New polygon

One application of the rules (a step) leads to the construction of a refined polygon. When we are talking about later steps the input to that step is called the old polygon and the output from it the new polygon. 

These new vertices are joined together, in the sequence matching the sequence of the edges and vertices of the old polygon, to form the new polygon. 

This first example has approximately twice as many vertices in the new polygon as in the old. We call it a binary scheme. If there had been three times as many it would have been a ternary scheme, and such generalisations will be discussed in a few pages time. In principle at each refinement we can multiply the number of vertices by whatever we choose, and this number is called the arity and denoted by the letter a. It is also called the dilation factor, which stems from generating function usage. 

The dyadic fractions are those which have a finite representation as binary numbers. Rationals which are not dyadic have a binary fraction representation which after a while repeats some pattern to infinity. Irrationals do not have any such pattern in their infinite binary representation nThis is not quite true, because some schemes have special rules at the ends which allow for definition of new vertices which don t have old vertices on both sides. Others do not, and then each new polygon covers a slightly shorter parametric range than the previous one. This distinction can be ignored until we come to the chapter on end-conditions at page 175. 

All the previous examples have the property that the corners are smoothed off the original polygon. It is also possible for a scheme to have the property that the new polygon has new v-vertices which lie exactly at the corresponding old polygon vertices. If this is true after one step it will also be true after two, or three, or more, and indeed it is true, by induction, of the limit curve. 

Stencils The first obvious representation is the set of weight combinations for each of the types of vertex in the new polygon. 

We consider first ternary schemes which are both primal, in the sense that there is a vertex in the new polygon corresponding to each vertex of the old, and dual, in that there is an edge of the new polygon matching each each of the old. 

The question is whether we can bound the values of f(x + Sx) — f(x) in terms of the original control points, and the answer is yes , using the neat idea of a difference scheme, which relates the first differences of the new polygon to the first differences of the old. 

All of this can be applied easily, at least in the two-dimensional case, to give a complete parallel theory in which a polygon is a sequence of edges (lines) rather than a sequence of points. A new polygon can be created from an old one by taking linear combinations of the lines to make new lines, and the vertices of the new ones just pop out as the places where consecutive new lines intersect. 

When this is done at both ends, the matrix becomes a finite one which can be applied to a finite old polygon. The result is a new polygon with... 

For the cubic B-spline this can be achieved in an ad-hoc way by just retaining the first control point of the old polygon in the new polygon. This is very easily achieved and implemented, but it has the unfortunate effect that the curvature is always zero at the end of the limit curve. 

Carbon tubules (or nanotubes) are a new form of elemental carbon recently isolated from the soot obtained during the arc-discharge synthesis of fuller-enes[I]. High-resolution electron micrographs do not favor a scroll-like heUcal structure, but rather concentric tubular shells of 2 to 50 layers, with tips closed by curved, cone-shaped, or even polygonal caps. Later work[2] has shown the possibility of obtaining singleshell seamless nanotubes. 

New Zealand Flax. This is obtained especially from the leaves of Phormium tenetx. The fibres are united in bundles—which are readily disaggregated—and are very thin, uniform and smooth, with a peculiar appearance of rigidity the lumen is very distinct and occupies about one-third of the fibre (Fig. 80, Plate VIII). The ends are acute. The cross-sections of unbleached fibres are united in bundles which are polygonal with rounded angles they are only weakly joined and the lumen is rounded and free from contents. By iodine and sulphuric add, the raw fibres are coloured yellow, while bleached fibres assume a greenish or blue colour and then show marked flexibility. 

FIGURE 5.34 Generalized pH-kobs polygon. [Graphreconstructed from J. T. Cartensen, Drug Stability Principles and Applications, 2nd Ed., Marcel Dekker, New York, 1995. p 91.]... 

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