The influence of the surface shape of contacting bodies on the distribution of contact pressure between them and the stress-strain state of these bodies is studied in the paper. The case study covers contact interaction between a ball piston and a running track of a radial hydro volumetric transmission. The elastostatics problem is reduced to the contact of bodies of revolution: a sphere and a toroid. The profile of the axial section of the toroid is given in the form of a power function defined by two variable parameters. These parameters determine the distribution of contact pressure between these bodies and its magnitude. The regularities of this influence have been established. The dependency of the maximum value of the contact pressure and the equivalent von Mises stresses on varied parameters are established. Constructed parametric contact interaction models make it possible to perform multivariate calculations of the stress-strain state of contacting bodies. Analyzing the obtained results makes it possible to build a specialized research database. In turn, this database is the basis for the justification of project decisions when creating progressive structures, which include complex profile details. Accordingly, rational profiles of contacting bodies are determined according to the strength criterion.