Continuity at Mark Points by Eigenanalysis

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 

S is called the Subdivision Matrix. It has an interesting structure. Every column is a copy of the mask, but successive columns have their copies shifted down by the arity compared with their left neighbours. For example, the subdivision matrix for the Cubic B-spline scheme is 

Note that because we are ignoring what happens at the ends of the polygon, this has to be treated as an infinite matrix, and we have to be very careful to justify steps which are only known to apply to finite matrices. 

The interesting property of an infinite matrix is that, lacking a top left hand corner, it doesn t have a principal diagonal. Any diagonal can be taken as principal. Because we have chosen a binary scheme for the example, all diagonals look the same anyway, so it makes no difference for our analysis which one we choose. 

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. 

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