

The Cartesian ovals are simply defined with bipolar coordinates. However it is difficult to draw them by handwriting. In this paper, therefore, we have constructed a computational drawing system by means of XY plotter that works with PC 8001mkII. And then, the ovals will show some particular properties. First, the ovals change gradually with left and right eccentricities. Second, the set of the intersections of three circles forms the oval. Third, when the eccentricities change continuously without changing the director circle and the foci, the ovals fill between the ellipse and the director circle. Forth, there can be shown two kinds of evolutes of the ovals. The one is drawn as an evelope of the straight lines, and the other with paramatric equations. In this way, these properties of the oval can be shown for the first time by using computational drawing system.

Effect of pressure on soft magnetic materials

After a review of previous work, the authors show their results obtained in the study of the effect of pressure on the magnetization mechanisms of polycrystalline soft ferrites up to 1.5 kbar and in a temperature range from 263°K to 573°K. Three kinds of pressures were applied : hydrostatic pressure, lateral pressure inducing stresses along the director circles of toroids and pressure perpendicular to the plane surfaces of toroids. All the results concerning reversible and irreversible magnetization mechanisms under hydrostatic pressure are explained from an unique idea: the change in the topography of domain walls in the grains due to random stresses from the hydrostatic pressure because of compressibility anisotropy. That was confirmed by the effect of directional stresses and by a pressure induced hysteresis. The origin of the compressibility anisotropy was found in the closed porosity.

Change of wall topography in soft Ferrites by isotropic and anisotropic external pressures

Polycrystalline toroidal samples of soft ferrites have been submitted successively to hydrostatic and uniaxial pressures. The influence of these stresses on the initial susceptibility and the frequency spectrum is studied. Uniaxial pressures applied along the director circles of the toroid produce no effect on these parameters whereas uniaxial pressures directed along the generating lines and hydrostatic pressures involve a decrease of the initial susceptibility and an increase of the frequency relaxation. An interpretation of these phenomena is given, considering that these materials have four easy axis and negative magnetostriction. The domain walls prealigned along the director circles after thermal demagnetization align themselves along the easy axis the nearest to that of the local stress, changing the initial wall topography.

A Direct Derivation of the Equation of the Director Circle of an Ellipse

A Direct Derivation of the Equation of the Director Circle of an Ellipse

1758. The director circle of the general conic

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1449. A new form of the equation of the director circle of a conic

1449. A new form of the equation of the director circle of a conic

On Pedal Quadrics in Non-Euclidean Hyperspace

1. The following are well-known theorems of elementary geometry: Given any euclidean plane triangle, A0 A1 A2, and any pair of points, X, Y, isogonally conjugate q. A0 A1 A2; then the orthogonal projections of X, Y on the sides of A0 A1 A2 lie on a circle, the pedal circle of the point-pair. If either of the points X,- Y describe a (straight) line, m, then the other describes a conic circumscribing A0 A1 A2, and the pedal circle remains orthogonal to a fixed circle, J; thus the pedal circles in question are members of an ∞2 linear system of circles of which the circle J and the line at infinity constitute the Jacobian. In particular, if the line m meet Aj Ak at Lt (i, j, k = 0, 1, 2), then the circles on Ai Li as diameter, which are the pedal circles of the point-pairs Ai, Li, are coaxial; the remaining circles of the coaxial system being the director circles of the conics, inscribed in the triangle A0 A1 A2, which touch the line m. If Mi denote the orthogonal projection on m of Ai, and Ni the orthogonal projection on Aj Ak of Mi, then the three lines Mi Ni meet at a point (Neuberg's theorem), viz. the centre of the circle J. Analogues for three-dimensional space of most of these theorems are also known ‖.

711. [C1. 16. 5.] The Polar Circle of a Triangle is orthogonal to the Director Circle of an Inscribed Conic.

711. [C¹. 16. 5.] The Polar Circle of a Triangle is orthogonal to the Director Circle of an Inscribed Conic.

315. [L1. 17. e.] Systems of conics whose director circles have a common radical centre

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271.[K. 2. c ] The polar circle of a triangle is cut orthogonally by the director circle of any inscribed conic

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272. [K. 2. c ] Geometrical proof that the polar circle of a triangle is orthogonal to the director circle of the inscribed circle

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272. Geometrical Proof That the Polar Circle of a Triangle Is Orthogonal to the Director Circle of the Inscribed Circle

272. Geometrical Proof That the Polar Circle of a Triangle Is Orthogonal to the Director Circle of the Inscribed Circle

271. The Polar Circle of a Triangle Is Cut Orthogonally by the Director Circle of Any Inscribed Conic

271. The Polar Circle of a Triangle Is Cut Orthogonally by the Director Circle of Any Inscribed Conic