  A piecewise constant curvature manifold is a triangulated manifold that is
assigned a geometry by specifying lengths of edges and stipulating that for a
chosen background geometry (Euclidean, hyperbolic, or spherical), each simplex
has an isometric embedding into the background geometry with the chosen edge
lengths. Additional structure is defined either by giving a geometric structure
to the Poincare dual of the triangulation or by assigning a discrete metric, a
way of assigning length to oriented edges. This notion leads to a notion of
discrete conformal structure, generalizing the discrete conformal structures
based on circle packings and their generalizations studied by Thurston and
others. We define and analyze conformal variations of piecewise constant
curvature 2-manifolds, giving particular attention to the variation of angles.
We give formulas for the derivatives of angles in each background geometry,
which yield formulas for the derivatives of curvatures. Our formulas allow us
to identify particular curvature functionals associated with conformal
variations. Finally, we provide a complete classification of discrete conformal
structures in each of the background geometries.
