We use a large N -body simulation to study the characteristic scales in the density gradient profiles in and around halos with masses ranging from 10 12 to 10 15 h −1 M . We investigate the profiles separately along the major (T1) and minor (T3) axes of the local tidal tensor and how the characteristic scales depend on halo mass, formation time, and environment. We find two kinds of prominent characteristic features in the gradient profiles, a deep 'valley' and a prominent 'peak'. We use the Gaussian Process Regression to fit the gradient profiles and identify the local extrema to determine the scales associate with these features. Around the valley, we identify three types of distinct local minima, corresponding to caustics of particles orbiting around halos. The appearance and depth of the three caustics depend significantly on the direction defined by the local tidal field, formation time and environment of halos. The first caustic is located at a radius r > 0.8R200, corresponding to the splashback feature, and is dominated by particles at their first apocenter after infall. The second and third caustics, around 0.6R200 and 0.4R200 respectively, can be determined reliably only for old halos. The first caustic is always the most prominent feature along T3, but may not be the case along T1 or in azimuthally-averaged profiles, suggesting that caution must be taken when using averaged profiles to investigate the splashback radius. We find that the splashback feature is approximately isotropic when proper separations are made between the first and the other caustics. We also identify a peak feature located at ∼ 2.5R200 in the density gradient profile. This feature is the most prominent along T1 and is produced by mass accumulations from the structure outside halos. We also discuss the origins of these features and their observational implications.