On the Local Linear Independence of Generalized Subdivision Functions

Abstract

Characterizing the linear and local linear independence of the functions that span a linear space is a key task if the space is to be used computationally. Given a control net, the spanning functions of one spatial coordinate of a generalized subdivision surface are called nodal functions. They are the limit, under subdivision, of associating the value one with one control net node and zero with all others. No characterization of independence of nodal functions has been published to date, even for the two most popular generalized subdivision algorithms, Catmull–Clark subdivision and Loop’s subdivision. This paper provides a road map for the verification of linear and local linear independence of generalized subdivision functions. It proves the conjectured global independence of the nodal functions of both algorithms, disproves local linear independence (for higher valences), and establishes linear independence on every surface region corresponding to a facet of the control net. Subtle exceptions, even to global independence, underscore the need for a detailed analysis to provide a sound basis for a number of recently developed computational approaches.