Constant curvature

Dense Configurations of Linear Chains of Constant Curvature.. 55... 

The most austere representation of a polymer backbone considers continuous space curves with a persistence in their tangent direction. The Porod-Kratky model [99,100] for a chain molecule incorporates the concept of constant curvature c0 everywhere on the chain skeleton c0 being dependent on the chemical structure of the polymer. It is frequently referred to as the wormlike chain, and detailed studies of this model have already appeared in the literature [101-103], In his model, Santos accounts for the polymer-like behavior of stream lines by enforcing this property of constant curvature. 

In the case of the Porod-Kratky model, the polymer backbones have a constant curvature c0. Accounting for the polymer stiffness in generating the dense configuration of stream lines, the vector field used must have a homogeneous curvature field with a unique value cq in the entire simulation box T. In order to quantify the success in creating such a vector field, the deviation of the curvature from the ideal Porod-Kratky case, a volume integral has been used by Santos as a penalty function ... 

A number of equations have been proposed for use in the calculation of pressure drop in coils of constant curvature [Srinivasan et al (1968)]. The latter are known as helices. For laminar flow, Kubair and Kuloor (1965) gave an equation for the Reynolds number range 170 to the critical value. In terms of the Fanning friction factor, their equation can be written as... 

Gauss has shown [14] that Equation 9.6a is true for any surface with constant curvature H for which the reversible fluctuations do not break the relation... 

Equation 9.6 and Equation 9.7 require that any interface between two fluids in equilibrium must have the constant curvature H. Appearance of two points with different curvatures results in appearance of a corresponding AP, which forces substance transfer until equalization of H. This gives rise to a... 

Figure 14.23 Shape accuracy results for continous dieless forming, variable curvature workpiece at the top, and constant-curvature workpieces under three different forming conditions at the bottom. Shape accuracy to about 0.016-in.) is obtained in the top two results...

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 group T (l, m, n) can be realized as a group of isometries of a simply connected surface X of constant curvature, where ... 

We now consider another geometric viewpoint on (r, )-polycycles. In the above consideration, the curvature was uniform and the triangles were viewed as embedded into a surface of constant curvature. Consider now the curvature to be constant, equal to zero, in the triangle itself, and to be concentrated on the vertices, where r-gons meet. 

DeSt05] M. Deza and M. I. Shtogrin, Metrics of constant curvature on polycycles, Mathematical Notes 78-2 (2005) 223-233. 

The torsion of a curve describes its pitch a helix exhibits both constant curvature and torsion. Its curvature is measured by its projection in the tangent plane to the curve - which is a circle for a helix - while its torsion describes the degree of non-planarity of the curve. Thus a curve on a surface (even a geodesic), generally displays both curvature and torsion. 

Rather better is the constant curvature end-condition, which sets the second derivative constant over the first span. 

We can set up end conditions analogous to constant curvature in the end span by asking that the first derivatives at A and B should have the chord B — A as their mean, and that the fourth difference at B should be zero. 

Figure 3.4 Illustrations of the Langmuir isotherm, (a) Nondimensionalized form of the isotherm, (b) Effect of the curvature b at constant

The phenomenon of Dean vortices was first observed by Dean in a curved tube. The fluid in the central part is driven toward the external wall by the centrifugal force, which gives rise to a secondary flow. This results in the inward movement of the fluid near the wall and the outward movement of the fluid near the center (Fig. 3A,B). Curved tubes may be classified as torus, bends, helical coils (with a constant curvature), and spirals (with a variable curvature). Fig. 4 shows the helical coiled, spiral, and bend tubes. The hydrodynamics in a coiled tube can be characterized by a dimensionless number named after Dean and is defined as ... 

The infinitely long cylinder with no motion of the interface or of the fluid within the cylinder is, of course, a possible equilibrium configuration, in the sense that it is a surface of constant curvature so that the stationary constant-radius fluid satisfies all of the conditions of the problem, including the Navier-Stokes and continuity equations (trivially), as well as all of the interface boundary conditions including especially the normal-stress balance, which simply requires that the pressure inside the cylinder exceed that outside by the factor 1//a. The question for linear stability theory is whether this stationary configuration is stable to infinitesimal perturbations of the velocity, the pressure, or the shape of the cylinder. 

An approach somewhat related to the broken-path method, but more accurate, has been employed by Johnson (1972). It also makes use of curvilinear coordinates (s, p) chosen with constant curvature, and divides the intermediate region into several sectors. In each of these the variables s, p are separable. Calculations were done with the amplitude density technique, matching functions and derivatives at each sector s boundary. The potential was that of Porter and Karplus, and the local vibrational motion was assumed to be harmonic. Good agreement was found with Diestler s at low energies. 

This representation gives a useful way of understanding how the second derivative is related to the force constant (curvature) and how to assign the type of stationary point (sign). 

One of the commonest methods of smoothing a curve is to pin down a flexible lath to points through which the curve is to be drawn and draw the pen along the lath. It is found impossible in practice to use similar laths for all curves. The lath is weakest where the curvature is greatest. The selection and use of the lath is a matter of taste and opinion. The use of French curves is still more arbitrary. Pickering used a bent spring or steel lath held near its ends. Such a lath is shown in statical works to give a line of constant curvature. The line is called an elastic curve (see G. M. Minchin s A Treatise on Statics, Oxford, 2, 204, 1886). 

This study is another rather successful attempt to correlate the fluid properties in the critical region. We have chosen an isochoric equation of state with constant curvature to represent these properties and we... 

Omitting the remainder 0(Bq/82), being a correction for the non-constant curvature of the energy level, we get the result... 

One of the most significant results of those non-Euclidean geometries, which consider space to be continuous and isotropic, is that it must then also be homogeneous and of constant curvature. 

In order to define the energy-momentrun tensor, matter is assruned to be uniformly distributed, as in an ideal gas, with stellar velocities orders of magnitude less than the velocity of light. However, a stable Boltzmann distribution of this type cannot persist in an open system from which entire stars may escape. Einstein therefore proposed a spatially closed continuum of constant curvature. The cosmological constant served to define both the mean density of the equilibrium distribution and a radius of the closed... 

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