  A hyperbolic polygon is defined to be cyclic, horocyclic, or equidistant if
its vertices lie on a metric circle, horocycle, or a component of the
equidistant locus to a hyperbolic geodesic, respectively. Convex such $n$-gons
are parametrized by the subspaces of $(0,\infty)^n$ that contain their side
length collections, and area and circumcircle or "collar" radius determine
symmetric, smooth functions on these spaces. We give formulas for and bounds on
the derivatives of these functions, and make some observations on their
behavior. Notably, the monotonicity properties of area and circumcircle radius
exhibit qualitative differences on the collection of centered vs non-centered
cyclic polygons, where a cyclic polygon is "centered" if it contains the center
of its circumcircle in its interior.
