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A plane-parallel atmosphere is a model atmosphere that ignores the curvature of the underlying body, i.e., the atmosphere is modeled as if it is planar/flat. This can be a very effective simplification in meteorology (Earth or other bodies) and in stellar models. The simplification is especially suitable for models of (spherical) bodies where the atmosphere is a relatively thin layer around the body, i.e., "locally virtually flat". Its use is typical in 1D models. The simplification can make a significant difference in a model's practical tractability, in both for analytic models and computer simulations.
Perhaps the primary simplification is: when dealing with lines at an angle, i.e., neither vertical nor horizontal to the surface, the lines are treated as if the atmosphere is flat. For example, when calculating the effects of a beam of light passing through at such an angle, the light is treated as if it is passing through flat, parallel layers, any effects of the atmosphere-layer's curvature ignored.