  The center of gravity as an algorithm for position measurements is analyzed
for a two-dimensional geometry. Several mathematical consequences of
discretization for various types of detector arrays are extracted. Arrays with
rectangular, hexagonal, and triangular detectors are analytically studied, and
tools are given to simulate their discretization properties. Special signal
distributions free of discretized error are isolated. It is proved that some
crosstalk spreads are able to eliminate the center of gravity discretization
error for any signal distribution (ideal detectors). Simulations, adapted to
the CMS em-calorimeter and to a triangular detector array, are provided for
energy and position reconstruction algorithms with a finite number of
detectors.
