  We study C^2 weakly order preserving circle maps with a flat interval. In
particular we are interested in the geometry of the mapping near to the
singularities at the boundary of the flat interval. Without any assumption on
the rotation number we show that the geometry is degenerate when the degree of
the singularities is less than or equal to two and becomes bounded when the
degree goes to three. As an example of application, the result is applied to
study Cherry flows.
