  The standard formula for the change in the effective action under a conformal
transformation is extended to the case when the boundary is piecewise smooth.
We then find the functional determinants of the scalar Laplacian on regions of
the plane obtained by stereographic projection of the fundamental domains on an
orbifolded two-sphere. Examples treated are the sector of a disk and a circular
crescent. The effective action on a spherical cap is also determined.
