Subdivision schemes in computer-aided geometric design. (English) Zbl 0760.65012

Advances in numerical analysis. Vol. 2: Wavelets, subdivision algorithms, and radial basis functions, Proc. 4th Summer Sch., Lancaster/UK 1990, 36-104 (1992).
[For the entire collection see Zbl 0744.00036.]
This paper is more than a review, it is an excellent text of a unified theory of convergent subdivision schemes for approximation of curves and surfaces (and, in fact, hypersurfaces in arbitrary dimensions). The unifying tool is the study of an analytic function \(\Sigma a_ \alpha z^ \alpha\) associated to a subdivision scheme \(p^{k+1}_ \alpha=\sum a_{\alpha-2\beta}p^ k_ \beta\). These functions are Laurent series in one or several variables and allow the use of analytic techniques that yield convergence conditions and theorems. The proofs are careful and complete. The text has extensive bibliographic references for attributions of results but a reader will find no need to go to the sources.


65D17 Computer-aided design (modeling of curves and surfaces)
65-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to numerical analysis


Zbl 0744.00036