Curvature Circle

It is important to note the sign convention used here. The pressure is highest in the convex medium, that is, the medium where the centers of the curvature circles are placed. 

If the first plane is rotated through a full circle, the first radius of curvature will go through a minimum, and its value at this minimum is called the principal radius of curvature. The second principal radius of curvature is then that in the second plane, kept at right angles to the first. Because Fig. II-3 and Eq. II-7 are obtained by quite arbitrary orientation of the first plane, the radii R and R2 are not necessarily the principal radii of curvature. The pressure difference AP, cannot depend upon the manner in which and R2 are chosen, however, and it follows that the sum ( /R + l/f 2) is independent of how the first plane is oriented (although, of course, the second plane is always at right angles to it). 

Measured curvatures for a 100-mm laminate (a single closed circle on each curve) and eleven 150-mm laminates (a closed circle for the mean with a range for all other values shown) are shown in Figure 6-26. The quantitative comparison is only fair as might be expected for an approximate approach. However, the qualitative agreement as to mode of deformation is correct. 

The curvature at a given point on a surface is characterized by the maximum and minimum radii, R and R2, of the circles in mutually perpendicular planes perpendicular to the plane tangent to the surface at that point, best approximating the curves formed by the intersection of the surface with these planes [221], (Fig. 1). The principal curvatures, k and k2 are defined as... 

Fig. 5.4 Voltage-time curve for a p-type silicon electrode anodized galvanostatically at 0.1 mA cm"2 in 10% acetic acid. Silicon electrodes were removed from the electrolyte after various anodization times (filled circles) and the thickness of the anodic oxide was measured by ellipsometry (open circles). The curvature of the sample was monitored in situ and is plotted as the value of stress times oxide thickness (filled triangles). The bar graph below the V(t) curve shows a proposed formation mechanism. Galvanostatically a...

Rowland circle spect A circle drawn tangent to the face of a concave diffraction grating at its midpoint, having a diameter equal to the radius of curvature of a grating surface the slit and camera for the grating should lie on this circle. ro.lsnd, s3r-k3l ... 

Approximating the elongation as the arc of a circle of base X and height A gives a radius of curvature R of... 

Equation 8 is less satisfactory when it is applied to rates of oxidation. The open circles and squares in Figure 1 are plots of R0 against R 1/2 at 50° and 100° from Table I. Thus, when the Rt/2a term is neglected, R0 is proportional to R 1/2. However, when we incorporate the R /2a term (solid circles and squares), considerable curvature appears, corresponding to unexpectedly high rates of chain termination (larger as) at higher rates of initiation. 

Figure 5.2 Schematic 3-dimensional depiction of the entropy function S(U, A), showing the tangent plane (planar grid) at an equilibrium state (small circle). According to the curvature (stability) condition, the entropy function always falls below its equilibrium tangent planes, and thus has the form of a convex function. ...

Given a map M, its circle-packing representation (see [Moh97]) is a set of disks on a Riemann surface E of constant curvature, one disk D(v, rv) for each vertex v of M, such that the following conditions are fulfilled ... 

The method of caustics has also been used to study the formation of cracks and crazes formed by exposure of PMMA to solvents (259). ISO 4599 (260) has been developed to better control the application of stress using a jig having the curve of the arc of a circle for shaping the specimen and maintaining a set curvature during exposure to the agent. After a predetermined time the specimen is tested for tensile or flexural properties and compared to preexposure test values. ISO 4600 (261) uses the technique of impressing an oversized ball or pin into a hole drilled in the specimen to apply a strain. 

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 ... 

A grating ruled on a spherical surface combines the properties of the diffraction grating with the focusing ability of the optical surface. Such a device, with radius of curvature R, focuses spectra as images of the entrance (primaiy) slit on the circumference of a circle of diameter R, when the entrance slit is also located on the circumference of the circle.z... 

To derive the equation of Young and Laplace we consider a small part of a liquid surface. First, we pick a point X and draw a line around it which is characterized by the fact that all points on that line are the same distance d away from X (Fig. 2.6). If the liquid surface is planar, this would be a flat circle. On this line we take two cuts that are perpendicular to each other (AXB and CXD). Consider in B a small segment on the line of length dl. The surface tension pulls with a force 7 dl. The vertical force on that segment is 7 dl sin a. For small surface areas (and small a) we have sin a d/R where R is the radius of curvature along AXB. The vertical force component is... 

Figure 6.13 Normalized force between a microfabricated silicon nitride tip of an atomic force microscope and a planar mica surface in 1-propanol at room temperature [174], The tip had a radius of curvature of R w 50 nm. The different symbols were recorded during approach (filled circles) and retraction (open circles) of the tip.

Now that we know the position of the center of the curvature, we can calculate the points where the unit circle around the current element and the circle that describes the curvature intersect. This is illustrated in Figure 12.12. Let us assume that the center of the curvature is located at point (0, r). This simplifies the calculation of the two points of intersection. We have to calculate the two point of intersection Pi and P2. The following two equations describe both circles. 

Figure 12.12 The values along the line of constant illumination can be obtained by calculating the intersection between the unit circle and the circle that describes the curvature at the current processing element. We have to obtain pixel values at positions Pi and l if the center of the curvature lies at position (0, r ).

The radius of curvature does not necessarily have to describe a circle. How-ever, if R describes a circular curvature, then we have two limiting possibilities, as shown in Figure 7.24. We get a sphere if the radii of curvature are equal, and... 

Moreover, flight clearance and curvature effects were also accounted for. Figure 9.37 indicates that, in this particular case, the simple Newtonian model provides a reasonable estimate, although it overestimates the rate of melting. Note that the predicted curve should approach the closed circles and triangles, which are the measured solid bed width at the melt film, rather than the open circles and triangles, which are the corresponding values at the root of the screw. As observed experimentally, the width near the root of the screw is reduced as a result of melt pool circulation. 

With the Shore methods the penetrating body is a cone with A and C the tip of the cone is flattened to a circle with 0.79 mm diameter with D it is rounded-off to a radius of curvature of 0.1 mm. The force is exerted by a spring, which for A, mainly applied to rubbers, is considerably less stiff than for C and D. Hard thermoplastics are mostly characterized by Shore D the values found are between 50 and 90 units. 

Application of Eq. 19 to the /7-HMX isotherm from simulations leads to the Us-Up curve shown in Fig. 11, where negative curvature in the simulation results is clearly evident (filled circles). While such behavior would be anomalous for metals, it is actually expected for pressures below about one GPa in the case of polyatomic molecular crystals, due to complicated molecular packings and intramolecular flexibility, and has in fact been reported for the high explosives pentaerythritol tetranitrate (PETN) where careful studies were performed for low levels of compression [77], By contrast, the experimental results for /3-HMX in the Us-Up plane do not exhibit significant curvature due to lack of data at pressures below about one Gpa [78], Thus, estimates of isothermal sound speeds, and hence isothermal bulk moduli, based on... 

Einstein [6] illustrated the curvature of space-time by considering two coordinate systems, K and K, with a common origin, one of them stationary and the other rotating (accelerated) about the common Z-axis, in a space free of gravitational fields, shown in Figure 2.4 on the right. A circle around the origin in the X — Y plane of K is also a circle in the X — Y plane of K. Measurement of the circumference S and diameter 2R in the stationary... 

Estimates of shifts of spectra in curved crystal geometries are often calculated for an ideal detector located on the Rowland circle. However, the detection surface is usually fiat and therefore cannot lie on the Rowland circle. Detectors located on a fixed length detector arm will additionally travel off the Rowland circle as the Bragg angle is scanned unless the crystal curvature is simultaneously scanned (which raises problems of stress hysteresis). Conventional shifts calculated for detection on the Rowland circle do not agree with shifts at a flat extended detector and this systematic error can be 100-200 ppm for any Johann curved crystal spectrometer. We have incorporated fiat surface detectors located off the Rowland circle into the general theory [18,17]. 

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