A large set of bow shock crossings (i.e., 1392) observed by 17 spacecraft has been used to explore the three-dimensional shape and location of the Earth's bow shock and its dependence on solar wind and interplanetary magnetic field (IMF) conditions. This study investigates deviations from gas dynamic flow models associated with the magnetic terms in the magnetohydrodynamic (MHD) equations. Empirical models predicting the statistical position and shape of the bow shock for arbitrary values of the solar wind pressure, IMF, and Alfvénic Mach number (MA) have been derived. Individual crossings have been rotated into aberrated GSE coordinates to remove asymmetries associated with the earth's orbital motion. Variations due to changes in solar wind dynamic pressure have been taken into consideration by normalizing the observed crossings to the average value 〈p〉 = 3.1 nPa. The resulting data set has been used to fit three-dimensional bow shock surfaces and to explore the variations in these surfaces with sonic (MS), Alfvénic (MA) and magnetosonic (MMS) Mach numbers. Analysis reveals that among the three Mach numbers, MA provides the best ordering of the least square bow shock curves. The subsolar shock is observed to move Earthward while the flanks flare outward in response to decreasing MA; the net change represents a 6-10% effect. Variations due to changes in the IMF orientation were investigated by rotating the crossings into geocentric interplanetary medium coordinates. Past studies have suggested that the north-south extent of the bow shock surface exceeds the east-west dimension due to asymmetries in the fast mode Mach cone. This study confirms such a north-south versus east-west asymmetry and quantifies its variation with MS, MA, MMS, and IMF orientation. A 2-7% effect is measured, with the asymmetry being more pronounced at low Mach numbers. Combining the bow shock models with the magnetopause model of Roelof and Sibeck (1993), variations in the magnetosheath thickness at different local times are explored. The ratio of the bow shock size to the magnetopause size at the subsolar point is found to be 1.46; at dawn and dusk, the ratios are found to be 1.89 and 1.93, respectively. The subsolar magnetosheath thickness is used to derive the polytropic index γ according to the empirical relation of Spreiter et al. (1966). The resulting γ = 2.3 suggests the empirical formula is inadequate to describe the MHD interaction between the solar wind and the magnetosphere.