  Simulating curvature due to gravity through warped surfaces is a common
visualization aid in Physics education. We reprise a recent experiment
exploring orbital trajectories on a precise 3D-printed surface to mimic
Newtonian gravity, and elevate this analogy past the status of a mere
visualization tool. We present a general analysis approach through which this
straightforward experiment can be used to create a reasonably advanced
computational orbital mechanics lab at the undergraduate level, creating a
convenient hands-on, computational pathway to various non-trivial nuances in
this discipline, such as the mean, eccentric, and true anomalies and their
computation, Laplace-Runge-Lenz vector conservation, characterization of
general orbits, and the extraction of orbital parameters. We show that while
the motion of a marble on such a surface does not truly represent a orbital
trajectory under Newtonian gravity in a strict theoretical sense, but through a
proposed projection procedure, the experimentally recorded trajectories closely
resemble the Kepler orbits and approximately respect the known conservation
laws for orbital motion. The latter fact is demonstrated through multiple
experimentally-recorded elliptical trajectories with wide-ranging
eccentricities and semi-major axes.
  In this first part of this two-part sequence, we lay down the formal basis of
this exposition, describing the experiment, its calibration, critical
assessment of the results, and the computational procedures for the
transformation of raw experimental data into a form useful for orbital
analysis.
