Computing the trajectories of particles in generic beamlines is an important ingredient of experimental particle physics, in particular regarding near-beam detectors. A new tool, Hector, has been built for such calculations, using the transfer matrix approach and energy corrections. The limiting aperture effects are also taken into account. As an illustration, the tool was used to simulate the LHC beamlines, in particular around the high luminosity interaction points (IPs), and validated with results of the Mad-X simulator. The LHC beam profiles, trajectories and beta functions are presented. Assuming certain forward proton detector scenarios around the IP5, acceptance plots, irradiation doses and chromaticity grids are produced. Furthermore, the reconstruction of proton kinematic variables at the IP (energy and angle) is studied as well as the impact of the misalignment of beamline elements.
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