Local Corner Cutting and the Smoothness of the Limiting Curve
Abstract
It was proved in an earlier work that corner cutting of any kind converges to a Lipschitz-continuous curve, but the question of how one might guarantee that the limiting curve be smoother than that was not considered there. Recently, Gregory and Qu GQ took up this question and established sufficient conditions for a certain systematic and local corner cutting scheme to give a limiting curve in C. Since GQ use the same parametrization of the successive broken lines that made the argument in B so simple, I became intrigued and took a look at what one might say in greater generality. Specifically, I looked for conditions under which continuous differentiability of the limiting curve could be inferred from the fact that the corners of the broken lines flatten out eventually. It is the purpose of this note to prove that the limit of any 'local' corner cutting scheme is in C provided the corners of the broken lines become increasingly flatter. A simple example is given to show that this condition is not necessary, while another simple example shows that, without 'localness', the condition is not sufficient, in general. Finally, as an application, Gregory and Qu's nice argument in GQ is redone.
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 01, 1988
- Accession Number
- ADA204092
Entities
People
- Carl R. de Boor
Organizations
- University of Wisconsin–Madison