A set of functions is presented which is orthonormal over the surface of a cylinder. The functions are useful for describing surface errors on the types of near-cylindrical optics found in x-ray and some extreme ultraviolet systems. In addition, the functions provide a convenient means of using surface deforma-tion data to separate rigid-body motions (misalignments) from surface errors. The functions are sinusoidal in the azimuthal direction and polynomial in the axial direction. In this paper, the functions are detailed in terms of the form of their corresponding surface errors. Also, some general relationships are given, which relate the surface errors defined by these functions to the resulting wavefront errors and focal plane errors. These relationships depend explicitly on the nature of the wavefront incident on the optic. This paves the way for describing the wavefront (either in a metrology configuration or in the real system configuration) in terms of the errors on the optic and the misalignments in the system. A specific example for calculating wavefront and focal plane errors is worked. Finally, some parallels are drawn between the use of these functions for describing near-cylindrical optics and the use of Zernike polyno-mials for describing conventional optics.