LINE-SEGMENT ON SURFACE

LINE SEGMENT ON SURFACE ( name name, , )... 

Of course, for a general polyhedron not all 2-faces must be 2D simplexes (i.e., triangles). For example, all 2-faces of a cube are squares. Furthermore, for a more general object of curved surface, the 2-faces (ordinary faces), 1-faces (edges) and 0-faces (vertices) of a polyhedron are replaced by the 2-cells (surface patches), 1-cells (line segments on the boundaries of the patches), and 0-cells (joining points of the line segments) of the object, respectively. 

Figure C3.6.2 (a) The (fi2,cf) Poincare surface of a section of the phase flow, taken at ej = 8.5 with cq < 0, for the WR chaotic attractor at k = 0.072. (b) The next-amplitude map constmcted from pairs of intersection coordinates. ..,(c2(n-l-l),C2(n-l-2),C2(n-l-l)),...j. The sequence of horizontal and vertical line segments, each touching the diagonal B and the map, comprise a discrete trajectory. The direction on the first four segments is indicated.

The contour plotting program then searches each cell on the x. grid defined by the data to determine, by interpolation, the location of points on the cell edges that should be part of the contour lines for each intensity level. A suitable coordinate transformation then maps the points located in this manner onto the plotting surface where they are Joined by straight line segments. 

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... 

Widom [9] realized the importance of this problem for statistical mechanics and showed that the centers of the particles of a hard disk gas, in an equilibrium position, are not uniformly random distributed. The available area for a nevt particle power series in particle density 6 = Nnr2/A, where N is the number of adsorbed panicles, r their radius and A the total area of the surface. The coefficients of the series terms are identical up to the second power of 9 for the equilibrium and the RSA models. The differences in the higher powers coefficients lead for RSA to jamming for Op = 0.76, 0.547 and 0.38 for the ID (segments on a line), 2D (disks on a surface) and 3D (spheres on a volume), respectively, while for the equilibrium configurations the close-packing occurs at 9 = 1, 0.91 and 0.74, respectively. 

For three dimensions, the AR construction algorithm is similar to the one described above—with the added possibility that we can find a (one-dimensional) connector on the AR that is described by a DSR. Glasser et al. (1992) defined conditions under which DSRs appear on the AR along with a direct method for finding the feed addition rate, q. While the conditions for DSRs appear to occur infrequently, examples have been constructed in the space of conversion, temperature, and residence time where the DSR was a prominent part of the AR. Nevertheless, Hildebrandt and co-workers conclude that most ARs will consist only of CSTR and PFR surfaces. In dealing with n-dimensional problems, Hildebrandt and Feinberg noted that the AR boundary is defined by line segments and PFR trajectories, with at most n structures needed to define a point on the AR boundary and n + 1 structures needed to define an interior point of the AR. Thus, for three-dimensional problems, at most three parallel structures (PFRs, CSTRs, DSRs) are needed to define any AR boundary point. 

The conclusion to be drawn from this (simplistic) illustration is, of course, that the only useful measure of a set of points defining an ordinary surface in is the area. Interestingly, the measure of this same set of points with a yardstick that has a lower dimension than the area (i.e. a line segment) yields an infinite (divergent) measure, while a yardstick with a higher dimension than the area (i.e. a volume) leads to a zero measure. Therefore, the practically useful yardstick corresponds to a transition from divergent measures to zero measures. This same feature, as we shall see below, characterizes the Hausdorff dimension. The definition of the Hausdorff dimension is based on the concept of the Hausdorff measure, which itself makes use of the notion of the (5-cover of a set. 

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