Row eigenvectors

Once we have computed the matrix of column-eigenvectors V we can derive the corresponding row-eigenvectors U using eq. (31.13) ... 

When the scheme is not primal, the scheme maps each vertex into an edge. However, the unit row eigenvector as a stencil maps edges into vertices, and so the product of the mask with the unit row eigenvector maps vertices into vertices. 

These have an odd number of a factors in the mask. However, the unit row eigenvector also has an odd number of a factors, and so the product has an even number, and so we can determine the amount of artifact in the limit curve in exactly the same way. 

For ternary schemes which are both primal and dual there is a short cut. There are two mark points, and we can determine the unit row eigenvector for each of them, thus giving the values at the limit points corresponding to both the vertices and the mid-edges. 

The polygons are shown as lines the dots are at points of the limit curve, using a unit row eigenvector obtained by convolving that of the normalised scheme [l,4,l]/6 with the prefix [l,2,l]/4. 

Again, the polygons are shown as lines and the limit points, using an appropriate unit row eigenvector, as dots. We see that the limit points do what we expect, but the polygon shows no sign of converging towards them. 

Row eigenvectors giving points and derivatives on the limit curve. 

The interpolation degree can also be expressed in terms of the presentation of the unit row eigenvector as a polynomial in S2. Since the unit row eigenvector is often shorter than the mask, this might be slightly advantageous. 

Because the unit row eigenvector is not a linear function of the mask, the quasi-interpolation degree is not necessarily a linear constraint in design space. 

If what is required is a polygon with points actually on the limit curve, then these can be constructed by multiplying the polygon constructed in the previous paragraphs by the unit row eigenvector. 

This is to evaluate the limit curve points corresponding to the control points, using the row eigenvector of unit eigenvalue, and also the first derivatives, using the next row eigenvector. These are then used to make a Hermite cubic interpolant, which converges as the fourth power of the number of refinements. 

The corresponding row eigenvectors are the same length as the rows, and so the derivatives depend only on the first few old control points at the start of the polygon. 

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. 

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