Control polygon

Because each vertex of the refined control polygon is a weighted mean of vertices of the original, the construction of a refined control polygon can be expressed in the form... 

Clearly the eigenvalue 1 is dominant. Its column eigenvector in the original matrix is a column of all Is19, which means that all the points in this piece of the control polygon will be at the same place in the limit. 

We can also ask the support analysis how many points influence one span of the limit curve, the piece corresponding to one edge of the control polygon. This turns out to be one fewer. Call it m. The value will be 4 for the cubic B-spline. 

A slightly surprising result is that the limit curve of polygon P(z) under the binary scheme with mask 2ak(l + z2)/2z is identical to the limit curve of polygon aP(z) under the scheme whose mask is 2ak+i. It is a B-spline curve, but with a different control polygon. 

Finally the issues are considered of what end conditions to support, and whether to offer the application of preliminary modifications to the control polygon to make the overall system more ergonomic for the curve designer. 

The number of refinements is first worked out from the required precision and the initial control polygon. These refinements are then carried out, but only in the smallest possible region around the place where the evaluation is to be made. Doing it everywhere requires excessive computation and storage space. The number of control points needed is only the number required for the evaluation of the points and derivatives at the end of the required span. 

The idea of adjusting the original control polygon before starting any refinement was introduced in the previous chapter in the context of end conditions, but it can be used more widely. In particular we can often use an approximating subdivision scheme to interpolate a set of given points. 

The unit row eigenvector is a stencil which gives a point on the limit curve in terms of the original control points. The product of a circulant matrix, E, all of whose rows are equal to that eigenvector, with the control polygon, P, gives a sequence of points, Q, on the limit curve. 

This implies that we could determine a control polygon whose limit curve would interpolate all the points of Q. All we have to do is invert E and multiply Q by it. This is not in fact practical for two reasons. The first is that we do have to worry about end conditions to make E finite. The second is that although E is a narrow-banded matrix, its inverse is typically completely full. It is therefore much cheaper to solve the system EP = Q for P than either to invert E or to multiply Q by it. 

E.Cohen and L.L.Schumaker Rates of Convergence of control polygons. CAGD 2(1-3), pp229-235, 1985... 

Approximation is only a method of harmonic shape control by the position of control vertices of a control polygon. Decreasing the distances between the curve and the control points is not a goal. Approximation replaced the troublesome curve definition by point-tangent pairs at that time. 

A similar curve description was achieved by Philip De Casteljeau at the French firm Citroen. Nevertheless, the method of approximation of the control polygon was linked to the name of Bezier in the literature even if the description of the curves uses functions other than Bernstein polynomials. 

In the 1950s, to develop a mathematical representation for the autobody surface, Pierre Bezier, at Renault in France, first published his work on spline that is represented with control points on the curve, which is now commonly referred to as the Bezier spline. Figure 2.9a illustrates a Bezier curve in solid line with four control points, 1,2,3, and 4, and its control polygon in dashed line, and Figure 2.9b illustrates two B-spline curves, each with multiple Bezier arcs, in solid, dash, or dot line, with a unified mechanism defining continuity at the joints. 

Fig. 2 Study area, digital elevation model and subcatchments used for the hydrological simulations, along with the location of the gauging stations of precipitation, air temperature and stream-flows. Yellow polygons indicate the location of the four subcatchments selected for carrying out the hydrological simulations during the control and the future scenarios described in Sect. 6...

In a double-blind, placebo-controlled study in eight healthy men, zolpidem 10 mg produced statistically significant postural sway in the tandem stance test, and triazolam 0.25 mg was statistically significant only as defined by the polygonal area of foot pressure center (23). Zolpidem, which has a minimal muscle-relaxant effect, produced more imbalance than triazolam, which is known for its muscle relaxant effect. The authors suggested that in the use of hypnotics, sway derives from suppression of the central nervous system relevant to awakening rather than from muscle relaxation. 

The bile canaliculus is formed as a bile capillary by means of a groove-like canal in the intercellular space, bounded by 2 adjacent liver cells. The bile canaliculi have no walls of their own, but are surrounded by a special zone of the cell membrane (so-calledpericanalicular ectoplasm). Their diameter amounts to 0.5-1.0 pm. They are interconnected and form an extensive polygonal network. The surface area of the bile capillaries is increased by microvilli, which show great functionally determined variability. The canalicular membrane constitutes 10% of the total plasma membrane in the hepatocytes. Similar to the pericanalicular ectoplasm, the hepatocytes contain contractile microfilaments and other components of the cytoskel-eton. These canaliculi are supplied with carrier proteins and enzymes to control bile secretion. (2,34)... 

Because the basis functions sum to unity, the coefficients transform as points, and are called control points and they are typically visualised by drawing the polygon which joins them in sequence. 

Artifacts What features can be seen in the limit curve which cannot be controlled by choice of the initial input polygon ... 

The basis function is the limit function resulting from cardinal data, where all vertices of the polygon have value zero except for one. Clearly there is one such basis function for each control point in the polygon, but in uniform schemes, where the weights in the weighted means do not depend on position in abscissa space, all of these basis functions have the same shape. They are just translates of each other, and so there is only one shape, which we call the basis function. 

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. 

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. 

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. 

---