The configuration-factor concept developed in the theory of thermal radiation to analyze the radiant interchange between diffusely emitting surfaces is extended in such a way that the directional dependence of the radiation emitted by engineering surfaces can be considered. It is assumed that the emissivity of a surface may vary between the angle bins determined by the angular quadrature of the discrete ordinates method. This leads to generalized configuration factors for which, as in the case of the common configuration factors, reciprocity relations, a distributive rule and a sum rule for enclosures can be defined. The generalized configuration factors are calculated analytically for basic spatial transitions in regular and annular cylinders with r-z geometry. The analytical expressions are evaluated numerically and configuration-factor algebra is used to determine the configuration factors for all transitions between the surface elements of a cylindrical system. The generalized configuration factors are used to calculate the absorbed energy densities in the walls of a cylinder with nondiffuse gray surfaces. Different wall materials are considered, the calorific brightness of which is described by experimental directional emissivities. The resulting absorbed energy densities are compared with those obtained by using direction-independent emissivities.