Several projects in radioastronomy plan to use large static cylindrical reflectors with an extended lobe sampling a sector of the rotating sky. This study provides the exact mathematical expression of the transit time of a celestial object within the acceptance lobe of such a cylindrical device. The mathematical approach, based on the stereographic projection, allows one to study the optimisation of the position and orientation of the radio-reflector, and should provide exact coefficients for the spatial Fourier Transform of the radio signal along the cylinder axis.