Analysis and Design of Univariate Subdivision Schemes
‘Subdivision’ is a way of representing smooth shapes in a computer. A curve or surface (both of which contain an in?nite number of points) is described in terms of two objects. One object is a sequence of vertices, which we visualise as a polygon, for curves, or a network of vertices, which we visualise by drawing the edges or faces of the network, for surfaces. The other object is a set of rules for making denser sequences or networks. When applied repeatedly, the denser and denser sequences are claimed to converge to a limit, which is the curve or surface that we want to represent. This book focusses on curves, because the theory for that is complete enough that a book claiming that our understanding is complete is exactly what is needed to stimulate research proving that claim wrong. Also because there are already a number of good books on subdivision surfaces. The way in which the limit curve relates to the polygon, and a lot of interesting properties of the limit curve, depend on the set of rules, and this book is about how one can deduce those properties from the set of rules, and how one can then use that understanding to construct rules which give the properties that one wants.
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abscissa algorithm Analysis and Design applied artifact barycentric coordinates basis function Berlin Heidelberg 2010 binary scheme chapter circulant matrix coefficients column eigenvectors components control polygon converge convex hull corresponding cubic B-spline defined Design of Univariate determined diagonal divided difference scheme dual schemes eigenanalysis eigencolumns eigencomponents eigenvalue enclosure end-conditions exactly factor factorisation finite four-point scheme Geometry and Computing given gives higher arities Hölder continuity initial polygon integer interpolation degree invariant subspaces joint spectral radius kernel Laurent polynomial limit curve limit points linear combination lower bounds mark points Matrix Norms multiplied Non-Stationary Schemes non-zero norm nth root old polygon original control points original polygon pieces polynomial of degree primal properties quadratic B-spline Sabin sampling second derivative sequence span spline Springer-Verlag Berlin Heidelberg square stencils step subdivision curves subspaces support width symmetric ternary scheme unit row eigenvector Univariate Subdivision Schemes upper bound v-vertices vector weighted mean z-transform zero