  A realizer, commonly known as Schnyder woods, of a triangulation is a
partition of its interior edges into three oriented rooted trees. A flip in a
realizer is a local operation that transforms one realizer into another. Two
types of flips in a realizer have been introduced: colored flips and cycle
flips. A corresponding flip graph is defined for each of these two types of
flips. The vertex sets are the realizers, and two realizers are adjacent if
they can be transformed into each other by one flip. In this paper we study the
relation between these two types of flips and their corresponding flip graphs.
We show that a cycle flip can be obtained from linearly many colored flips. We
also prove an upper bound of $O(n^2)$ on the diameter of the flip graph of
realizers defined by colored flips. In addition, a data structure is given to
dynamically maintain a realizer over a sequence of colored flips which supports
queries, including getting a node's barycentric coordinates, in $O(\log n)$
time per flip or query.

Computational Geometry