Space filling curve

To calculate the space-filling value for a specific compound, one has to know the radii of the atoms and the lattice constant. Neither of these is needed for the construction of a space-filling curve of a crystal structure type it is sufficient to know the point positions of the atoms and the axial ratios. The curve is based on a hard sphere... 

This equation describes the middle section (0.225 < e < 4.44) of the space-filling curve for the sphalerite type structure plotted (with log scales) in Fig. 4.22. 

In the p vs. e diagram every structure type is generally characterized by its own individually shaped space-filling curve. The space-filling curves, however, of all binary structures belonging to one homeotect structure set coincide with one curve (see 3.9.2). 

Figure 4.22. Space-filling curves for compounds having the NaCl, CsCl, NaTl or ZnS type structures. For the different structures, the ratio

See the space filling curve characteristic for this structure type in Fig. 4.22. 

Space-filling curves of the (u,v) domain) Space-filling curves on the surface... 

Five-Axis Tool Path Generation, Fig. 12 Curvilinear space-filling curve (Makhanov 2008)... 

The harmonic oscillator is of course an integrable system. When the underlying system is a chaotic Hamiltonian system, the trajectories from any collection of initial conditions become disordered (even while preserving the energetic constraints defined by their initial states). The hope (in some cases, naive) is that typical trajectories computed using a symplectic method applied to such a chaotic system provide space filling curves on the = E perturbed energy surface. 

If, in Figure 2.4, one takes the part of I4 that corresponds to the interval [0,1/3] of the initiator, and scales this part up horizontally and vertically by three, one obtains h. In the limit n -> 00, however, this same scaling up of any segment of / would reproduce the von Koch curve itself. Furthermore, like the Peano-Hilbert space-filling curves, the von Koch curve is nowhere differentiable. All of these curves have a topological dimension equal to unity a simple stretching operation (or rectification ) transforms them into an infinite straight line. 

Sagan, H. (1994). Space-Filling Curves. Springer-Verlag, New York. 

Peano Guisepe (1858-1932) It. math., founder of symbolic logic, non-Euclidean geometry, known for construction of space-filling curve named after him (Peano s axioms)... 

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