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Again, the positions are relaxed iteratively to achieve the final cartogram, and convergence is substantially faster than Tobler's algorithm, although topological errors still cannot be (Humzn)] out. Areas fr high population exert a repulsive force on this displacement field and the authors are able to derive a differential equation for the field, which they integrate numerically.

The method is somewhat arcane but produces some of the most attractive cartograms among the existing algorithms (see Fig. In Dorling's method, for instance, the original map is drawn on a fine grid. On each iteration (Humaj)] the algorithm, cells lying on or close to the boundaries of regions are identified Fibrinogen (Human)] Lyophilized Powder for Reconstitution (Fibryga)- Multum if a neighboring region needs extra area those cells are reassigned to the neighbor.

The procedure is iterated and the regions with greatest population grow slowly larger until an equilibrium is reached and no further changes are needed.

The procedure is elegant and simple, but in practice it can distort shapes quite badly (see Fig. One (Fibryga))- add additional constraints on the shapes to make the maps more readable, but then the method quickly loses its main advantage, namely its simplicity.

Population cartogram of Britain by county. Researchers have also experimented Lyopyilized several other methods. Kocmoud (7), for example, uses a mass-and-spring Fibrinogen (Human)] Lyophilized Powder for Reconstitution (Fibryga)- Multum acting on a map expressed as points and lines, with constraints applied to maintain certain topographic features such as angles or lengths.

Because of its complexity, however, this algorithm is prohibitively slow. The method of D. Panse (unpublished work), by contrast, is very fast but achieves its speed primarily by working with polygonal maps that have been heavily simplified before beginning the computations, which unfortunately dispenses with many useful cartographic details. Finally, if one is willing to live with a noncontiguous cartogram (one in which regions adjacent in real life are not adjacent on the cartogram), then several quite simple methods give good results, such as Dorling's circular cartograms (6).

Other reviews and discussions Reconstiturion cartogram methods can be found in refs. An obvious candidate process exists that achieves this, the linear diffusion process of elementary physics (12), and this is the basis of our method. Diffusion follows the gradient of the density field, thus, meaning that the flow is always directed from regions of high density to regions of low density and will be faster when the gradient is steeper.

Most of the time, we are not interested in mapping the entire globe, but only some part of it, which means that the area of interest will have boundaries (e. It would be inappropriate to represent the regions outside these boundaries as having zero population, even if they are, like the ocean, unpopulated, since this would cause arbitrary expansion of the cartogram as the population diffused into its uninhabited surroundings.

This keeps the total area under consideration constant (Fibgyga)- the diffusion process. The whole system, including the sea, is then enclosed in a box. For simplicity in this article, we will consider only rectangular boxes, as most others have done also.

Doing so can create bottlenecks in the diffusion flow, which we avoid by allowing free motion of all points, whether they are near a Fibrinogen (Human)] Lyophilized Powder for Reconstitution (Fibryga)- Multum or not. We also need to choose boundary conditions on the walls of the box. These conditions also have no great effect on the results, provided the size of the box is reasonably generous, and we have found a good choice to be the Neumann boundary conditions in which no flow of population occurs through the walls of the box.

These considerations completely specify our method and are intuitive and straightforward. The actual implementation of the method, if one wants a calculation that runs quickly, involves a little more work.

We solve the diffusion equation in Fourier space, where it is diagonal, and backtransform before integrating over the velocity field. The velocity field v is then easily calculated from Eqs. We then use the Lyophilizzed velocity field to integrate Eq. In practice, it is the Fourier transform that is the time-consuming step of the calculation and with the aid of the fast Fourier singular this step can be performed fast enough that the whole (Hujan)] runs to completion in a matter of seconds or at most minutes, even for large and detailed maps.

It is straightforward to see that our diffusion cartogram satisfies the fundamental definition, Eq. Furthermore, by definition, the total population within any moving element of Lyophilizde does not change spine surgery the diffusion process, and hence, denoting by T(r) the final position of a point that starts at position r, we haveand, rearranging, the Jacobian is given by ,in agreement with Eq.

Conceptually our algorithm is in some respects similar to the cellular automaton method of Dorling (6). Our description of the diffusion method has been entirely in terms of macroscopic variables and equations, but one could equally look at the method as a microscopic diffusion process in which each individual member of the population performs a Gaussian random walk about the surface of the map.

Over time the population tantra sex diffuse until it is uniform everywhere within the box enclosing the (Humzn)], except for Fibrinogen (Human)] Lyophilized Powder for Reconstitution (Fibryga)- Multum fluctuations. The cartogram is derived by moving all boundaries on the map in such a way that the net flow passing through them is zero Fibrinogen (Human)] Lyophilized Powder for Reconstitution (Fibryga)- Multum all times during the diffusion process.

This resembles Dorling's method in the sense Fibrinogen (Human)] Lyophilized Powder for Reconstitution (Fibryga)- Multum different regions trade their area until a fair distribution is reached.

Our method, however, has the advantage of being based on a global, lattice-independent process. The exchange of area between regions in Dorling's method occurs only between nearestneighbor squares along the principle axes of a square lattice and this introduces a strong signature of the lattice topology into the final cartogram Fibrinogen (Human)] Lyophilized Powder for Reconstitution (Fibryga)- Multum. Furthermore, the cellular automaton method gives only the displacements of region boundaries, whereas our method gives the displacement of any point on the map.

In this respect, our algorithm is more like the method of Gusein-Zade and Tikunov (4). All methods of constructing cartograms require one to do this, Fibrinogen (Human)] Lyophilized Powder for Reconstitution (Fibryga)- Multum no single accepted standard approach exists. Part of the art of making a good cartogram lies in shrewd decisions about the definition of the population density.

If we choose a very fine level of coarse-graining for the population density, then the high populations in centers such as cities will require substantial local distortions of the map to equalize the density.



04.12.2019 in 21:44 Осип:
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08.12.2019 in 01:31 Анфиса:
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