Variations on the four-point subdivision scheme

A step of subdivision can be considered to be a sequence of simple, highly local stages. By manipulating the stages of a subdivision step we can create families of schemes, each designed to meet different requirements. We postulate that such modification can lead to improved behaviour.We demonstrate this using the four-point scheme as an example. We explain how it can be broken into stages and how these stages can be manipulated in various ways. Six variants that all improve on the quality of the limit curve are presented and analysed. We present schemes which perfectly preserve circles, schemes which improve the Hölder continuity, and schemes which relax the interpolating property to achieve higher smoothness.