Primal binary schemes

Suppose that the scheme is a primal binary one. The mask has an odd number of entries, to. Thus the effect of the original control point, Pj, being moved reaches as far after one iteration as (to — l)/2 new points from Pj, or (m — l)/4 old points. At the second step the mask being applied has the same numbers but is at a narrower abscissa scale, and so the second step extends the influence by one half of this, = (to — 1 )/8, and all subsequent steps give a total series of ((to— l)/4)(l + l/2 + 1/4. ..) which is easily summed to give (to — l)/2. 

By looking at the extent of influence of one control point after 0,l,2,oo refinements, in the cubic B-spline scheme we can see that the refined polygons converge towards the basis function, and the last non-zero entry converges towards the end of the support region. 

Thus moving a single control point influences at most a piece of the limit curve which stretches (m — l)/2 old spans on each side, m was odd and so this is an integer. 

Beyond this point the control point has no influence. We should not therefore be surprised to find a discontinuity of some derivative at such places. Note that the sum of the series converges from below, and the amount of influence actually felt at the limit of the series has diminished to zero by the time that the edge of the support is reached. The original control point can influence some derivative on the near side of the final point, but not the point itself or anything beyond it. 

We have made a big assumption so far, that the subdivision process converges to a limit curve. We shall discover later that it is possible to have a scheme which is not convergent. However, these take deliberate perversity to construct, and the support analysis applies even to such monsters. 

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What the choice of a diagonal does is to imply a labelling, giving a correspondence between a sequence of points of the old polygon and a sequence of the refined one. In particular it implies a mark point which is an abscissa value which maps into itself under the map from old abscissa values to new ones. In the case of a primal binary scheme, the mark point is at a point of both new and old polygons. In the case of a dual scheme the mark point is at a mid-edge in both old and new. 

This procedure can be applied to any primal binary scheme, although it may be necessary to imagine higher dimensions than 3 in order to keep applying the principle of suppressing successive dominant eigencomponents. 

We consider first binary schemes primal and dual need not be distinguished. If the symbol has d+ 1 factors of (1 + z)j2 then... 

The space of binary uniform stationary schemes is therefore understandable in terms of two components one contains all primal schemes, with an odd number of entries in the mask, which is a linear combination of odd degree B-splines the other all dual schemes, with an even number of entries in the mask, which is a linear combination of even degree B-splines. In principle each is only a countably infinite-dimensional space, and in practice each is only a finite dimensional space because we shall not wish to include B-splines above some maximum degree in order to keep the support limited. 

We consider binary primal schemes, for which the generic scheme can be represented as... 

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