Osculation. Points

Possible types of intersections between A and 0 as a function of time for different ratios of the rate coefficient. (A) Acme point,/ = 0.5 (B) Euler point,/ =1 (C) osculation point,/ =2 (D) triad... 

At the osculation point, p = 2. Experimentally, we are better able to observe peak and interseetion values than osculation at infinity, because tail analysis can be marred by experimental error. 

When q = 0, the Hugoniot curve represents an adiabatic shock. Point 1 (Pi, p ) is then on the curve and Y and J are 1. Then [(l/pt) - (Ilp2) = 0, and the classical result of the shock theory is found that is, the shock Hugoniot curve osculates the adiabat at the point representing the conditions before the shock. 

Figure 4.4 a. Formal definition of curvature in two dimensions, for concave and convex curves, b. Definition of curvature in two dimensions, depending on the curve s osculating circle which is drawn at point B on the curve, merging as much as possible with the section of the curve around B. The value of radius of curvature, Rb is sufficient to characterize the shape of the curve around B. The same procedure leads to Ra at point A. The curvature at A is larger than the curvature at B. 

In this analysis the most interesting points of the two osculating orbits are of course the apocenter of the inner orbit and the pericenter of the outer orbit. The corresponding distances are A = at( I et) = apocenter distance of the inner orbit, and Pe = oe( 1 — ee) = pericenter distance of the outer orbit. On the other hand, by continuity, the condition (32) remains satisfied and thus we can write that the zone of possible motion of figures as Figure 7 also contains at all times the points (R, A%) and (.Pe,Ai). 

The cusps which have just been described are called single cusps in contradistinction to double cusps or points of osculation in which the curves extend to both sides of the point of contact. These are what Cayley calls tacnodes. The differential coefficient has now two or more equal roots and y has at least two equal values. The different branches of the curve have a common tangent. 

To distinguish cusps from points of osculation compare the ordinate of the curve for that point with the ordinates of the curve on each side. For a cusp, y and the first differential coefficient have only one real value. 

The differential that characterizes non-Euclidean space is known as curvature (Lee, 1997). In two dimensions it describes how a smooth curve deviates from linearity. The curvature, which varies from point to point, is specified in terms of the osculating circle of radius R and centred on the perpendicular to the tangent at p, and which follows the curve in the vicinity of p. On an infinitesimal scale each point on a curve has a unique osculating circle. 

An intersection of two temporal concentrations, for instance Ca(0 and c t), means that these concentrations can be considered equal at some point in time, that is, cx t) = Cb (t). It is a well-known mathematical fact that phase trajectories do not intersect or merge nevertheless, the temporal trajectories may well intersect. A special case is osculation, in which not only the concentrations but also the temporal slopes coincide cx t) =CB(t) and dcx t)/dt = dc g t)/dt. 

During their analysis, Yablonsky et al. (2010) found a number of special points, which they named Acme, Golden, Euler, Lambert, Osculation, and Triad (AGELOT) points. Table 11.1 summarizes some characteristics of these points. For more details see Yablonsky et al. (2010). 

A change of sequence indicates a point of intersection between curves. For example, in Table 11.2, between D1 and D2, the values A = C and B ,ax change position, so these curves intersect at the Acme point with/ = 0.5 the final value of domain D1 and the starting value of D2. Note that the same value sequence is obtained for the intervals l/2special point is that where the A = C and B ax value curves intersect, namely at the unique value of p that satisfies the transcendental equation... 

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