Barycentric Subdivision - Repeated Barycentric Subdivision

Repeated Barycentric Subdivision

When approximating a mathematical function or a surface by a spline, the accuracy of the approximation is usually determined by the piece size — the bigger the pieces, the larger the error. Thus it is often necessary to split large pieces into smaller ones, in order to achieve a prescribed accuracy.

In theory, BCS could be used for that purpose, since it has the property that the longest edge of any piece is smaller than the longest edge of the original polytope by a factor less than . Therefore, by applying BCS sufficiently many times, the largest edge can be made as small as desired.

However, in practice BCS is not well-suited for that purpose. For one thing, each application after the first one multiplies the number of simplices by . BCS also multiplies the degree of each original vertex by, and the degree of each edge by . Moreover, the BCS will split all simplices, even those that are already small enough. Finally, each BCS stage also makes the simplices not only smaller but "skinnier", i.e. it tends to increase their aspect ratio (the ratio between the longest and shortest edge). For all these reasons, in practice one rarely applies more than one round of BCS, and other subdivision schemes are used instead.