This paper is the third in a series of papers on the diffraction and imaging properties of generalized curvilinear diffraction gratings. Here we study classes of plane uncrossed diffraction gratings which have the common property that their Fourier transform patterns consist of sharp-line caustics. These types of generalized gratings are referred to as ‘Euclidian diffraction gratings’, because the condition for the formation of line caustics is equivalent to the global vanishing of the gaussian curvature of the grating line pattern function. The analysis is applied to the laser material processing system discussed in our last paper [1].