Deriving the accumulation density function with parabolic trajectories of particles for arbitrary lines

Ideal particle accumulation on an arbitrary line, with a positive or a negative y-intercept, is studied in this paper. Particles are assumed to be emitted from a fixed-point location at the same emission speed onto the surface of the ground, a wall, or a ceiling depending on their relative positions. Since particles acted on by gravity travel in space via parabolic trajectories, by grouping parabolas, the accumulation of particles on such surfaces can be deduced. An analytical equation, namely the accumulation density function, is subsequently established to represent particle accumulation on each such surface. This accumulation function provides a basis for simulating accumulation of materials on surfaces in diverse application areas, including agricultural spraying irrigation, seeding, chemical vapor deposition, and spray painting.