Curvature plane curves

The radius of curvature R of a plane curve at any point P is the distance along the normal (the perpendicular to the tangent to the curve at point P) on the concave side of the curve to the center of curvature (Figure 1-33). If the equation of the curve is y = f(x)... 

F. Cao and L. Moisan. Geometric computation of curvature driven plane curve evolutions. SIAM J. Numer. Anal., 39(2) 624-646, 2001. 

The curvature at any point of a plane curve is the rate at which the curve is bending. Of two curves AG, AD, that has the greater curvature which departs the more rapidly from its tangent AB A (Fig. 93). In passing from P (Fig. 94) to... 

K. Mikula and D. Sevcovic. A direct method for solving an anisotropic mean curvature flow of plane curves with an external force. Math. Methods Appl Sci., 27(133 1545-1565, 2 0 04. 

Expressions for the curvature vector and curvature of a plane curve corresponding to different representations are ... 

Mr. Lamarle, generally considering the helicoids generated by a plane curve which is driven uniformly along a fixed line located as its axis, while it also turns uniformly around this same line, has searched for those which could satisfy the condition of a constant mean curvature, and he found thus, in addition to the skew helicoid with central axis, four other surfaces. These five helicoids correspond respectively to five of the equilibrium shapes of revolution, which are the plane, the sphere, the unduloid, the catenoid and the nodoid. As for that which corresponds to the cylinder, it is the cylinder itself. 

Mr. Mathet made known, in 1863, a method by means of which one can form the differential equation of all surfaces with zero mean curvature which pass through a given plane curve. 

Picture a small element of a curved interface between a liquid a and a vapour p, having two radii of curvature r, and (Fig. 3.6). These radii are defined by taking two planes at right angles to one another, each of them... 

The model of the shell under consideration is therefore described by the fact that its mid-surface is identified with a plane domain, while at the same time the curvature of the shell is not in general zero (see Section 1.1.3). Let tp G Hq(0, 1), and n be the normal to the curve y = tp x), x G (0,1). Then the condition of mutual nonpenetration for the crack faces can be written as follows ... 

So, we consider the shallow shell with the distances on the mid-surface coinciding with those on the plane. At the same time the curvatures are not equal to zero, in general. The shells like these are called the weakly curved plates. 

Jets discharging dose to the plane of the ceiling or wall are common in ventilation practice. The presence of an adjacent surface restricts air entrainment from the side of this surface. This results in a pressure difference across the jet, which therefore curves toward the surface. The curvature of the jet increases until it attaches to the surface. This phenomenon is usually referred to as a Coanda effect. The attached jet or, as it is commonly called, wall jet, can result from air supply through an outlet with one edge coincident with the plane of the wall or ceiling fFig. 7.27). Jets supplied at some distance from the surface or at some angle to the surface can also become attached (Fig. 7.28)... 

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... 

When the volume dV2 of the liquid evaporates, the volume of the vapor increases by dVt the two partial differentials refer to the same mass of substance. Thus (3 V2/d Vl)P2 = —Pi ip2, Pi and p2 being the densities of the two phases. Integration of the equation (3p1/3p2)K1 = P1/P2 affords- p0 = (p,/p2) (p2 - Pa)-The pressure p0 is that on both sides of a plane liquid surface. Pressure p2 is different from p0 whenever the liquid surface is curved. If its two principal radii of curvature are/ and/ 2, then... 

Fig. 5.1 A schematic projection of the 3n dimensional (per molecule) potential energy surface for intermolecular interaction. Lennard-Jones potential energy is plotted against molecule-molecule separation in one plane, the shifts in the position of the minimum and the curvature of an internal molecular vibration in the other. The heavy upper curve, a, represents the gas-gas pair interaction, the lower heavy curve, p, measures condensation. The lighter parabolic curves show the internal vibration in the dilute gas, the gas dimer, and the condensed phase. For the CH symmetric stretch of methane (3143.7 cm-1) at 300 K, RT corresponds to 8% of the oscillator zpe, and 210% of the LJ well depth for the gas-gas dimer (Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M. /. Phys. Chem. A 105, 9284 (2001))...

If the specimen crystal is curved, there will be a range of positions where the diffraction conditions are satisfied even for a plane wave. The rocking curve is broadened. It is simple to reduce the effect of curvature by reducing the collimator aperture. For semiconductor crystals it is good practice never to mn rocking curves with a collimator size above 1 mm, and 0.5 mm is preferable. Curved specimens are common if a mismatched epilayer forms coherently on a substrate, then the substrate will bow to reduce the elastic strain. The effect is geometric and independent of the diffraction geometiy. Table 2.1 illustrates this effect. 

For the Eikonal theory to be valid, the distortions must be suffieiently small that the eoneept of a ray is retained. The criterion is that the radius of curvature of the reflecting planes does not exceed a critical value R. approximately equivalent to an angular rotation of the Bragg planes by half the reflecting curve width in an extinction distance. The critical radius of curvature is thus... 

The standard relation which gives the radius of curvature of a curve in the X-Z plane (1) is as follows ... 

With this idea in mind, the horizontal surface in Figure 6.3b can be taken as a reference level at which Ap = 0. Just under the meniscus in the capillary the pressure is less than it would be on the other side of the surface owing to the curvature of the surface. The fact that the pressure is less in the liquid in the capillary just under the curved surface than it is at the reference plane causes the liquid to rise in the capillary until the liquid column generates a compensating hydrostatic pressure. The capillary possesses an axis of symmetry therefore at the bottom of the meniscus the radius of curvature is the same in the two perpendicular planes that include the axis. If we identify this radius of curvature by b, then the Laplace equation applied to the meniscus is Ap = 2y/b. Equating this to the hydrostatic pressure gives... 

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