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This paper presents a novel formulation to reduce radiation integrals to line integrals. Such a reduction is exact for Kirchhoff aperture radiation integrals and physical optics (PO) scattering from flat soft/hard (perfectly conducting) plates, illuminated by a spherical source, but can be effectively extended in an approximate version to more general configurations. The advantage of our approach is that the integrand of the line integral along the rim of the radiating surface is free from singularities and can be easily integrated at all the observation aspects, including geometrical optics shadow boundaries. Conversely, at those aspects, existing formulations exhibit, in the integrand, a pole singularity that renders the numerical integration inaccurate or time consuming, since it requires adaptive integration routines. This was a main concern in the use of this kind of approach for the time reduction in the numerical calculation of aperture/scattering radiation integrals, which is overcome by our approach. Also, the novel result presents a neat ray interpretation which is physically appealing and allows for the heuristic extension of the approach to non-exact cases (e.g., arbitrary impedance boundary conditions or curved surfaces) using standard ray approximations. Beside the already known boundary diffraction wave (BDW), which is an incremental wave excited by the incident field and arising from the rim of the surface, a further term called incremental geometrical optics (IGO) is introduced. This novel term is an elementary portion of the direct field arising from the source and impinging at the observation point; it is able to cancel the BDW singularity thus rendering the whole integrand smooth. For the sake of simplicity, the BDW+IGO theory is here presented with reference to the simplest scalar case of aperture radiation.