Osculating circle

For the particular case of an axisymmetric surface r(z), the curvature is the sum of the radius of the osculating circle (in the plane shared by the surface normal) and the curvature of r(z) in two dimensions ... 

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. 

Figure 4.6 Description of three-dimensional curvature using plane geometry concepts. Each of the perpendicular planes contains a portion of the arc that intersects the curved surface. The radii of curvature (the radii of the osculating circles) are designated R, and R2 and the arc lengths are designated as x and y, respectively. The radian angles are a, and a2. If the curved surface is moved outwards by a small amount dz, the surface area increases by increasing the arc lengths to (x + dx) and (y + dy).

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. 

For comparison purposes, experimental data are usually normalized by dividing the storage and loss modrdus by G sc, the radius of the osculating circle of the Cole-Cole plot at the origin. Any departure from monoexponential decay is represented by deviations from the semicircle. This behavior is schematically shown in Figure 9.2, where nonexponential behavior occurs at elevated values of the angular frequency. 

Mean anomaly Mean rate of motion of a body in an ellipse, relative to the center of the osculating circle. 

The local shape of the interface in each of the 3D images provided in Fig. 30 can be described by the probability densities of the mean and Gaussian curvatures—Ph(H) and PxiK), respectively—and can be calculated from P(H,K) [72]. The curvature is arbitrarily chosen to be positive if the center of the osculating circle resides within the I microphase of the copolymer or the PB phase of the polymer blend. To facilitate comparison, Ph(H) and Pk K) have been scaled in the same manner as described in Sect. 4.4. E was equal to 0.070 nm for the copolymer and 0.136 for the blend. The Ph h) and Pk k) determined from the two bicontinuous morphologies shown in Fig. 30 are displayed in Fig. 31 and exhibit surprising similarity. 

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