Original polygon

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. 

Consider first the case of integer support width. If the support width is even, the end-point will have an integer label and correspond to an original control point. If the support width is odd, the end-point will have a halfinteger label and correspond to a midedge of the original polygon. 

That place is given by multiplying the original polygon by the normalised eigenrow, which gives (Z + 4A + B)/6. 

However, the symmetric component gives something more awkward. The limit has different third divided differences on left and right, which means that there is a discontinuity of third derivative. The size of this discontinuity is given by the fourth divided difference of the original polygon. 

The sequence of 6x used for taking this limit is conveniently the sequence of polygon edges at successive refinements of the original polygon. 

Note that this doubling of the density is not the refinement of the subdivision step, but a doubling of the density of the original polygon. A doubling of the amount of work done in collecting data. 

If the original polygon is sampled from a sinusoid of spatial frequency ui = 1/m where m is the number of polygon points per complete cycle, so that P0[j] = cos(2njcj), j G Z, the result of multiplying by the sampling matrix can be expressed as... 

Half the difference of the two eigenrows is a binary mask for the artifact [1, —4, 6, —4,1]/16, and half the sum a binary mask for the signal [1,4, 6,4,1]/16. Multiplying the original polygon by these therefore gives the required measures. 

It is not too surprising that the ternary and binary quadratic B-splines have the same amounts of signal and artifact. They are the same curve, depending in the same way on the original polygon. 

We therefore have an important property, that of step-independence, to consider. A scheme is step-independent if the original polygon and the n-times refined polygon have the same limit curve. I.e., if for all values of n they have the same limit curve without the implementation knowing the value of n. 

There is a lot of subtlety here. Although it is still possible to define a basis function as being the effect of moving one point by an infinitesimal amount, this applies only to one specific original polygon. As soon as you start editing it, the basis functions change. 

A non-stationary scheme does not have the necessary eigenvectors to apply the above directly to the original polygon. However, in cases where the scheme converges adequately fast towards its own limit, the eigenvectors of the limit scheme can be used with good accuracy after a relatively small number of refinements. How many such refinements are needed has to be determined for each scheme individually. 

At each subsequent refinement step, a further shortening takes place, and the limit curve is significantly shorter in parameter space than the original polygon. 

It is far from obvious to the curve designer, who may well want to constrain the position of the end of the curve, how the polygon should be constructed to achieve a particular end-point for the limit curve. It would be much nicer if we could in some way arrange for the limit curve to reach exactly the end of the original polygon. 

A second is to modify the original polygon once and for all before starting any subdivision. This usually involves adding extra control points, (whose positions depend on the original ones) but may also involve moving some control points. The new control points can loosely be called fake points. 

Once the curves network is created, the model is ready to generate NURBS surfaces (fig. 8). This can be done automatically or manual. Automatic surface generation doesn t need to draw a curve, while manual surface generation can completely maintain the flow line of the original polygon surface. Manual generation of surfaces is related to the network of curves. 

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