  We present a detailed study of the viewing angles of the LMC disk plane. We
find that our viewing direction differs considerably from the commonly accepted
values, which has important implications for the structure of the LMC. The
discussion is based on an analysis of spatial variations in the apparent
magnitude of features in the near-IR color-magnitude diagrams extracted from
the DENIS and 2MASS surveys. Sinusoidal brightness variations with a
peak-to-peak amplitude of approximately 0.25 mag are detected as function of
position angle, for both AGB and RGB stars. This is naturally interpreted as
the result of distance variations, due to one side of the LMC plane being
closer to us than the opposite side. The best fitting geometric model of an
inclined plane yields an inclination angle i = 34.7 +/- 6.2 degrees and
line-of-nodes position angle Theta = 122.5 +/- 8.3 degrees. There is tentative
evidence that the LMC disk plane may be warped. Traditional methods to estimate
the position angle of the line of nodes have used either the major axis
position angle Theta_maj of the spatial distribution of tracers on the sky, or
the position angle Theta_max of the line of maximum gradient in the velocity
field, given that for a circular disk Theta_maj = Theta_max = Theta. The
present study does not rely on the assumption of circular symmetry, and is
considerably more accurate than previous studies of its kind. We find that the
actual position angle of the line of nodes differs considerably from both
Theta_maj and Theta_max, for which measurements have fallen in the range
140-190 degrees. This indicates that the intrinsic shape of the LMC disk is not
circular, but elliptical, as discussed further in Paper II. [Abridged]
