Hyperbola conic section

Conic Sections The cui ves included in this group are obtained from plane sections of the cone. They include the circle, ehipse, parabola, hyperbola, and degeneratively the point and straight line. A conic is the locus of a point whose distance from a fixed point called the focus is in a constant ratio to its distance from a fixea line, called the directrix. This ratio is the eccentricity e. lie = 0, the conic is a circle if 0 < e < 1, the conic is an ellipse e = 1, the conic is a parabola ... 

The cross section of a right circular cone cut by a plane. An ellipse, parabola, and hyperbola are conic sections, coordinates... 

If there is a finite point in the xy plane which is symmetrically, situated with respect to a conic section, the latter is called a central conic (ellipse, circle, hyperbola). If this point is taken as the origin, the equation of the conic contains no terms of the first degree in x and y. The parabola is a non-central conic, since the centre is at an infinite distance. 

Conic section—A conic section is a figure that results from the intersection of a right circular cone with a plane. The conic sections are the circle, ellipse, parabola, and hyperbola. 

In either Laue method, the diffraction spots on the film, due to the planes of a single zone in the crystal, always lie on a curve which is some kind of conic section. When the film is in the transmission position, this curve is a complete ellipse for sufficiently small values of 0, the angle between the zone axis and the transmitted beam (Fig. 8-12). For somewhat larger values of 0, the ellipse is incomplete because of the finite size of the film. When 0 = 45°, the curve becomes a parabola when 0 exceeds 45°, a hyperbola and when 0 = 90°, a straight line. In all cases, the curve passes through the central spot formed by the transmitted beam. 

Assuming it is nondegenerate, the form of the conic section is determined by its discriminant, — 4AC. If the discriminant is positive, it is a hyperbola, and if it is negative, an ellipse or a circle. A discriminant of zero implies a parabola. The coefficients D, E, and Fhelp determine the location and scale of the figure. 

We have already considered parabolas of the form (y — yo) = k(x — xq), in connection with the quadratic formula. Analogous sideways parabolas can be obtained when the roles of x and y are reversed. Parabolas, as well as ellipses and hyperbolas, can be oriented obliquely to the axes by appropriate choices of B, the coefficient of xy, in the conic section Eq. (5.19), the simplest example being the 45° hyperbola xy = 1 considered above. Parabolas have the unique property that parallel rays incident upon them are reflected to a single point, called the focus, as shown in Fig. 5.8. A parabola with the equation... 

Conic Section Geometric form that can be defined by the intersection of a plane and a cone examples are a circle, an ellipse, a parabola, a hyperbola, a line, and a point. 

Kepler s principal contribution is summarized in his laws of planetary motion. Originally derived semiempir-ically, by solving for the detailed motion of the planets (especially Mars) Ifom Tycho s observations, these laws embody the basic properties of two-body orbits. The first law is that the planetary orbits describe conic sections of various eccentricities and semimajor axes. Closed, that is to say periodic, orbits are circles or ellipses. Aperiodic orbits are parabolas or hyperbolas. The second law states that a planet will sweep out equal areas of arc in equal times. This is also a statement, as was later demonstrated by Newton and his successors, of the conservation of angular momentum. The third law, which is the main dynamical result, is also called the Harmonic Law. It states that the orbital period of a planet, P, is related to its distance from the central body (in the specific case of the solar system as a whole, the sun), a, by a. In more general form, speaking ahistorically, this can be stated as G M -h Af2) = a S2, where G is the gravitational constant, 2 = 2n/P is the orbital frequency, and M and M2 are the masses of the two bodies. Kepler s specific form of the law holds when the period is measured in years and the distance is scaled to the semimajor axis of the earth s orbit, the astronomical unit (AU). 

Fig. 24. Dupin cyclide and perfect focal-conic domain construction, (a) Vertical section showing layers of the structure thick lines indicate the ellipse, hyperbola, Dupin cyclide, and central domain, (b) Focal-conic domain showing structural layers with a representation of the arrangement of the molecules within one of them.

Fig. 5.4.3. (a) Geometry of a pair of focal conics. (After Friedel. ). b) A section of a) in the plane of the hyperbola, showing parts of Dupin cyclides. (After... 

The choice of elementary shapes starts with the straight line. Straight lines are applied as edges of flat, ruled, or other surfaces. They also serve as important reference entities such as centerlines, etc. Arcs are elements of compound lines, sections for fillets, and serve many other purposes. In addition to lines and arcs, analytical lines are the conics such as circles, ellipses, hyperbolas, and parabolas (Figure 4-1). 

Fig. 8.30 Focal-conic defect structure in SmA A pair of cones with a common elliptical base and a hyperbola connecting cone apices (a) cross-section of the upper cone by plane ABC with gaps between lines (Dupin cyclides) indicating the smectic layers (b) filling the space of the sample by cones of different size (c)...

Smectic A phases in which the layers are not uniformly parallel to the glass slides confining the sample (i.e. not in a planar orientation) are characterized by fan-like textures (Fig. 5.10b), made up of focal conics (Fig. 5.12). A focal conic is an intersection in the plane of a geometric object called a Dupin cyclide (Fig. 5.13), which results from lamellae forming a concentric roll (like a Swiss roll) being bent into an object based on an elliptical torus of non-uniform cross-section. The straight line that would define the rotation axis of the torus is distorted into a hyperbola in the Dupin cyclide. 

Fig. 1.22 The geometry of focal conic units, (a) the simplest geometry, with a circular cross-section and a straight, central discilination line, (b) the more general geometry, with an ellipse and hyperbola, (c) a slanting section of a focal conic unit, showing the origin of the characteristic focal conic fan texture...

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