Conic section parabola

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 ... 

In the conics section we discussed the property of parabolas. Since one of its conjugates is at infinity, it will form a perfect image from the infinite conjugate at focus. This is the basis for a single mirror telescope. This telescope is either used at this prime focus, or sometimes a folding flat mirror (Newtonian telescope) is used to fold the beam to a convenient location for a camera or the... 

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. 

Parabolic—Shape based on the parabola, a conic section from mathematics. 

This reflects the fact that all parabolas have the same eccentricity, namely 1. The eccentricity of a conic section is the ratio of the distances point-to-focus divided by point-to-directrix, which is the same for all the points on the conic section. Since, for a parabola, these two distances are always equal, their ratio is always 1. 

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... 

Equation (8.12) is the equation to the Condon parabola in the (v",v ) plane, in a form that is convenient to compute and to draw. For analysis, it may be more convenient to write it in the standard form for a conic section, namely. 

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). 

The Marcus parabolas (Fig. 14.27d) represent a special section (along the collective variable) of the hypersurfaces passing through the eonieal intersection (parabolas Vr and Vp). Each parabola represents a diabatie state, so a part of each reactant parabola is on the lower hypersurface, while the other one is on the upper hypersurface. We see that the parabolas are only an approximation to the hypersurface profile. The reaction is of a thermal character, and as a consequence, the parabolas should not pass through the conical intersection, because it corresponds to high energy, instead it passes through one of the saddle points. 

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. 14.27. Electron transfer in the reaction DA D+A , as well as the relation of the Marcus parabolas to the concepts of the conical intersection, diabatic and adiabatic states, entrance and exit channels and the reaction barrier. Panel (a) shows two diabatic surfaces as functions of the f i and I2 variables that describe the deviation from the comical intersection point (within the bifurcation plane cf., p. 312). Both surfaces are shown schematically in the form of the two intasecting paraboloids one for the reactants (DA), and the second for products (D A ). (b) The same as (a), but the hypersurfaces are presented more realistically. The upper and lower adiabatic surfaces touch at the conical intersection point, (c) A ma-e detailed view of the same surfaces. On the ground-state adiabatic surface (the lower one), we can see two reaction channels I and 11, each with its reaction barrier. On the upper adiabatic surface, an energy valley is visible that symbolizes a bound state that is separated from the conical intersection by a reaction barrier, (d) The Marcus parabolas represent the sections of the diabatic surfaces along the corresponding reaction channel, at a certain distance from the conical intersection. Hence, the parabolas in reality cannot intersect (undergo an avoided crossing).

Figure 30. Dupin s cyclides and their deformations, (a) A toroidal Dupin cyclide and its circular lines of curvature y and /, the planes of which intersect along the axes A (joining O and Q) and A, respectively. Two tangential planes along two symmetrical circles T intersect along A. (b) A parabolic Dupin s cyclide, with its two focal parabolae P and P, its circular curvature lines y and /, and its axes A and A, (c) Three parallel, but separated, parabolic Dupin s cyclides. The lines represent sections by planes x=0,y = 0 and = 0 at the top of the figure, two conical points are present, as in (b), but the spindle-shaped sheet has been removed (Rosenblatt et al.

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