Complex Eigenvalues for Binary Subdivision Schemes

TL;DR

This paper investigates the conditions under which binary subdivision schemes for curves can have complex eigenvalues, providing theoretical proofs, a specific example, and insights into their geometric implications in CAD.

Contribution

It proves when convergent schemes with palindromic masks can have complex eigenvalues and identifies the minimal mask size for such schemes.

Findings

A scheme with complex eigenvalues achieving the minimal mask size.

Complex eigenvalues are theoretically possible in certain convergent schemes.

Numerical and geometric analysis explains why such schemes are not yet used in CAD.

Abstract

Convergence properties of binary stationary subdivision schemes for curves have been analyzed using the techniques of z-transforms and eigenanalysis. Eigenanalysis provides a way to determine derivative continuity at specific points based on the eigenvalues of a finite matrix. None of the well-known subdivision schemes for curves have complex eigenvalues. We prove when a convergent scheme with palindromic mask can have complex eigenvalues and that a lower limit for the size of the mask exists in this case. We find a scheme with complex eigenvalues achieving this lower bound. Furthermore we investigate this scheme numerically and explain from a geometric viewpoint why such a scheme has not yet been used in computer-aided geometric design.