Focus 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 ... 

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... 

Other conic sections exist paraboloidal mirrors, ellipsoidal mirrors, and hyperboloidal mirrors. In paraboloidal mirrors, all rays (from infinity or not) converge at the same focus. In ellipsoidal mirrors, all rays emanating at focus... 

A conic section or cotdc is the locus of a point which moves so that its distance from a fixed point (called the focus) is in a constant ratio (called the eccentricity) to the distance from a fixed straight line (called the directrix). 

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. 

Directrix—The fixed line in the focus directrix definition of a conic section. 

Focus—point, or one of a pair of points, whose position determines the shape of a conic section. 

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... 

The equations for conic sections can be expressed rather elegantly in polar coordinates. As shown in Fig. 5.9, the origin is defined as the focus and a line corresponding to x = d serves as the directrix. Recall the relations between the Cartesian and polar coordinates x = rcos 9, y = rsin6>. The point Fwill trace out the conic section, moving in such a way that the ratio of its distance to the focus r to its distance to the directrix d — x = d—rcos9 is a constant. This ratio is called the eccentricity, e (not to be confused with Euler s 6 = 2.718...) ... 

The outline of the remainder of this contribution is as follows. In Section 3.7.2, we discuss radical anion dissociation in solution, in which a conical intersection has an important impact on the ground state reaction barrier, rate constant and reaction path, all of which are also influenced by nonequilibrium solvation. The excited electronic state conical intersection problem for the cis-trans isomerization of a model protonated Schiff base in solution is discussed in Section 3.7.3, focusing on the approach to, and passage through, the conical intersection, and the influence of nonequilibrium solvation thereupon. Some concluding remarks are offered in Section 3.7.4. We make no attempt to give a complete discussion for these topics, but rather focus solely on several highlights. Similarly, the references herein are certainly incomplete. We refer the interested reader to refs [1-9] for much more extensive discussions and references. 

We have already mentioned in the Introduction (Section 3.7.1) the importance of conical intersections (CIs) in connection with excited electronic state dynamics of a photoexcited chromophore. Briefly, CIs act as photochemical funnels in the passage from the first excited S, state to the ground electronic state S0, allowing often ultrafast transition dynamics for this process. (They can also be involved in transitions between excited electronic states, not discussed here.) While most theoretical studies have focused on CIs for a chromophore in the gas phase (for a representative selection, see refs [16, 83-89], here our focus is on the influence of a condensed phase environment [4-9], In particular, as discussed below, there are important nonequilibrium solvation effects due to the lack of solvent polarization equilibration to the evolving charge distribution of the chromophore. 

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